How much will bond prices fall if and when interest rates go up? With many government bond yields straying into negative territory or teetering on the brink, surely this asset class now only offers the prospect of painful loss in the years ahead? Or maybe not?

You can get an intuitive feel for how big bond losses – or potential gains – can be using a bond price calculator.

And some of the results may seem a bit, well, weird…^{1}

Let’s run through a few simulated examples of how a range of hypothetical bonds could move in response to changes in market rates.

**A note on confusing bond terminology**

Just to clear a few things up before we start.

**Interest rates**

When people talk about bond prices falling due to rising interest rates, they’re not talking about central bank interest rates like the Bank Of England’s Bank Rate. They’re talking about the market interest rate for a bond. Each and every bond is subject to a market interest rate that is the sum of supply and demand for that particular bond. The market interest rate is the return investors demand for tying up their wealth in that bond, and it fluctuates in line with the market’s view of factors such as inflation, the bond’s credit rating and maturity date, other macro-economic forces and, yes, the influence of central bank interest rates.

**Bond yields **

There are many different types of bond yields. Commentators often bandy about the term ‘yield’ as if it’s a unified concept that everybody understands. When I talk about yield in this piece, I’m referring to the yield to maturity (YTM), also known as the redemption yield. This is the annualised return you’d expect to receive if you invest in a bond and hold it to maturity (accounting for its market price and the remaining interest payments, which are assumed to be reinvested at the same rate). It’s the go-to yield to use when comparing similar bonds (for example gilts) that vary by price, maturity date, and coupon rate.

### Scenario #1: Interest rates rise by 1%

Say we own a newly minted 30-year government bond and interest rates shoot up by 1%, with our bond’s yield rising in turn to 2%. We can use a bond price calculator to survey the damage using the following specs:

**30-year bond**

- Face value: £100
- Coupon rate
^{2}: 1% - Market rate: 2%
- Years to maturity: 30

Dial that scenario into the calculator and it tells us the bond price falls from £100 to £77.52.

Capital loss: **-22.5% **

From our perspective here in June 2020, 30-year gilt yields have come down 1% in a year, so it doesn’t seem beyond the realms of possibility that they could rebound back, given time.

(Note: the size of your loss also varies depending on the speed of the interest rate change – we’ll come back to that.)

Now let’s replay the interest rate rise but this time with a 5-year bond:

**5-year bond**

- Face value: £100
- Coupon rate: 1%
- Market rate: 2%
- Years to maturity: 5
- Price falls to: £95.26

Capital loss: **-4.7%**

Okay, that’s a much less harrowing number. It also explains why many investors have moved to shorter-dated bonds as interest rates tumbled over the years.

The trade-off is that shorter-dated bonds offer ever less downside protection as interest rates continue their journey to the centre of the Earth. (We’ll come back to that, too.)

Let’s look at the middle ground with a 10-year bond:

**10-year bond**

- Face value: £100
- Coupon rate: 1%
- Market rate: 2%
- Years to maturity: 10
- Price falls to: £90.98

Capital loss: **-9.0%**

Tough but not awful. Stick in an instant 2% interest rate rise though (not likely, but bear with me) and the capital loss is **-17.2%**.

### Cheaper prices = higher yields = recovery mode

Lower bond prices aren’t all bad news. Sure a chunk of your portfolio will get taken to the woodshed for a whalloping. But at least in you’ll be able to buy new bonds at higher yields.

In time, reinvesting your income into those now-cheaper bonds will offset some of the pain of that initial bond market beating.

You can use a duration calculator to see how long it would take you to make good the capital loss by reinvesting your interest payments into higher-yielding bonds after a rate rise.

Turns out the 10-year bond in my example scenario gets back to breakeven after about 9.5 years. After that point, your higher-yielding holdings would put you in profit, relative to the old bond and assuming interest rates remained stable.

The five-year bond takes just 4.9 years to breakeven.

It’s a long 25 years for the 30-year bond.

### Scenario #2: Interest rates fall by 1%

So far, so traumatic. But what if interest rates are forced down even further as central banks suck up bonds with their QE 2020 giga-Dyson?

**30-year bond**

- Face value: £100
- Coupon rate: 1%
- Market rate: 0%
- Years to maturity: 30
- Price rises to: £130

Wait for it…

Capital gain: **30%**

That’s an equity-like gain in the puff of a recession – and enough to offset a lot of stock market pain if you’re packing a large slug of long bonds.

This is why many investors hold long bonds and they aren’t mad to do so. They don’t see historic lows as an unbreakable floor. They think interest rates can fall further. Long bonds will make big gains if they do.

Notice how the 30% gain is larger than the equivalent -22.5% capital loss from the 1% rate rise scenario. Long bonds become more potent at ultra-low and negative rates. That’s what makes them so tempting even in the face of interest rate risk in the other direction.

A rapid **2% yield drop** would mean a **70% gain** for our 30-year bond. You could buy a lot of cheap equities for that, if you could stomach rebalancing into a tanking market.

Before you drool your way to your broker’s screen, note though that interest rates don’t tend to move that hard and fast for long bonds. During the coronavirus crash, for instance, the SPDR 15+ Year Gilt ETF (average maturity 29 years) spiked just 12% as equities dive-bombed.

Is a -1% yield possible for long bonds over time? Well, long-dated inflation-linked UK bonds have drilled down to near -3% yields.

Finally, the 30-year bond is again less lethal if rates rebound in the opposite direction. You’d take a **-39.4% loss** if interest rates rocketed by 2%.

What happens if we go for a short bond?

**5-year bond**

- Face value: £100
- Coupon rate: 1%
- Market rate: 0%
- Years to maturity: 5
- Price rises to: £105

Capital gain: **5%**

That shallow 5% gain demonstrates that short bonds won’t do much to stabilise your portfolio if equities plummet and central banks keep firing their bazookas. The upside for short bonds is limited, especially at this end of the interest rate spectrum.

**10-year bond**

- Face value: £100
- Coupon rate: 1%
- Market rate: 0%
- Years to maturity: 10
- Price rises to: £110

Capital gain: **10%**

Our compromise 10-year bond puts in a decent but not pyrotechnic show. If rates fell 2% it would **gain 21.1%**.

Again, the downside drop is amplified for intermediate bonds relative to its losses when interest rates rise, but the effect is muted in comparison with 30-year bonds.

### Scenario #3: Ultra-low interest rates

The long bond effect is magnified in a low interest world (where this post certainly belongs).

Let’s cut the coupon rate down to 0% and model a 1% fall into negative yield country.

**30-year bond**

- Face value: £100
- Coupon rate: 0%
- Market rate:
**-1%** - Years to maturity: 30
- Price rises to: £135.09

Capital gain: **35%**

That 35% capital gain compares with a 30% gain for the higher-yielding 30-year bond in our earlier interest rate drop scenario.

The lower-yielding long bond **gains 82.8%** on a -2% drop in rates, versus 70% previously.

So don’t believe bonds are necessarily firing blanks.

But what happens if we point this thing in the other direction?

You guessed it. A lower-yielding bond is more dangerous than its higher-yielding cousin when rates rise.

Imagine a 30-year bond with a 0% coupon rate, issued at the nadir of a zero-rate world that was on the turn…

**30-year bond**

- Face value: £100
- Coupon rate: 0%
- Market rate: 1%
- Years to maturity: 30
- Price falls to: £74.14

Capital gain: **-25.9%** (vs 22.5% previously)

Worse, a 2% rise would expose you to a **-45% loss** (vs -39% previously).

It now takes 30 years to breakeven according to the duration calculator, because with no coupons the impact upon return is driven solely by capital gains – with a 0% coupon you don’t have any interest payments to invest into higher-yielding bonds to accelerate you to breakeven.

That’s also why our 0% coupon long bond makes a big 35% capital gain when rates drop – it doesn’t receive any coupon payments that cause it to start reinvesting into lower yielding bonds after the interest rate fall.

The upshot is that lower yielding bonds are more sensitive to interest rate changes. They’ll show bigger losses and gains as we enter the negative yield underworld and the effect is particularly pronounced with long bonds.

For more on the counterintuitive impacts of interest rate changes on bonds, read this excellent piece on bond convexity from *Portfolio Charts*.

### Scenario #4: Rate rise impacts are affected by time

What if the 1% interest rate rise happens after you’ve held our example bond for one year?

**30-year bond**

- Face value: £100
- Coupon rate: 1%
- Market rate: 2%
- Years to maturity:
**29**(previously we calculated the rise to maturity 30 years away) - Price falls to: £78.08

Capital loss: **-22%** (vs 22.5% previously)

Hahaha… -22%? Why, tis but a scratch! While we’re hopping about on one financial leg, just note that interest rate rises are less scary the longer it takes for them to gently waft upwards.

### How quickly do market interest rates move?

All my examples have shown an instantaneous drop in interest rates. That isn’t very likely. Rates fluctuate daily. They will drift up or down over months and years.

What we fear most though is big interest rate rises, so let’s conclude with some of the nastiest examples I can unearth using the UK government bond data I can access.

- The worst year for gilt losses in the low interest rate world (i.e. post-Great Recession) was 2013. 15-year gilts took a -9.6% real return loss that calendar year, according to the Barclays Gilt Equity study (BEG).

- 10-year gilt yields rose
**around 1%**that calendar year according to this aggregation of Bank of England data by*Data Hub*.

- 30-year gilt yields rose
**about 0.5%**that year according to the Debt Management Office’s records.

- 1994’s -13.8% was the worst calendar year return for 15-year gilts since the BEG study started tracking them in 1990.

- The 10-year gilt yield rose just
**over 2%**from 1 January to 30 September that year (*Data Hub*again).

The worst post-war year for gilts came with the -29% loss suffered by 20-year gilts when stagflation was all the rage in 1974 (BEG). Using a bond price calculator and an 11% guesstimate for the coupon rate on 20-year bonds in 1974, that implies a rough(ly) **4.75% increase** in market interest rates that year.

If somebody out there has access to more accurate data, I’d love to hear more.

### Unbroken bonds

Interest rate rises as violent as those I’ve simulated are possible. Shorter-dated government bonds will shrug off those hikes better than long government bonds.

But the capacity of bonds to protect diversified portfolios against a crash is far from exhausted at low interest rates, except in as much as short bonds run increasingly out of puff the lower we go.

The extreme volatility of long bonds in this environment suggests we may need to think about them in a new way.

Would you be interested in an asset that’s negatively correlated to equities that could help offset a market crash – but which entails big-kahuna level risks of its own?

If long bonds are too risky to properly belong in the defensive part of the portfolio, then what if a 5%-10% allocation was carved out of the equity side?

That is how the Permanent Portfolio works. With cash acting similarly to short bonds, long bonds provide the best protection against a deflationary recession, while equities are for growth and gold for when nothing else does it.

A risk-portfolio allocation to long bonds could also make sense for somebody whose holdings are dominated by extremely risky equities (think risk factors, emerging markets, and sector bets) or even a young adventurer who would-be all-in on 100% equities but would also be happy to have the best dry powder to hand when the market crashes again.

Personally, I’m happy to keep holding intermediate gilts as a muddy compromise between knowing that interest rates could go either way and needing some decent crash protection for my portfolio.

I recommend playing with a bond price calculator for yourself though, as an easy way of visualising more ‘What If?’ scenarios.

Take it steady,

*The Accumulator*

**Bonus appendix: Bond funds, duration and bond price calculators**

It’s simplest to use duration as an approximate guide to your bond fund’s prospects when its market interest rate changes.

As a rule of thumb, a bond fund (or bond) with a duration of 7 will:

- Lose 7% for every 1% rise in its yield.
- Gain 7% for every 1% fall in yield.

Whatever your bond fund’s duration number, that’s roughly how big a gain or loss you can expect for every 1% change in its yield. The duration number should be published on the fund’s home page.

However, duration is a moving target. Duration increases as yields fall (and vice versa) which means losses and gains are amplified the lower we go. Again, as we saw earlier that super-charges the volatility of long bonds in particular, and the same goes for long bond funds.

Still, this stuff only really sunk in for me once I started running my bond fund numbers through the calculator.

First go to your bond fund’s home page. Look up its average coupon and average maturity metrics.

**Vanguard UK Gilt ETF – interest rate falls by 1%**

- Face value: £100
- Coupon rate: 3.1% (fund’s average coupon)
- Market rate: 2.1%
- Years to maturity: 19.7 (fund’s average maturity)
- Price rises to: £116

Capital gain: **16%**

This happy 16% gain is a little more than implied by the fund’s average duration of 15 (we’d expect a 15% lift) but this brings me to a good point about all the calculations I’ve used in this piece.

They cough up results to however many decimal places but the equations whirring away in the background use a ‘best fit’ process. They ‘guess’ at the final value and then modify it until further iterations don’t make much difference.

The bottom line is that these calculations aren’t precise answers but they are close enough.

Inputs matter, too. If I change the ETF’s coupon rate to 3.05% then the calculator hands me a 15% gain. So perhaps Vanguard rounded the average coupon number up and that threw the calculator off.

Similarly, a newly minted bond with a 1% coupon won’t behave quite the same as its secondary market equivalent with a 2% coupon.

Nevertheless the calculators help illustrate what we’re in for – even though they have to use a little guesswork.

- If you want to understand the maths behind the calculator a tiny bit better, see these musings by
*The Investor*on a potential bond market crash from… gulp… 2012! (You see why we keep warning that people have feared a bond market correction for donkey’s years? [↩] - Assume a semi-annual interest payout in every example. [↩]

Great bit of analysis.

From a practical point of view i requested a final salary pension transfer value about a year ago.

The figure was about 37 the annual value.

If i get one again this year, will it be 40 / 45 / 50?

If bond yields are zero then a lot of numbers end up heading to infinity and beyond.

(If i had transferred out last year my 37 times might now only be worth 30 (?) So double bonus for – except that i won’t be cashing in)

For many simple bonds, you can convert yields to a clean price very accurately using the Excel PRICE function:

Bond price =PRICE(settle date, maturity date, coupon, YTM, redemption value, frequency, day count basis)

So to price a Gilt, say the UKT 4.75% 7-Dec-2020, for settlement tomorrow (T+1) at a yield of 0.19%, you would use the following

Bond Price = PRICE(“23-Jun-30″,”7-Dec-30”, 4.75%, 0.19%, 100, 2, 1) = 147.19

Redemption value is 100, frequency is semi-annual so 2, day count basis is Act/Act which is 1 in Excel.

It will screw up a bit on edge cases: short first coupon, leap years, redemption date is not a good business day etc but so do most internet calculators. If you want absolute accuracy you need to roll up your sleeves and start using things like QuantLib XL (https://www.quantlib.org/quantlibxl/).

Very timely article. Thank you very much.

Hewlett Packards HP12 calculator is very good for accurate Bond price calculations.

For a (US$) view of the performance of the permanent portfolio (aka Harry Browne) over the last 25 years see: https://retireearlyhomepage.com/reallife20.html

I should add that for those not familiar with the work of John Greaney (at link I gave above) he tracks performance including implementing the, so-called, “4% rule”

Looking at USTs is probably a better measure of what can happen to developed market govt bonds than Gilts (which is sort of a captive market these days).

These are some of the biggest retracements in 10-year UST yields over the past 40 years

May83-May84: 3.90% (10.10%-14.00%)

Mar87-Oct87: 3.00% (7.20%-10.20%)

Oct93-Nov94: 2.80% (5.20%-8.00%)

Oct98-Jan00: 2.60% (4.20%-6.80%)

Dec08-Apr10: 1.95% (2.05%-4.00%)

May13-Dec13: 1.40% (1.60% -3.00%)

Jul16-Mar17: 1.30% (1.35%-2.65%)

Sep17-Nov18: 1.20% (2.05%-3.25%)

Obviously when you had much higher yields (10%+), you got much larger yield moves since there is a geometric, aswell as additive, element to how rates move.

I’ve put a few more moves in the last decade to give an idea of how recent moves higher have been smaller. The fastest recent move was in May 13, the “Taper Tantrum”, when Fed chairman Benanke implied they would taper down QE. USTs moved about 1.4% higher in 6 months.

That’s very helpful. Thank you ZX.

Nice article; interesting to see some numbers worked through in the examples. How does the “yield curve” fit in with this though? Presumably so long as market interest rates evolve along the path predicted by the yield curve there’s no “surprise factor”, and pricing shouldn’t change: the yield curve is telling you what future expectations the market is already pricing in. But if market interest rates (or expectations of where they’ll be in future) drift away from what the yield curve projects, those would be the sort of change which produce these duration-related pricing changes. Or does it actually work completely differently and the yield curve is all about something else?

I’d like to see negative coupon perpetual bonds, We would have to pay for storage!

They would have positive value if

Price = coupon/yield

And 2 negatives cancel out

And the less negative the yield the higher the value?