Are you a stock picker who wonders whether you’re beating the market?

Or perhaps you’re a passive investor keen to see how the returns from your DIY ETF portfolio compares to Vanguard’s *LifeStrategy* fund?

Whatever the motivation, you’ll need to track and compare your performance versus active and index funds to know for sure.

And that means comparing your returns calculated using the same methodology that they use.

If you think you’re a great investor but really you’re lagging the market by 3% a year, it will have a disastrous impact on your wealth compared to if you’d used index funds.

That’s not to mention the many hours wasted in fruitless research and so on (unless you happen to enjoy it.)

### Many unhappy returns

Now I’ve warned before that there are good reasons not to closely track your returns.

For starters, if stock picking is your hobby, then finding out you’re in the ‘D’ League – less Neil Woodford, more Woody Woodpecker – will be thoroughly depressing.

True, you might end up richer if you then give up on active investing and turn passive – although that partly depends on what hobby you replace active investing with. (Sailing boats don’t come cheap…)

But if you’ve wrapped your self-worth up in your ability to beat the market, be prepared to take an emotional beating when you discover the dumb money has actually beaten you…

More subtly, passive and active investors alike can be provoked into overtrading or excessively fiddling with their portfolios by paying them too much attention.

You might even scared out of shares altogether, should you begin to more closely follow their ups and downs.

You could even get sick with stress.

Behavioural scientists have shown we hate losses much more than we love gains.

Yet as Nassim Taleb explained in *Fooled by Randomness*, on any given day it’s a coin toss as to whether the market is up or down.

This means looking at your performance every day is a guaranteed downer.

You’ll be twice as miserable as you are happy!

### It all adds up

With willpower, you can accurately track your returns and avoid looking at your numbers every day – or even every month.

You can make up the rules.

Just don’t kid yourself by also making up your returns.

I have met many people who don’t take account of all the new money they add to their portfolio – yet they talk confidently about how well they’ve done through their active investing decisions in growing their wealth.

Only when you grill them do they admit to “top-ups” and the like.

This is all nonsense from the perspective of comparing performance.

Hint: If your portfolio starts the year at £10,000, you add £2,000, and at the end of the year you have £12,000, you did not earn 20%!

Your return was diddlysquat.

### You fooled you

You may decide not to obsess about returns, or even to ignore them altogether.

Good for you!

But if you don’t know your returns, don’t presume you do when somebody asks.

That includes you, when you ask yourself whether you’d be better off with index funds instead of active investing.

If you haven’t properly tracked your returns – taking into account all the new money you add, as well as all the capital gains and dividends – then don’t pretend you’re the next hot hedge fund manager.

You just don’t know.

### Unitize your portfolio to track your returns

Still want to know how you’re doing?

I believe the best way to track your returns is to **unitize your portfolio**.

Sure you can use online portfolio tools or work out various numbers on-demand, but I think it’s better to take charge for yourself so you understand not just the numbers, but what is driving those numbers.

Unitization is the method used by fund managers who must account for money that flows in and out of their open-ended funds.

As it’s the industry standard method of measuring returns, unitization means you can compare your performance with any existing fund.

You can also compare a unitized portfolio’s performance against a benchmark such as an index.

And unitization encourages you to keep decent records – also important if you’re trying not to kid yourself.

As physicist Richard Feynman warned: *“The first principle is that you must not fool yourself, and you are the easiest person to fool.”*

### Why open-ended funds are called unit trusts

So what is unitization?

You are probably already familiar with unit pricing when it comes to funds.

If not, here’s a quick refresher.

When you invest your money into an open-ended fund, you buy a certain amount of units in that fund with your money.

For instance, let’s say I have £18,420.58 in Legal & General’s European Index Trust.

The L&G website tells me how it calculates this:

Number of units I own: 6,564.712

Unit price Buy/Sell: 280.6p/280.6p

Value: £18,420.58 (i.e. 280.6 pence x 6,564.712 units)

All investors in the L&G fund would see exactly the same unit price. However they will own different numbers of units, depending on how much they have invested.

Whenever investors put additional money into the European Index Trust, L&G creates new units at the prevailing unit price.

The new money buys the right number of units at that price for the money invested.

For example, if the unit price were 280.6p, then investing £5,000 would buy you 1,781.9 additional units.

The new cash you’ve invested now comprises part of the assets of the unit trust, which offsets the creation of those new units.

The fund’s total Assets under Management have increased, but the returns haven’t changed just because new money has been added – and this is confirmed by the unchanged unit pricing.

Finally, the fund manager deploys your extra money to buy more holdings in order to keep the fund doing what it says on the tin.

In the case of this example, he or she buys more shares to match the European Index.

### Unit prices and new money

The crucial point is that adding new money does not change the price of a unit.

Only gains and losses on investments, dividends and interest, and costs that are charged against the portfolio’s assets will change the price of a unit.

For example, if the companies owned by the European Index fund rose 10% in value, then the unit price would be expected to increase by 10%, too.

So here the new unit cost would be:

280.6*1.1 = 308.66p.

Measuring changes in the value of units like this – as opposed to measuring the total monetary size of the fund – enables the manager to maintain a consistent record of performance.

This record is not distorted by money coming in and going out.

Also, when an investor wants to cash out from the fund, there’s no confusion about what percentage of the assets they own or anything like that.

They’ll own a certain number of units. To cash out, they sell them back to the fund manager at the prevailing unit price.

For instance, let’s say you own 600 units.

At 280.6p per unit, you’d cash out with:

£2.806 x 600 = £1,683.60

By **unitizing your portfolio**, you can use the same principle to measure your own returns – whether you’re saving and investing extra cash or you’re withdrawing money from the portfolio.

### Interlude: A pep talk

So far, so boring.

Well, we are talking about accountancy here!

I’ll level with you. Things aren’t going to get any more exciting.

However the good news is that unitizing your portfolio is a very easy process.

Read on to discover how to do it.

### How to unitize your portfolio

For something that has the letter ‘z’ in it, unitization is very simple to do.

Admittedly, trying to retrospectively see your unitized performance over previous non-unitized years is a nightmare.

It’s such a nightmare, in fact, that I didn’t bother trying when I started to track my returns properly a few years ago, and I wouldn’t suggest you do, either.

If instead you take a leaf out of the Khmer Rouge’s generally best-avoided playbook and declare today to be “Year Zero” for your portfolio then it’s straightforward to get started.

**1. First decide on an arbitrary unit value**

The first thing you need to do is to decide what one unit in ‘your fund’ is worth.

It’s a totally arbitrary decision, as it will just be used as the base for future return calculations.

Many people choose £1.

I chose £100 for egomaniacal reasons.

**2. Calculate how many units you currently have**

As it’s Year Zero for unitizing your returns, you need to work out how many units you currently have.

This is based on the total value of **everything in the portfolio** you’re tracking, together with the unit value you just came up with.

You simply divide the former by the latter.

Let’s say your portfolio is £50,000 and your unit value is £100.

This means you have 500 units to begin with, like so:

£50,000/£100 = 500 units

Make a note of this number, together with your arbitrary unit value.

I track all this on a Web-enabled spreadsheet that constantly tells me what my portfolio is worth right now, what one unit is worth, and how many units I have.

From this it also works out and tells me my returns over various periods.

But if you want you can just write these numbers in a notebook. It’s only when you add new money that further calculation is mandatory.

**3. If you don’t add new money you can now easily track your returns**

Let’s say you never did add or remove another penny from your portfolio.

You know how many units you have, and you know the starting unit value.

You can now easily work out your unitized returns.

For example, let’s say your portfolio increases to £60,000.

The unit value is now:

£60,000/500 = £120

Your return to-date is the change in unit value.

£120-£100/£100 = 20%

However you hardly needed to unitize to see that…

**4. Adjust your total units as you add new money**

The whole point of unitizing is to properly take into account money added or removed from the portfolio.

Every time you add new money, you need to calculate and take note of the value of one unit.

For instance, let’s say that when your portfolio hits £60,000 you decide this investing lark is a piece of cake, and so decide to add in another £6,000.

The unit value before the additional cash is added, as above, is £120.

Now we need to calculate how many units our new money buys:

New money added / unit cost = number of new units

£6,000/ £120 = 50 new units

This means your £66,000 portfolio now comprises 550 units.

Again, you note it down ready for next time.

**5. Keeping ‘buying’ new units as you add money**

This process is simply repeated over your investing lifetime.

Let’s say the value of your portfolio increases to £69,850, and you decide to add a full ISA contribute of £15,240.

First:

Unit value = Portfolio value / number of units

Unit value = £69,850/550

Unit value = £127

Number of new units that £15,240 buys:

New money added/ unit cost = number of new units

£15,240/£127 = 120 new units

With the new ISA money your portfolio is now worth £85,090, and is comprised of 670 units (that is, 550+120).

As always you note down the total unit number for next time.

**6. What happens when you remove some money?**

When money is removed entirely from the portfolio, the principle is exactly the same as when money as added – the number of units changes as consequence, but not the unit value.

You are effectively ‘selling’ units in your own fund to free up the cash. Obviously this procedure doesn’t change the returns you have achieved on the mix of assets you happen to hold.

For example, let’s say your portfolio continues to motor on and breaks through six figures to hit £100,165.

At this point you get collywobbles (I told you there was a downside to tracking your returns) and you decide to spend £10k of it while you’ve still got your teeth.

You know from your records that your portfolio currently consists of 670 units.

This means the unit value currently is:

£100,165/670= £149.50

You decide to remove £10,016.50 from the portfolio to keep the sums simple:

£10,016.50/£149.50 = 67 units

After the withdrawal you have £90,148.50 in your portfolio represented by 603 units (670-67).

**7. Work out your unitized return at any point**

At any moment in time you can see exactly what your return is by looking at your unit value.

For instance, let’s say that after all of the above, your portfolio ends the year with a value of £90,450.

Your unit value is:

£90,450/603= £150

So your unitized return since you unitized your portfolio is:

£150-£100/£100 = 50%

This is the return that you can compare with trackers and other funds and benchmarks that report their returns over the same period.

Let’s say at the end of next year a unit was worth £160.

You started the period with a unit value of £150. So your return over the year is:

£160-150/150= 6.7%

You see how easy it is to calculate and note down your annual return figures every year, for instance?

### Actually it all seems a lot of work

It’s really not, once you’ve set it up properly.

My spreadsheet tells me my current portfolio and unit value at any time.

A sheet also tracks money added and subtracted over the year, and calculates the number of units added or subtracted when I do so, which gets added to the ongoing tally.

At the end of the year I simply record all the relevant values for my records.

Then, on the first trading day of the New Year, I hard-update the spreadsheet with my starting unit value and the total number of units.

This makes it easy to see and record my discrete total return figures every year.

### When should I calculate my returns? And what else?

Up to you. I take regular snapshots, and as discussed above I work out the annual figures as I go.

I also work out things like my Total Expense Ratio (TER) – based on another sheet tracking all my trading costs – and my total portfolio turnover.

If you take monthly snapshots then you can track your volatility, and start to look into other more esoteric ratios if you want to, such as the Sharpe Ratio or the Information Ratio.

I don’t bother. (Despite having written over 3,000 words on it here, I’m neither an expert nor an enthusiast for data crunching!)

### What about Net Present Value or Internal Rate of Return or CAGR etc?

There are many different ways of calculating returns. They all have value in different circumstances. And they usually deliver different numbers.

The advantage of unitization is it gives you the appropriate numbers to compare with any fund or index out there. That’s most important when tracking your own performance.

### What if I have multiple dealing accounts, SIPPs, and so on?

If I were you I would track all the different holdings on one spreadsheet.

I’d then unitize the returns on this entire portfolio, and also track expenses, portfolio turnover, and other interesting figures across the entire piece.

This is exactly how I measure my own total returns across half a dozen different platforms and brokers.

There’s not a lot of point in tracking the returns in a SIPP separately from returns in an ISA, in my view.

Ultimately it’s all your net worth. They’re just different baskets.

However if you did want to track how particular accounts are doing – perhaps because you employ different investing strategies in one versus another – then you could create separate spreadsheets to follow them.

You’d also have to track separate money flows in and out of each them, and generate unitized return figures for each ‘pot’ of cash.

Remember, you’d still want to unitize the entire portfolio to properly track your overall returns (rather than, say, averaging the returns on the different accounts, as this would not account for the different amount of money in each – though you could create a weighted average I suppose).

### What about dividends, spin-offs, dealing fees, stamp duty, rights issues and so on?

None of this matters, so long as all the incoming money (from sources such as dividends and spin-offs) and expenses (such as dealing fees and stamp duty) are retained or paid from within the portfolio.

The number of units you own doesn’t change because you were paid a dividend – no more than if one of your shares went up by 20p.

But your units do now represent more assets, in the form of that extra dividend cash. That increases the value of each of your units.

Similarly, dealing fees and stamp duty are no different to seeing the price of one of your holdings go down, in terms of their impact on unit value.

Expenses subtract from your total assets, and therefore they reduce the value of a unit.

**The golden rule:** It’s only when external money is added or when you take money out of your portfolio that you need to adjust the number of units you own. Such additions or withdrawals are as if an investor was buying or selling units in your fund.

Of course none of this is to say that dealing fees and dividends don’t matter from an investing and return standpoint. They matter a great deal.

The point is simply that unitization is capturing their impact on returns, so you don’t need to adjust for them. They are not ‘extra’ money added or subtracted.

### XIRR, time-, and money-weighted returns

From previous discussions elsewhere, I know some people prefer to plug numbers into Excel than to unitize and track their portfolios.

As I understand it – because they have told me so – such people usually get very similar results by using Excel’s XIRR function.

I won’t go into the details of using XIRR here because, frankly, I’ve never done it.

There are various tutorials on the Web.

However you should note that while the results you get from unitization and XIRR may often appear very similar, they will sometimes diverge markedly.

That’s because they are calculating different kinds of return.

- Unitization gives what is known as a
**time-weighted return**– all time periods are weighted equally, irrespective of how much money is invested when. Unitization tells you the underlying investment performance, and strips out the impact of money flowing in and out.

- XIRR gives what is known as a
**money-weighted return**– this means that time periods in which more money is invested have more of an impact on the overall return than equivalent time periods in which less money is invested.

XIRR calculates the Annualized Internal Rate of Return on a portfolio. The gist is that you supply the XIRR function with a column in your spreadsheet that lists these cash additions and withdrawals. The function then uses an iterative process to hone in on your returns.

It makes sense for the fund industry to use unitization because a fund manager typically has no control over when money is added or removed from their fund – it comes from the fund’s customers – and also because it’s the choice and performance of the underlying assets that matters when evaluating how skillful a manager is. (The same reason I unitize my returns).

However some would argue that a money-weighted return like XIRR is at least as illuminating from the perspective of a private investor, since you are in control of money flows and what matters is your personal rate of return.

Many private investors do derail their results with poor market timing. By calculating and comparing both the unitized return and a money-weighted return you can spot the impact of poor (or perhaps good) market decisions timing on your portfolio.

The *Henry Wirth* Investing blog explains:

- If your Money Weighted Return Rate is greater than your Time Weighted Return Rate, then your market timing is adding value to your portfolio.

- If your Time Weighted Return Rate is greater than your Money Weighted Return Rate, then your market timing is subtracting value from your portfolio.

You should also read the excellent comments from John Hill that follow today’s post.

### Do it for yourself

Personally I think if you’re going to keep track of the money flows in and out of your account, you might as well go the whole hog and unitize your portfolio.

Once you’ve setup a spreadsheet to do the sums it’s very easy to stay on top of things, and you have the satisfaction of knowing your returns are directly comparable with those reported by fund managers.

That’s not to say there’s a right or wrong way to measure returns.

It all depends on context, and on knowing what you’re measuring and why.

Also remember that in itself a return figure tells you nothing about the volatility or risk you took to get those returns, nor about the maximum troughs (aka drawdown, or losses at the portfolio level) that you endured along the way.

But if you unitize your portfolio and keep records of, say, the daily unit price, then you can go ahead and calculate and track that sort of thing for yourself.

Indeed, once you’ve unitized your portfolio, you can go crazy if you want to and use it as the basis for calculating all kinds of extra stuff – such as the Sharp Ratio that might help you understand if your returns are down to skill or risk taking.

Comments on this entry are closed.

Superb post TI – just the sort of simple but effective system I’m looking for. I will very likely use this idea in my own spreadsheets. I’ve been looking for a way to effectively track how each additional quarterly/monthly top-up of money to my portfolio performs in its own right over time, as well as my portfolio over all. Unitizing each lump sum seems like a very easy way to achieve this – rather than having to worry about how each proportion of the top-up, distributed across multiple funds, is performing.

I feel I posted too much on the last thread so I’ll keep quiet unless anyone encourages me otherwise!

Great post – something to think about – I’ll be putting a new tab in my finance s/s to try this method. At first glance it’s at a higher level than I’ve used before and I’m uncertain about these imaginary “units” rather than something more real.

I’m not a great fan of percentages in such calculations but they are used a lot.

However…your example:

500 units, with each unit being valued at a notional £100 = £50,000.

Portfolio values rises to £60,000 so your 500 units are now worth £120 each. 20% gain.

You add £6,000 of new funds with a unit price (your imaginary unit, not the unit of the fund you are buying) of £120, so you “buy” 50 more units.

You now have a portfolio worth £66,000 consisting of 550 of your imaginary units.

Your gain is still 20%. £66,000 / 550 = £120 per unit vs a “face value” of your unit of £100.

I would have said your gain was still £10,000 on a, now, £66,000 portfolio – 15%. The new money dilutes your overall percentage (but not £) gain. It’s one of the reasons I prefer to deal in £, not %. Either 20% or 15% can be regarded as correct, it just depends on what question you’re trying to answer. But if someone says to me they’re 20% up and their portfolio is worth £66,000, I’m going to think they’re £13k better off. I wouldn’t naturally think to ask them how they defined their %.

If your £66k portfolio then fell to £56,000 I think you’d tell me that you’re 2% up (£56,000 / 550 units = £102). I’d say you were break-even as you’ve put £56,000 in and your portfolio is worth £56,000.

Oh dear…obvious mathematical error in my last post. Blush. Please delete penultimate para in your minds…

How you treat dividends does matter, in that when you compare your portfolio to an index, you need to decide if you want to compare to a total return index ( where dividends are accumulated) or not ( I.e. dividends are withdrawn from the portfolio). Eg there is the FTSE all share index, and the FTSE all share TR ( total return ) index. If you assume that dividends are held within the portfolio, then you need to compare to the TR index.

You can of course calculate both – Also known as accumulation units or income units.

Richard: Mathematical error excused, but I’m confused why you’d say you were 2% up – unit price is not a measure of how successful your portfolio is, simply what it costs to “buy in”, relative to when you first started it. £56,000 is the same as what you’ve invested overall, so you are neither up nor down.

However, what *has* taken place is that your latter £6000 investment has dropped into the red (50 units now valued at £5090.91 total), while the original £50k investment is still slightly up (500 units valued at £50909.09). So one investment is ‘saving’ the other – but overall you are still back where you started. Only now it costs slightly more than before to buy in, relative to your original investment.

@Moongrazer

Hmmm…not sure. You’re now keeping track of separate lots of purchases – which is what I do (but for real shares not imaginary units).

I read the post as the unit price was the single number you needed to measure your portfolio’s return.

@all — I’ve just written 3,000+ words on this, so excuse me if I have a bit of a break from writing long replies.

However please note I fully expect(ed) people to turn up in the comments saying *this* does or doesn’t measure *that*. 🙂

There are myriad ways to calculate different kinds of returns. To draw attention to that, I included the section on “What about Net Present Value, or IRR, or CAGR and so on?”

It is an explicit nod to the fact that there are very many different ways to measure return, which are applicable in different circumstances.

So no, the article did not say: “the unit price was the single number you needed to measure your portfolio’s return”. 🙂

What it is saying is that it is the single number you need to track your portfolio’s unitized return, and hence over time to compare it with other funds, indices, benchmarks and so on.

If you want to work out some other kind of return in some other way for your own purposes — such as, for some reason, what percentage you happen to have more in total than when you started, irrespective of fund flows — then knock yourself out. 🙂

You can’t compare that to a return quoted by a tracker fund or Neil Woodford though.

@Nigel — Well really what you’re saying is “don’t compare a total return portfolio with a capital-only index” or similar, which is quite correct.

However from the point of view of unitization, all that matters is whether you keep the dividends in (/reinvest your dividends) or take it out (/you spend your dividends).

If you leave the money in, that’s fine.

If you take out/spend the dividends, you’ll need to track those withdrawals and reduce your unit count as you do so.

@Richard

The way I see it (if I think I’ve read TI’s post correctly), the aim of the exercise boils down to being able to illustrate “I invested £X on this date and this is how well that particular sum is doing now.”

If I did that with each of my individual funds every time I topped up my investment, I’d have to add 9 new entries to a spreadsheet every time (apportioning the lump sum to each according to weightings), in order to track what that particular investment was doing. To say nothing of what would happen with all the convoluted scenarios we were discussing in the previous blog post of selling old funds and buying new ones.

With this approach, each time I top up, I am locking in that new investment according to how my portfolio is doing at the current time. If my portfolio is currently in profit, then the new investment will have to do that much better to pull its weight. If my portfolio is currently in the red, then the new investment is going to (hopefully) yield a bigger reward in the future.

This, for me, solves a problem I’d been thinking about for a while. Each year I invest money with the intention of it being locked away for 25 years, and then after that time withdrawing the profit and living off that for a year (I hope!), whilst also leaving the original sum invested for another 25 years. The problem I needed to solve was how to differentiate money invested in year 1 vs. money invested in years 2, 3, 4, etc. so I could work out whether year 1 had done sufficiently well enough in 25 years’ time that I could live off the profit alone. Maybe thats a bit of a bonkers way of looking at it.

Nonetheless unitizing appears (to me) to be the answer to that, since I now have a way of crystalising the value of any given investment at the point in time it goes into my portfolio!

@Moongrazer — No, that’s wrong and I’m a bit dismayed if I’ve given that impression. (Are other people reading it this way? If so tell me where/why please so I can edit it).

Unitization is a method to track your whole porfolio’s performance when there is cash going in and out of the account, used by the industry, and hence best adopted if you want to compare your returns with the industry, in my view.

Perhaps this Wikipedia entry will help:

http://en.wikipedia.org/wiki/Unit_valuation_system

The reason I am a fan is I want to know how well my portfolio is performing in a directly comparable way to other funds. The best way to do that is to use the same system they use. If you want to figure something else out, then there are other methods to turn to such as IRR or NPV that are not the subject of this post. 🙂

TI is right to stress that there are many different ways of looking at returns. However, I do not think that unitisation is one that gives you a view of your “portfolio” return, its intention is to tell you about the return of your underlying assets in a way that can be compared with other indexes and funds. It does not tell you if your timing of inflows and outflows messed up the whole apple cart by selling low and buying high.

Consider this simple example…

1/1/2014 Invest £1000 in portfolio (1000 units at £1 each)

1/7/2014 Portfolio is now worth £2000. Yippee! You invest another £1000 (total of 1500 units at £2 each)

31/12/2014 Whoops! Portfolio has fallen to £1500. (total of 1500 units at £1 each).

I think most people would expect that their measure of “return” would be negative.

Yet for the year your unitised return is 0%. Your IRR however is -32%. And from a cash perspective, you have invested £2000 and have now only got £1500.

If you want to know if you are making any money or if your crap market timing is hurting you, then use IRR (or other versions thereof). If you want to compare your portfolio constituents with Neil Woodford independent of your market timings then use unitisation.

“There are myriad ways to calculate different kinds of returns.”

I think that just about sums everything up!

As long as we’re very clear about the question we’re answering everything should be fine. But if I want to compare “my” return to “your” return we need to be very clear that we’re talking the same methodology.

I’m pretty sure it is just my misreading of it. I think I’m going in a different direction with the maths than this particular system is meant to do.

Re-reading, where I went off on a tangent was the bit when you talk about working out your actual rate of return by using the present unit value (current portfolio value divided by total units) versus the original unit value.

In which case I appear to be in Richard’s camp in thinking that’s a bit strange – as he illustrated, if your investment dropped back to £56,000, the value of your portfolio would be equal to the money you put in, but your unit price would be £56,000 / 550 = £101.82 (ish), not £100, so it would be ‘up’ 1.82%, even though your portfolio’s current value is no higher or lower than the original money that went in.

I’m certainly not disputing its use in comparing your portfolio’s return to an industry index – but if that’s the case, then suddenly industry indices look a whole lot less useful (or I am still not getting something).

For me, tracking how my entire portfolio has performed versus the market over time, while interesting (and certainly useful to see if I’m off the mark), is less useful than seeing how each individual lump sum is currently doing in its own micro existence.

But unitizing still seems to work for my purposes (if not the ones you intend) – because when I put in new amounts of money, I can see how those individual sums go up and down easily (I could look at individual monies put into many differents real funds, but that’s just more time consuming).

Really, the maths behind my problem just requires me to know the fraction of my portfolio that my new investment is occupying versus the portfolio as a whole. Units is just a nicer way of representing it.

Sorry for the long ramble!

@helfordpirate:

Useful summary. Think that clears up the distinction for me, if I’m honest. So IRR is the manner in which I should measure how my portfolio is doing versus the money I put in, Unitization is what I need to see how my portfolio stacks up against other market indexes.

One can come up with quirky examples that ‘break’ all the different ways of calculating returns, I’ve noticed. And people always do when these discussions come up! 🙂

Unitization seems particularly suspect when small sums are involved relative to the cash inflows/outflows, and when the portfolio is liquidated. But in practice I find it perfectly usable, which isn’t a surprise given the financial industry — passive and active — also does, too.

Yes, I think you should be unitizing your returns. 🙂

At the end of the day, I’m not a statistics junkie and certainly not a statistics expert. Unitization struck me as a sensible and surprisingly simple way to track returns over time and deal with the cash I am regularly adding when it was first introduced to me several years ago, and the fact that it enabled direct comparison in an industry standard way was very appealing.

@Moongrazer

I think I got off lightly in my example. Poor Helfordpirate’s lost £500, had a -32% IRR and yet Mr Woodford’s telling him he’s breakeven.

Is it too late to retrain as a fund manager…?

@Richard

That gave me a good chuckle. Touché sir!

Having a set-up of different platforms and differing monthly inputs; I was looking for a solution to this problem at the end of last year. My research ended up taking me down the excel XIRR solution. But interesting to see another option – if only to double check my original spreadsheet calculations are correct!

After a quick test I see that there’s a minor difference to the XIRR v’s Unit figure – which is not surprising. (but good to know it’s only a minor difference)

And I’m guessing it is due to the that fact that I currently only check my Total Portfolio amounts (from the platforms/etc) some days after those monthly installments have been made. And I don’t record the Total Portfolio amount the day before the monthly installments – to give a better figure to calculate the Unit Value that will be used the following day. (As the XIRR formula only needs to know only the dates when the various inputs/outputs happened, and then the current Total Portfolio amount)

Would I be correct in my assumption for the small % difference between the 2 systems or am I barking up a wrong tree? (Hope the above makes sense 🙂 )

Thanks TI for clear explainations and revealing a different solution to the problem. And will be playing around further with this idea.

I’m struggling with the maths in thie following section. If unit value is£143 yet starting value was £160 then that’s a £17 loss ain’t it?:

“Let’s say at the end of next year a unit was worth £143.

You started the period with a unit value of £160. So your return over the year is:

£160-150/150= 6.7%

You see how easy it is to calculate and note down your annual return figures every year, for instance?”

@Mikkamakkamoo — Oops, sorry, my fault! It’s a typo, I was going to show an example loss calculation but then swapped it for a gain as the ‘minus’ sign is never very clear online so I thought it looked confusing. Not as confusing as leaving old numbers in! *red face*.

Fixed now. Thanks very much for the spot!

There is a major difference between XIRR and “unitization” as neatly described by The Investor, as is hinted at by the examples above from Moongrazer and helfordpirate.

XIRR gives what is known as a “money-weighted return” – time periods in which more money is invested have more of an impact on the overall return than equivalent time periods in which less money is invested.

Unitization gives what is known as a “time-weighted return” – all time periods are weighted equally, irrespective of how much money is invested and when. So this is just the underlying investment performance, with all the money in, out and inbetween stripped out.

Think about the another example:

– You invest £1,000 on 1 January 2014.

– Your portfolio goes up by 10% in the first six months of the year.

– The first contribution is therefore worth £1,100 on 1 July.

– This is a licence to print money, you think! You invest another £1,000 on 1 July.

– Your portfolio goes down by 5% in the second six months of the year.

– The first contribution is therefore worth £1,045 on 1 January 2015.

– And the second contribution is therefore worth £950 on 1 January 2015.

– So the whole portfolio is worth £1,995 on 1 January 2015, less than your £2,000 investment.

The unitized return is 4.5% (1.05 x 0.95 – 1), reflecting the fact that the underlying investment actually did just fine, leaving aside your poor timing.

The XIRR return is negative (at -0.33%), because you had more invested during the second six months when returns were much lower, reflecting the fact you’d have more money now having left it in your bank account all along.

Neither return is wrong, but you need to appreciate that time-weighted returns and money-weighted returns serve difference purposes.

Time-weighted returns tell you about relative investment performance only. You can directly compare the time-weighted return to that of any given index, fund or other investor’s time-weighted returns over the same period, and get a clue as to whether or not your investment strategy is doing what you expect it to be doing (whether that be accurately tracking an index, capturing a risk premium, or delivering active outperformance).

Your portfolio’s time-weighted return of 4.5% may actually have been quite respectable, if your benchmark returned only 3% over the year as a whole. Perhaps you’re good at picking your investments, though you’ll need a lot longer than a year to be sure!

Money-weighted returns tell you about the actual performance of your money taking into account the good, bad or indifferent timing of your contributions and withdrawals as well. No matter the validity of your investment strategy, that’s may be all for nought if you’re constantly buying high and selling low. And you need to take into account your actual returns to see if your money is on track to get you to your financial goals.

Clearly, -0.33% per year isn’t going to grow your money in a hurry, but perhaps the timing was just unlucky this time around.

Generally speaking, over a long enough time and with enough contributions and/or withdrawals, you would expect any good, bad or indifferent timing that is due just to luck (or lack thereof) to average out, and money-weighted returns to end up being very similar to time-weighted returns. But, even then, in some cases, that may require more time than you have, if the luck (or lack thereof) is big enough!

And if the timing of the contributions and/or withdrawals is not down to luck at all, but deliberate investor behaviour, then money-weighted returns may differ markedly to time-weighted returns (see the DALBAR studies, for example, that show that actual US investors have consistently earned lower returns than the market, due to buying high and selling low).

Perhaps it is worth also pointing out that you can use money-weighted returns for comparative purposes, but only if you apply the exact same money-weighting to whatever it is that you’re comparing your portfolio to – this involves calculating how many units in each and every comparator each of your contributions or withdrawals would theoretically have bought or sold on the same dates, and the end value of the sum of those units. Needless to say, that can be an awful lot of theoretical calculations.

On the other hand, to calculate time-weighted returns you need to know (or be able to calculate) exactly what your portfolio was worth at the time of each and every single contribution and withdrawal, which can also be easier said than done.

So if you’re intent upon making meaningful comparisons of a portfolio’s performance relative to any other, there tends to be quite a bit of work to do either way!

while i’ve gone for using IRR instead of unitization (let’s not go into that), what struck me is that you’re counting cash held with your broker/platform as part of the portfolio, and i’ve never counted cash as part of the portfolio (i.e. the portfolio is just all the funds, shares, and so on, that i hold).

your approach has the advantage of making the returns look worse if an investor foolishly holds lots of cash with their broker (earning minimal interest) while markets soar (not that i’ve ever done that … well, not any more :)).

and it may make the calculations simpler – assuming you add or remove cash from your broker account less frequently than you buy or sell investments, or receive dividends.

@TI Thanks for this very well explained method of unitization and all the helpful coaching in the preamble. I did get a bit stuck at point 3

Whoops! My emoji crashed out the rest of my comment! I re read point 3 and got it. The comments were another matter… Isn’t is more a matter of avoiding comparing apples with pears? 🙂

Just to say that I use a programme call Fund Manager (https://www.fundmanagersoftware.com). It’s not the easiest programme to get to know, but once set up I just add my transactions, and then I can monitor the performance according to all the above options, and more. I think ‘Time-Weighted Return’ is similar to the unitization, but I’m a little beyond my pay grade here.

The programme is also nice as, again if set up right, it will work out capital gains and makes rebalancing a doddle. It’s a little more than I need in general, and a lay persons guide to the programme would be very nice, but it does make adding data easier, and any later analysis simple to do (and provides a lot of performance measures to do it). I can recommend it.

I’ve now created a nice, simple spreadsheet to complement my other (fund-specific) sheet. It contains entries for each investment sum I make into my portfolio, how many units of Moongrazer Fund that has purchased, and the IRR of that specific sum relative to the whole portfolio.

I then have total IRR for the portfolio (calculated across all the sums invested) so I can see clearly how my portfolio is doing for me personally, and the Unitized Return (current unit value / inception unit value) so I can compare how the choices I make inside portfolio are affecting it versus the market and other funds.

So I have a peaceful coexistence of Unitized Return and IRR that should hopefully suit my needs. I guess I’ll find out in years to come once the additions to my portfolio start going in. 🙂

@TI Great article; thanks.

In para 3 “track your performance” may be a seed of some of the confusion in the comments. Maybe it would be more accurate to say something like “*compare* your performance. And to compare your performance against industry funds and indexes, you need to use the same performance measurement tequnique as them”. I can see the post being a useful one in the archive so heading the confusion off at the pass might be worth the effort.

@John Hill — Superbly clear explanation that beats mine, thank you. (Fancy a gig writing the odd article for Monevator? Please contact me if so!) You’ve done such a good job explaining XIRR in fact that I just rewrote the section in my post and took the liberty of adding a few of your words, with a note to readers to jump down here and read the rest of them.

Thank goodness for John Hill’s excellent post above: I thought I was going to have to dust-off the mathmo books to explain the difference.

I struggle slighlty (as a poster above notes) with defining the edges of the portfolio. I do include cash in a brokerage account (and have an asset allocation for cash). I also include the cash float in a Santander current account. But I exclude the cash in my Lloyds account. I don’t include my rental property or mortgage, although I consider it a working asset (unlike my dwelling).

It comes down to what you want to measure, as stated above. There’s not much I can do about property allocation but I can play with the more liquid assets. I focus on allocation rather than return as my actions are determined by it.

In terms of measuring progress, I forecast retirement income, since that is the goal.

If I were a hobbyist active investor then I’d focus on unit return as a measure of my hobby.

@NearlyThere — I think I see what you’re saying, and have mad a tweak according — thanks. However I do want to push back at the implication (perhaps?) that unitization isn’t enabling you to *track* your returns. It is, but it is just one of a few different ways of doing so.

Perhaps it is my active bent that makes unitization so attractive to me. Clearly I need to see that my nefarious stock picking activities are adding value. If you’re a pure passive investor, then money-weighted returns might be a superior first port of call as you’re comparing more directly with just having the money in a bank account or similar.

That said, this discussion has made me appreciate more that I should probably track both in an ideal world. Possibly a future project!

Thanks very much for posting this, TI and also to people commenting.

I think I’ll give this a go and have a play around.

From a purely mathematical point of view, unitization is just the reciprocal of the profit/loss of your portfolio since inception, when flowing money in or out of it.

In other words, you could achieve the same calculation by multiplying any newly-added (or removed) funds by the reciprocal of the portfolio’s growth since inception, and treating that value as what you added to the portfolio. So adding money when the portfolio has grown 10% means you actually added £money/1.1 – thus £1000 would be treated as having added £909.09 at inception.

But I suspect measuring in units is a good deal easier than befuddling yourself with time-adjusted investment values!

What’s the objection to “comparing apples with pears”? Comparing apples to Tuesday now, that would be silly.

Great post, been reading Monevator for a while but never commented – but wanted to thank you for this 🙂

I’d been looking to track my returns recently and couldn’t get my head around the different methods – this makes it all clear to me, and the unitization method is nice and simple.

Now the hard bit is to ignore it 🙂

Interesting idea – thanks

We can also say – that the BIG gaps between investor (money weighted) returns and the average (unitised) return on a fund over time do not reflect investor mistakes (or genius) but simply the shape (‘path’) that the fund price takes over time.

This can be tricky to get ones head around – so I wrote a couple of plain English articles on this matter (one on savings and one on drawing down money) for investor’s chronicle last year.

Hope this is of interest

Hi

Very good and timely article for me as I’m trying to get to grips with tracking my investment returns.

Does anybody have an working example spreadsheet to get me started?

Thanks

Paul

Sorry for the ignorance but for passive i. e. lazy investors, isn’t Morningstar’s free portfolio tracker more than sufficient for calculating pretty much all of the above?

@TI

Thank you for this piece. I didn’t know there was XIRR at hand, and this addresses my problems :). I had tried to come up with something like that. From what I understand having played with it for a while, it calculates the returns both time- and money-weighted which is exactly what everyone should do to calculate their portfolio’s internal returns. And then you can compare it with anything you please.

On a side note there is one comparison I find useful for some who drip feed their portfolio every month or every quarter. Divide your total contributions by the number of months, you now have the average monthly contribution. Plug it into a simple smooth growth calculator and it will show you what annualized return was necessary to get to the current value of your portfolio. Your XIRR will likely be different unless you really contributed the exact same amount every month.

Unitization complements XIRR in my view. It can give you an idea as to why your internal returns are or aren’t similar to annualized unit price change.

For future projections XIRR may prove not as useful as your averaged out unit price change unless you are actively playing on volatility (market timing 🙂 and expect to be as lucky or as unlucky as before.

Internal rate of return (IRR) should not be used to track portfolio return if you are adding significant amounts of cash to your portfolio. Unitization as explained in the article is likely to be much more appropriate, but not without drawbacks as others have commented on. The idea that IRR is the same as “Money Weighted Return” is plain wrong, probably some rubbish put out by a wealth manager who misunderstood a basic financial arithmetic lecture. This misinformation has unfortunately spread like a joke that went wrong, much as Sinclair’s “Quantum Leap” did.

In some circumstances, internal rate of return may approximate a “Money Weighted Return”, but this is not the general case.

The internal rate of return of a series of cash flows is simply the discount rate that makes the net present value of the flows zero. For example, you invest £1000 in a deposit account and a year later receive £100 interest and the year after that another £100 interest plus your money back £1000. The flows are -£1000 (money into deposit account), +£100 after 1 year, +£100 after 2 years and +£1000 return of capital after the 2 years. Unsurprisingly, the internal rate of return is 10%, more formally this is because the net present value of all the flows, discounted at 10% pa, is zero: -1000 + 100/(1 + 10%) + 100/(1 + 10%)^2 + 1000/(1 + 10%)^2 = 0

IRR and Excel’s dangerous XIRR function, makes perfect sense when used as above and you want to calculate a single value that represents the rate of return achieved after making a single point in time investment. It is however extremely dangerous when used with multiple investments and withdrawals and may not have any useful meaning when used in these circumstances.

As an admittedly extreme example, consider an initial portfolio investment of £1000 that does very well over the following year and you take £3600 out. Then on the second anniversary you invest another £4310, but disaster strikes and over the following year the entire portfolio value drops to only £1716. What is the internal rate of return? The answer is that there are 3 perfectly valid answers of 10%, 20% and 30%. i.e.

-1000 + 3600/(1+10%) – 4310/(1+10%)^2 + 1716/(1+10%)^3 = 0

-1000 + 3600/(1+20%) – 4310/(1+20%)^2 + 1716/(1+20%)^3 = 0

-1000 + 3600/(1+30%) – 4310/(1+30%)^2 + 1716/(1+30%)^3 = 0

This illustrates one reason why it can be inappropriate to ask what the internal rate of return is when you have multiple positive and negative cash flows. If someone really wants to know a “Money Weighted Return”, it should at least be calculated in a way that can only give a single result. I would suggest calculating the internal rate of return of each investment (each set of “units” purchased if unitizing) and then calculating the weighted average of those IRRs.

However, sometimes it is far better to accept that it is misleading to represent a population by a single average value, which is what we are doing when we try to distil the performance of a portfolio with multiple in and out cash flows down to a single rate of return.

A most excellent post and replies, thank you all.

@HRP

I agree that in some scenarios (huge volatility, short investment period) asking about average annual return could be inappropriate. I can’t see, however, anything wrong with multiple discount rates XIRR returns when operating on alternating inflows and outflows.

Let’s have a look at your example. Initial investment is £1000, then after a year there is a withdrawal of £3600. What happened to that money then during the following year? If it’s invested elsewhere at 10% and you’ve got £3960 after a year then it would be like leaving that money in your original investment and adding only £350 instead of withdrawing £3600 and adding £4310 after a year. This way we can make sense of the discount rate returned by XIRR. The same goes for 20% and 30%.

Arguments passed to XIRR can be amended to achieve a single result.

@HRP

I am essentially already doing as you suggest in my spreadsheet. I am isolating each payment I invest in my portfolio (along with portfolio ‘units’ purchased) and tracking its IRR separately from any others.

I am not yet concrete about how to represent outflows from my portfolio. Since I have a 20 year minimum horizon this won’t be an issue for some time, but I am curious about how to best represent it. At the moment there is a line for each payment into my portfolio along the lines of Sum Invested / Unit Price At Purchase / Units Purchased / Current Value.

Your point about not using the entire portfolio as a basis for IRR seems perfectly logical to me. A sum invested in year 1 is clearly going to have gone on a far bigger journey by year 21 than a sum invested at year 19.

On that basis I would probably ‘sell off’ the oldest units/purchases in my portfolio first. Or perhaps as time advances and I move closer to retiring I would evaluate which sums invested in my portfolio have achieved well thus far (i.e. met the target growth I am hoping to achieve) and crystalise them so as to stem their volatility, whilst leaving funds that have performed below target to hopefully grow a little longer.

Is there anything worth reading on crystalising/reducing volatility in older investments as one gets older? I know the more general thinking is “increase bond %, decrease equity %” but that seems like a broad-strokes approach and I get the feeling there must be something better if I am taking the time to track how each individual sum I invest is doing.

@Paul W,

“If it’s invested elsewhere at 10% and you’ve got £3960 after a year then it would be like leaving that money in your original investment and adding only £350 instead of withdrawing £3600 and adding £4310 after a year.”

You are solving a different problem – initial investment £1000, no withdrawal after 1 year, add £350 in year 2 and take out 1716 after 3 years – this conveniently has only 1 IRR. What if the money withdrawn after the first year was spent?

The point I was trying to make is that in certain situations there can be multiple internal rates of return and using XIRR and variants to calculate a money weighted rate of return is inappropriate in these circumstances. If someone blindly creates a spreadsheet of flows and uses XIRR to calculate an internal rate of return, XIRR will usually oblige with an answer, but that answer can be misleading if it is one of many. IMHO investors are better off steering clear of XIRR unless they also check that there are not multiple internal rates of return. If there is only a single initial investment, there can only be single IRR and as such XIRR is fine.

@Moongrazer, what you are doing sounds sensible, but it sounds as though you might be mixing up the calculation of a historical rate of return and a strategy to reduce forward volatility. The 2 are entirely unrelated.

I think there are 2 possible ways of doing the calculation. The first is the “first in first out” (FIFO) method, as you have described. This will give you 2 sets of IRRs and sometimes a partially closed out deal. One set of IRRs will be completely fixed, as earlier unit purchases are closed out. The other set of still open investments will have IRRs that will vary according to unit value. You could then calculate an average IRR by weighting according to the number of units. For example, if you received an IRR of 5% on 10 units, 10% on 3 units and -20% on 2 units, your weighted average IRR would be (5%*10 + 10%*3 – 20%*2)/(10 + 3 + 2) = 2.67%.

The other approach would be to keep every investment alive and apply the same percentage withdrawal across all of them. For example, if you are selling 5% of units and had made 2 investments, A of 20 units and B of 10 units, you would sell 5%*20 = 1 unit from investment A and 0.5 of a unit from investment B. Then calculate the weighted average IRR as with the FIFO method. This approach would give you a result that is closer to what people are aiming for with XIRR.

Personally I don’t do any of this. I doubt it would be of any use to me if I did as the investment risks I have taken have varied enormously over time. I also hate admin 😉

@Moongrazer, forgot to mention, in your case as you making multiple investments, but not taking money out, IRR is fine as well as there can only be a single IRR that will “fit” your situation. Just be careful with it if you start making withdrawals AND subsequent investments.

@HRP

“You are solving a different problem – initial investment £1000, no withdrawal after 1 year, add £350 in year 2 and take out 1716 after 3 years – this conveniently has only 1 IRR. What if the money withdrawn after the first year was spent?”

Then still one of the three discount rates return by XIRR has to be arbitrary chosen by the investor, because XIRR can’t know how much of the subsequently invested £4310 is/would be new money. It’s tricky to blindly use XIRR, I agree.

” If there is only a single initial investment, there can only be single IRR and as such XIRR is fine.”

Wouldn’t there be single IRR also when there are multiple investments followed by multiple withdrawals i.e. only one change of sign?

*returned and *arbitrarily

Hey guys,

I feel we’re overcomplicating things here.

XIRR copes fine with both money in and money out transactions and it gives you an equivalent (smooth) internal rate of return – that you would have required to hit the same end result.

The key concept here is that the the IRR can vary massively from the average annual return on the unitised fund prices over time where money is being put in (or withdrawn) over time.

This is due entirely to pound cost / Dollar cost averaging or it’s inverse.

Where your saving for the future – putting money in- then a big dip in your unit price followed by recovery will amplify your average returns whereas if prices race ahead and then fall back this will drag down your average returns.

As i said above – this can be a tricky concept to get ones head around but one of two plain English article that i wrote (for investor’s chronicle last year) to explain it – can be found here http://wp.me/p45vXF-ts

Hope this is of interest

Yeah Paul, I’ve seen your articles, but you clearly oversimplify things, seems like you didn’t read the comments here. With alternating withdrawals and investments things do get complicated to the point where XIRR does not give a single result because multiple are possible and some understanding is required to be able to draw conclusions and maybe pick one of these.

To Paul W.

Well I do try to simplify problems and communicate in plain English. As Einstein said, we should try to reduce all problems to their simplest form – but no further.

My concern is that some comments here overcomplicate the issue.

Are you really saying that (for a given start value, end value and series of inputs and withdrawals) there are multiple solutions to an IRR calculation ?

That would certainly be news to me.

@HRP

” You could then calculate an average IRR by weighting according to the number of units. For example, if you received an IRR of 5% on 10 units, 10% on 3 units and -20% on 2 units, your weighted average IRR would be (5%*10 + 10%*3 – 20%*2)/(10 + 3 + 2) = 2.67%.”

But that would be unit-weighted average regardless of unit prices paid. Had the prices differed significantly you would get a result very different from money-weighted-average and this is ultimately what we invest – money.

And time of investment for every lot of units also matters, so in no way this average is correct.

I think I know how it could be done… I hope the formatting will be ok. If anyone can see a fault in this, please let me know.

Imagine such a series of cash flows:

date, number of units flow, unit price, cash flow

2009-01-01 -500 1 -£500

2010-01-01 200 1.2 £240

2011-01-01 -300 0.9 -£270

2012-01-01 600 1.5 £900

To calculate XIRR on this would be incorrect because there is a withdrawal before a subsequent investment and XIRR can give multiple results in such cases. We can, however, take advantage of the knowledge XIRR doesn’t have, that is prices paid and returns achieved in between of cash flows reflected in unit prices. This series of flows can be hence rearranged to always get a single result. To do that we need to keep the same time of investment for every lot of units ever held. We move every closed deal to the end of the table. For the above table the amended table would be:

2009-01-01 -£300

2011-01-01 -£470

2012-01-01 £900

I only moved the 200 units bought for £200 and sold for £240, held for a year, to the end of the table. Now we can use XIRR to calculate the IRR for the amended table and we get 8.95%. Does it make sense?

@paul claireaux, “That would certainly be news to me.”

I have found it news to many people, including many who should really know better. Look at the example I posted here – 4 cash flows, resulting in 3 different IRRs. I have seen many problems with the misuse of IRR, including use outside of investing.

@Paul W. “Wouldn’t there be single IRR also when there are multiple investments followed by multiple withdrawals i.e. only one change of sign?” No, you can still get multiple IRRs, although in this case you would tend to see one reasonably sensible solution and other silly looking ones, i.e. large and negative. e.g. -3000, -2000, 5000, 1000 annual cash flows have a “reasonable” looking IRR of about 11% and silly ones around -119% and -259%.

“But that would be unit-weighted average regardless of unit prices paid.” I think that is fair, but weight by capital invested if you prefer. I have a bit of an issue with that though in that capital invested further back in time is usually worth more than it is today – there might be an argument for uplifting by at least the risk free rate, or inflation as it is easier to look up. There is no right answer.

To HRP.

I must say that I feel that your comments are confusing what was otherwise a useful article.

The Investor has outlined us a useful way to determine a unit price on our portfolios over time. And from this we can work out what the average annualised return on that portfolio has been so as to compare it to a Unit Trust or OEIC.

We have then explored how, where we put money in OR take money out of our portfolio over time – our effective return (IRR) is often a very different number. And I’ve explained this in plain English in my blog. if you believe that to contain flaws – please show where they exist.

This marked difference between the average annual growth on a fund price and our personal effective return is due to the well known effects of pound cost averaging (and reverse pound cost averaging on withdrawals)

Now, you claim that there are multiple IRRs for a given set of input and output values.

Perhaps you’re simply stating that each payment in (or withdrawal) from a portfolio will have a different rate of return?

Well yes of course – but that’s different to stating that there are multiple IRRs on a series of transactions. There is only one.

If you still claim otherwise I’d suggest you try to prove it – with a simple example.

I will try to make this as simple as possible. If you have multiple input and multiple output cashflows, there can be more than one IRR. The example I posted is about as simple as I can think of:

Cashflow -1000, +3600, -4310, +1716. IRRs for these flows are 10%, 20% and 30%. Easily proved:

-1000 + 3600/(1+10%) – 4310/(1+10%)^2 + 1716/(1+10%)^3 = 0

-1000 + 3600/(1+20%) – 4310/(1+20%)^2 + 1716/(1+20%)^3 = 0

-1000 + 3600/(1+30%) – 4310/(1+30%)^2 + 1716/(1+30%)^3 = 0

If you are having problems with the concept of a function resulting in one of many values, consider the square root of 4, answer 2, but -2 is also valid.

To HRP

Thanks for that lesson in square roots!

As it happens I quite like maths riddles but prefer slightly trickier ones than that. So let me offer you a riddle in return.

What’s the square root of minus 4?

(answers on postcard)

Now to your example.

I have to say you’ve picked a ‘curious’ example to make your point.

Your portfolio starts with an injection of £1,000 followed by a withdrawal of £3,600 the following year. In other words you would be in ‘debt’ !!

So, yes of course, if you allow your portfolio to go into debt AND you assume the same IRR on the debt as on your growth! – there are multiple ‘theoretical ‘IRRs’

But seriously – is this curious example useful to readers of this educational blog?

Let’s make the example a REAL one.

Try setting your initial investment to £10,000 (to keep the balance positive)

Then in subsequent years:

1. Withdraw £3,600

2. Add £4,310

3. Withdraw £1,716 and finally

4. Withdraw £13,177 (which is your final value in the fund)

Now tell us all how many IRRs apply to this series?

My excel software only gives one answer but I’m open to challenge (;-)

Thanks

Paul

@paul claireaux

“Your portfolio starts with an injection of £1,000 followed by a withdrawal of £3,600 the following year. In other words you would be in ‘debt’ !!” This claim is unfounded as you have no knowledge of the performance of the investment. For example, if the investment had 10 bagged to £10,000 a withdraw of £3,600 would leave a remaining portfolio value of £6,400.

“Now tell us all how many IRRs apply to this series?”. 2 real valued IRRs. +10.00% and -187.73%. Assuming that the portfolio is not geared, it is safe to discard the second IRR as the portfolio is effectively capped at a 100% loss.

I had a look at your blog link by the way and it seems fine to me. No mention of internal rate of return though.

@Paul W

I have given this a bit of thought and I think a reasonable way to come up with a sensible portfolio return value is to divide the total current value of units sold by the cash invested. For example, taking Paul Claireaux’s 5 cashflows and adding (inventing) some unit prices gives this (rounded to 2dp, hopefully formatting will not look too horrible):

cash received unit price units bought total units FV units sold cash invested

-10000 100.00 100.00 100.00 10000

3600 110.00 -32.73 67.27 4251.52

-4310 90.00 47.89 115.16 4310

1716 125.00 -13.73 101.43 1783.37

13177 129.91 -101.43 0.00 13177.00

Total cash invested is £14,130

Total value of units sold (at current unit price) is £19,211.89

Return is £19,211.89 / £14,130 – 1 = 34.26%.

Annualised return over 4 years is (1 + 34.26%)^(1/4) – 1 = 7.64%

The 7.64% compares with 6.76% annualised unit price change and 10.00% for the internal rate of return. Had all cash been invested up front and units held until the end, all 3 rates of return would be the same.

The value I have suggested is a form of something called the Modified Internal Rate of Return (MIRR). It is superior to IRR in a number of ways, including being guaranteed to have only a single value, and I think may be a good metric to use to track portfolio return and compare with an alternative investment. I have not used it for this so could be wrong, but have used MIRR as a forecasting tool in project planning where it is recognised as being superior to IRR. You could compare the MIRR of your portfolio with that of a benchmark of similar risk, e.g. a tracker with the same timed investments and withdrawals.

@The Investor

Sorry if I have added confusion to your article, which I think is good by the way. Please feel free to delete my comments as I will not in the least be affronted. I have found thinking about the issues for calculating a rate of return for a portfolio with multiple inputs and outputs useful in any case. Some form of MIRR seems sensible to me and I am not sure why I did not suggest it in the first place!

To HRP

Aha and now we agree – the only sensible answer is 10% p.a.

As to my own blog – I don’t publish articles on IRRs for reasons well proven by this discussion. My site is about helping non experts get to grips with some basics and know where to find useful help.

I would say the excellent Monevator site is for more advanced DIY investors.

And yes, I have explained the potential upside and downside to your returns from pound cost averaging in plain English.

Now what about that riddle?

Any idea about the square root of minus 4?

Paul

@ paul claireaux

“Aha and now we agree – the only sensible answer is 10% p.a.”

On your example, I agree, but I do not agree (and have demonstrated with a simple example) that in general there is only one answer.

Whether 10% is “sensible” is not clear either. I would say that this may have flattered the return achieved because it over values the interim cash withdrawals, which is another potential problem with IRR.

“Now what about that riddle?

Any idea about the square root of minus 4?”

+/- 2 times the square root of minus 1, as I am sure you are aware.

To HRP.

Glad to be on the same page at last and yes, I accept that some random theoretical alternatives can be thrown up with IRR but do they mean anything to ordinary investors?

And yes, I agree that the IRR (with monies flowing in and out) can be very different to the average annual return on units over time and that’s precisely the lesson I offer in that article.

I’ll post the second in that series (also appeared in Investor’s Chronicle) anon.

That’s where we’re in the drawdown phase with our money – taking it all out over time – where of course the unit price path effect is the opposite to our wealth building phase.

Signing off for now – it’s been fun – and I can see you understand your numbers.

Paul

For those not aware (the square root of minus one is a complex number – otherwise known as an imaginary number known by the letter ‘i’ and it points at 90 degrees up from the normal number line . . . but that’s another story)

IRR of 10% in Paul’s example overvalues the withdrawals as if they were still invested at the same rate. This is the problem with all interim withdrawals. Each of them could be reinvested at a different rate than what’s assumed by IRR.

This is probably why it’s better long term that I’m just tracking each individual sum I invest from the time it is invested (with its own level of return that is accurate to the single sum invested). If you’ve unitised your portfolio you need only know what the current unit price is, and that automatically feeds into working out the current value of every individual investment.

Admittedly, if you invest small amounts each week or month that could get rather fiddly, but at least you have the granular data for any more complex calculations you care to do.

That said, generalising your regular investments over the course of a year down to a single lump sum would probably give good enough accuracy to be acceptable – yes, that skews in favour of sums invested later in the year but if we’re talking 20-30 year time frame minimum, I think it’s inconsequential.

To Paul W

I agree that all your investment inputs will normally enjoy a different return to their respective maturities.

And that’s where the IRR can be useful.

It shows your effective smoothed return on all your investments and if that’s no more than say 2% or 3% p.a. over the long term – it begs the question of whether all this risk and hard work is worth it – and whether you might be better leaving it on deposit with the bank or in some cautiously managed fund.

@paul

I am saying that a wthdrawal of £3600 in our example is discounted at the same rate by XIRR function. In other words: XIRR assumes that the subsequent investment of £4310 contains only £350 of “new” money. It gives the same result as if there was no withdrawal and only another investment of £350 which is £4310 – 1.1x£3600. It is a problem because in real life you would for example keep the £3600 in your savings account and have a different return over that year.

Hey Paul W.

I’ll have to make this my final word as my writing is getting behind now.

The XIRR function doesn’t assume anything.

it simply takes the cash-flows you provide to work out what ‘equivalent’ (compound and smooth) rate of return would give the same answer as your outcome.

Best wishes.

@Paul W, I understand exactly what you mean and I agree with you. The problem with IRR is that it assumes the return on cash is the same as the IRR, which is likely to be nonsense. My previous MIRR comment suffers from the same problem, so best ignore it was illegible and I have thought of a better way anyway!

A very simple calculation that can be done is to divide the total outflow of cash by the total inflow, e.g. for flows -10000, 3600, -4310, 1716, 13177 the return is

(3600 + 1716 + 13177)/(10000 + 4310) = 1.2923

That is a rate of 29.23% over 4 years, or 6.62% annual equivalent rate. That problem with this calculation is that it is the opposite of IRR – it understates the return/cost of cash (assumes it is zero). What is a good rate to use for the return on cash? It dawned on me that a good rate to use would be either RPI or CPI inflation. The return on cash roughly tracks inflation and RPI/CPI numbers are readily available from the ONS.

So to calculate the rate of return using CPI adjusted cash, do this:

1) Look up the CPI figures at the time cash flows (assume these are 100, 105, 109, 112, 113)

2) Discount the invested cash to the date of first investment using the CPI. So 10000 stays as it is and the 4310 becomes 4310*100/109 = 3954.13.

3) Forward value cash withdrawn to the point of the final withdrawal (or valuation date). 3600 becomes 3600*113/105 = 3874.29, 1716 becomes 1716*113/112 = 1731.32 and the final 13177 stays as it is.

4) Calculate the return as before (total output)/(total input)

(3874.29 + 1731.32 + 13177)/(10000 + 3954.13) = 1.3460

That is 34.6%, equivalent to 7.71% per year.

To compare with unitisation and IRR, simply adjust the calculated return by inflation or adjust the unit growth rate and IRR by inflation. e.g. IRR is 10% and the CPI went from 100 to 113, so IRR adjusted for inflation is

(1+10%)*(100/113)^(1/4) – 1 = 6.69%

This is a very simple calculation to do. It just requires looking up CPI/RPI for each cashflow. If that is too much hassle, just do it once per year as Moongrazer suggested. If net cash was invested over the year, discount the total amount for the year and treat is as input, if net cash was withdrawn, forward value it and treat as output.

@HRP

I am afraid you don’t get correct annualized return because the cash flows’ dates were ignored in your calculations. They have to be taken into account even if you assume no inflation.

@Paul W

I appreciate what you are saying, but it really depends on what interpretation is put on “money weighted return”. My proposed definition results in a convenient sleight of hand which means that the dates are actually immaterial. My definition is the weighted average growth, where the weights are the initial investment amounts. Would you agree that that would be a fair “money weighted return”? If not, ignore the rest of this comment!

Possibly the easiest way to explain why the dates are immaterial is by continuing the example. The approach is to consider the investment as a set of matching buy/sell trades and then calculate the weighted average growth on those pairs of trades.

If you assume unit prices of 100, 110, 90, 125, 129.91. The actual prices do not matter, but the last must equal the final cash flow divided by the remaining number of units in order to sell off all units at the end. You can then consider the investment to consist of a series of 4 disposals. The first 3 disposals covering the initial £10,000 and the last covering the later £4,310:

Disposal 1) 32.73 units bought for 3272.73, sold for 3600, growth 3600/3272.73 = 1.1

Disposal 2) 13.73 units bought for 1372.80, sold for 1716, growth 1372.80/1716 = 1.25

Disposal 3) 53.54 units bought for 5354.47, sold for 6955.87, growth 6955.87/5354.47 = 1.30

Disposal 4) 47.89 units bought for 4310, sold for 6221.13, growth 6221.13/4310 = 1.44

To get the weighted average growth, for each trade multiply the growth by the amount invested, add them all up and divide by the total amount invested. But multiplying the growth by amount invested gives the final trade amount, the output cash flow for the trade. In effect then the weighted average growth is the sum of the output cash flows divided by the sum of the input cash flows, which is what I suggested in the first place.

Guys, really, take it offline – you undermine the value of this BLOG with your nitty gritty theoretical debates.

Book a skype together and discuss if you really think they’re worth any more chat.

@paul

Firstly you didn’t see a problem where I pointed it out for you, then you made your final comment. Now you come back patronizing people who wanted a deeper understanding than yours and developed a discussion where it started. There are plenty of entries and comments on these blog, just stop reading these. You are winding people up by saying they undermine this blog with their comments, how ridiculous is that. And you have your own blog to worry about.

@paul claireaux

Is this your blog?

If The Investor thinks I am “undermining the value” of his blog no doubt he will say so, delete my comments and ask me to desist, which I will do happily. I do not see it as any business of yours to determine the suitability of mine or anyone else’s comments.

@The Investor, please let me know whether you think my comments are of any value and whether you would like me to continue posting. I have found thinking about this a useful exercise and will happily take it elsewhere if you prefer (I might do that anyway). My comments have certainly strayed away from unitization towards ways to calculate money weighted return, which I do appreciate was not the focus of your article.

To Paul W

Our debate was quickly resolved and resulted in a plain English explanation of IRR for this community.

The debate now between yourself and HRP is something else and looks like it could run for months with no conclusion!

As a blogger myself I understand the pain of seeing one’s carefully crafted work ignored by side discussions that go nowwhere.

Well I can’t say I’ve really understood many of the last few comments but suffice to say, based on the original article, I have now successfully unitized my portfolio and it was swimmingly simple – thanks TI!

I’ve come late to this party but here’s my opinion:

@ HRP and Paul W – thank you for your contributions. Though complex and difficult for me to understand that’s due to my own limitations not yours. You’re trying to help people and I appreciate you taking the time.

@ Paul C – please do not presume that you know what’s best for other people.

Thanks for that advice Accumulator.

I think weenie sums the debate up nicely above.

A pretty amazing thread on IRR that sheds more light on HRP’s and Paul W’s discussion:

http://www.bogleheads.org/forum/viewtopic.php?f=10&t=115030

Intense but illuminating.

A bit late on this one. Not sure if anyone still looking. Couple of things:

* Would SIPP tax relief count as newly added cash? Or part of the growth? Since it can appear many weeks or months after initial contributions, it’s quite difficult to track

* I presume it makes sense to include all assets & debts (mortgage, credit card) in the unitization calculation, since it’s one’s net worth which is the bottom line?

@algernond

Re: SIPP tax relief – yes, it would count as newly added cash, as you would presumably use it to purchase more funds within your portfolio. It isn’t growth in its own right as it is simply more of your money coming into the portfolio rather than anything your existing investments have returned.

Yes, it’s new cash but waiting for it is a problem with many SIPPs. That’s one reason why using a simple insured PPP can be a better idea for many savers who don’t want or need the massive fund choice of SIPPs

You get your tax relief invested on day 1.

@Algernond

I also treat the SIPP tax relief as funds inflow, although I don’t unitise my portfolio as discussed on this thread.

However I only treat the basic rate tax relief as funds inflow.

So, for example:

I invest £8,000 into my SIPP and buy £8,000 of funds.

A couple of months later my broker reclaims the basic rate tax of £2,000 and I buy more £2,000 funds.

At the end of the tax year HMRC give me back my £2,000 higher rate tax. I bank this and do not treat it as part of my current SIPP (even though it essentially lowers the costs of my units for unitisation purposes).

Of course, if I invest that £2,000 back directly into my SIPP I would treat it as new funds.

Assuming my funds remain at their purchase price my portfolio looks like:

£10,000 of funds for £8,000 cash in.

£2,000 increase in my cash asset held outwith my SIPP.

I’m happy to see things like this without anything more complex. But others may differ. If this thread has taught me anything it’s that everyone measures their personal performance differently!

I keep track on my net worth using individual pots – essentially house, cash, SIPP, ISA, other funds/shares. I don’t unitise at all, but if I did I think I would only do so within the SIPP, ISA, funds/shares (as three calculations, not one big one).

Again care needs to be taken. For example I’m 100% invested in equity in my SIPP, but also have cash elsewhere. So my asset allocation is 100% equities in my SIPP but much less within the net worth as a whole. Many finance blogs, etc, tacitly assume you have your whole net worth (ex house) within SIPPs, ISAs etc, and advice on asset allocation within this. I step back and view my net worth more holistically.

Hope this ramble helps!

@algernond — Personally I track what I consider my investment portfolio via a unitized process, but I do have cash outside of this (earmarked for a house deposit some day) plus 1-2 other small assets that I don’t include.

As discussed above, my primary goal with unitization is to see how I am doing versus the market and other active investors.

It doesn’t make a lot of sense for me to distort this with a big house deposit — or a mortgage or whatever, were I too have one — given that allocation to those assets would be driven by different parameters/goals.

(Currently my portfolio is by far the largest portion of my assets, as it happens, but that’s by the by. Also, if I decided to hold a big slug of cash for strategic investment reasons (which I could well be doing was cash readily yielding say 4-5% right now) then that would stay in the unitized portfolio. It’s the intention/purpose/thinking of the money that drives what I consider my investment portfolio, not the asset class, if that makes sense 🙂 ).

As others have said, if you’re more trying to see how your net worth is increasing generally over the years, it may well be more appropriate to use another return measure.

I find this interesting – I have thought about unitising my portfolio, but never really got round to it. I decided a while back to use a very simplistic approach to things so I could track my performance against my Financial Advisers.

I take a snapshot of the value of the portfolio on the 1st day of the tax year (be it ISA / Pension etc.).

At the end of the tax year, i take a snapshot of the value again.

I then add the amount i invested over the year to the original value to give me my base cost – any value above that is my return.

I apply this to both the ones I manage and the ones my FA runs, and it means we are on the same page simply. So far its not done too badly (he has beaten me this year for the first time!).

@Accumulator thank you for your comments. The Bogleheads link was interesting. I have a fair amount of experience of people getting strange results with the Excel XIRR function. It can be dangerous because management decisions can be turned on an XIRR result. For example, one course of action might imply a higher XIRR than another, which can push a decision in the wrong direction if the calculated IRRs are wrong or misleading.

The final suggestion I made for calculating a money weighted return is actually very straightforward and does not rely on Excel XIRR. all it requires is adding up inflation adjusted investment values, adding up the inflation adjusted withdrawals and dividing withdrawals by investments. I don’t believe it is any more difficult to understand or implement than unitisation and could be done with the addition of a few extra columns in a spreadsheet.

As an example, consider an active investor putting money into US equities since the end of 2008. Using the Vanguard provided returns here https://www.vanguard.co.uk/documents/adv/literature/tactical-asset-allocation-equity-and-bond.pdf it is possible to work out the £ increase in unit values of US equities from the end of 2008 to the end of 2014. This works out as 2.35 times, or 135% (15% AER). If the investor made these investments:

2008 10000

2010 10000

And these withdrawals:

2013 20000

2014 16774

Then the return without inflation adjustment is (20000 + 16774) / (10000 + 10000) = 1.8387, or about 84% (10.7% AER).

To correct the cash flows for a reasonable return on cash, using the CPI index figures:

2008 109.5

2009 112.6

2010 116.8

2011 121.7

2012 125.0

2013 127.5

2014 128.2

Calculate the 2008 value of cash invested:

2008 10000

2010 9375

And the 2014 value of cash withdrawn:

2013 20110

2014 16774

The return is then (20110 + 16774) / (10000 + 9375) = 1.9037 times, about 90% (11.3% AER)

A descriptive explanation of the above might help to understand why the calculated return value is reasonable. Imagine the investor starting off with £19,375 at the end of 2008. £10,000 is invested in US equities and £9,375 left on deposit. By the end of 2010, the £9,375 has grown to £10,000 and this is fully invested in US equities. The investor sells £20,000 of equities at the end of 2013, after a 28% increase over the year, and puts the cash on deposit. By the end of 2014, the deposit has grown to £20,110 and the remaining US investment is worth £16,774. Total portfolio value at the end of 2014 is £36,884 compared with £19,375 at the end of 2008. Divide one by the other to get the return.

The Excel XIRR figure comes out as 2.25 times, 14.5% AER. This is misleading – there is no way that the investor made a return of 125% on the 20000 invested. The IRR figure is so high because it discounts the return on cash and the return on the investments by the same amount, which is ludicrous in this example as the return on cash (CPI inflation) was much lower than the return on US equities.

I might now go and unitise my SIPP and calculate the money weighted return. It might show up whether my activity has added any value!

I should point out that most of my investments are in fixed income and that IRR is an extremely useful and important metric. For example, the yield to maturity of a bond is the IRR and I use IRR all the time in evaluating bond investments. So I am not against IRR, or even Excel’s XIRR function. I just consider IRR to be unsuitable as a portfolio return metric, amongst other things.

Hopefully this article is not long forgotten, and someone is still checking the comments. Understanding the logic above (which has been very well explained, thanks Investor), you always need to calculate the current total before you then start any calculations for adding/sub-ing.

I’d be very interested to find out what level of accuracy people use for finding out the current total? On the exact moment just before purchased? Day before purchased? Week? Month?

I’m trying to get that balance of unitizing effectively (with reg. monthly updates on a mix of dates) without constantly checking 3 different accounts, and keeping manual input in Excel to a minimum. E.G. updating only once a month would be perfect, but market volatility will cause errors to creep in.

Appericate any thoughts, or corrections where my understanding is wrong.

@IC – for my own simplification purposes, my current total is calculated on the last working day of the month. This obviously means it’s far less accurate than if I say used the exact moment I made my purchases but I’m using the unitization as merely a “guidance” of how my overall portfolio is doing.

Updating consistently just once a month is enough for me.

@weenie – thanks for the info.

Just want to quietly say, credit due, I’m loving your advice and opinion. This article is icing. Suffice to say I’ve finally escaped the niggling torpor of my long held list of 20 active funds and their eye-poking charges, and switched to a Monevator-balanced list of 7 passive. It’s taken 2 years of thinking about it (i’m kidding, it’s probably 3) but finally in progress, I feel sure sense of relief. I say in progress because Best Invest have made a hash of the switch, but I was half expecting that. Anyway, thanks for the help. Unitization ahoy, can’t wait for the weekend 😉

Hi guys

Great article. I have a question if i may…..

when you value your original portfolio do you include cash not invested ie just sitting in isa account???

At last I’ve got round to unitising my portfolio following your guide here. I‘ve gone back over 9 years of records for my isa and sipp and can see how my highish yield portfolio has performed.

Only now it has struck me that comparing it to the FTSE100 is not a good like for like, as any dividends have always been reinvested. While historical prices are easy enough to download for the FTSE100, I think I need a to compare against a total return FTSE100 index. However I haven’t found a website with daily downloadable data. Can anyone suggest where this data is available……freely?

I’ve just found this article and tracking my portfolio has always been a huge error filled mess. I track all things financial in google sheets, does anybody have an example spreadsheet/template that could assist me with this? I’ve tried to make one myself but im awful at excel.

“However if you did want to track how particular accounts are doing … then you could create separate spreadsheets to follow them.

…

Remember, you’d still want to unitize the entire portfolio to properly track your overall returns…”

Any thoughts on how best to do this? I’ve unitised three separate accounts, with total value and unit value at the end of every calendar month for each, but I’m not sure how to combine them into a single total unit value for each month.

My first thought is it’s something like this, but I’m not sure if this would be correct:

(

(Fund1Total£ * Fund1UnitValue)

+ (Fund2Total£ * Fund2UnitValue)

+ (Fund3Total£ * Fund3UnitValue)

) /

(Fund1Total£ + Fund2Total£ + Fund3Total£)

After trying some examples out in spreadsheets, I think that maybe using “total number of units” instead of “total £” in the above equation makes more sense. So:

(

(Fund1NumUnits * Fund1UnitValue)

+ (Fund2NumUnits * Fund2UnitValue)

+ (Fund3NumUnits * Fund3UnitValue)

) /

(Fund1NumUnits + Fund2NumUnits + Fund3NumUnits)

But I’m not sure how to verify whether this is accurate or not. Any thoughts appreciated!

Fabius, you’ve unitised each of your separate portfolios by looking at the value of the portfolio at different periods, assigning a number of units, then adjusting the units for every cashflow change.

To unitise your combined portfolio you need to do the same but you have to combine the different total values of each investment into a total portfolio value and combine the different cashflows into a combined portfolio cashflow. You then assign a unit value and adjust for every cashflow change throughout the entire portfolio.

Just came across this really useful article. I must admit I haven’t all of the comments but if anyone has the basic template of a spreadsheet which they are willing to share I would really appreciate it. I’d even be open to paying for it.

@TI – maybe be a bit of a money spinning oppty for you?