# How to calculate bond yields

A few people emailed to ask how I calculated the yield on the RBS Royal Bond.

Hey presto! This post will tell you everything you need to know about calculating bond yields, whether for government or corporate bonds.

First I’ll remind you of the basic kinds of rates or yields, then we’ll look at how to calculate them.

### The main types of bond yield

There are three main yields applicable to dated bonds:

#### Coupon rate

This is the interest rate the bond initially pays on issue. It’s invariably given in the name of the bond.

For instance Treasury 5% would have a coupon of 5%. i.e. If you bought it when it was first issued — before the price began to fluctuate in the market — you’d get 5% interest annually for the life of the bond (ignoring costs). The coupon rate is also known as the interest rate.

#### Running yield

Bond prices fluctuate in value as they are bought and sold in the secondary market. As their price changes, so does the running yield that the (fixed) coupon delivers on the (variable) price paid.

The running yield is also called the flat yield or the interest yield.

#### Redemption yield

Bonds with a fixed lifespan pay back their nominal face/par value when they mature. If you buy a bond at less than par and hold to maturity you’ll make a capital gain. If you pay more than par, you’ll make a capital loss.

The redemption yield adjusts the running yield to take this gain or loss into account. It’s the most important yield calculation in most circumstances.

It is also called the yield to maturity (YTM).

### How to calculate the running yield

The running yield is very easy to work out.

Let’s say Treasury 5% has five years to maturity and is currently selling for £120. It has a nominal value of £100.

We can see from its name that it pays a coupon of 5% of £100, so £5 per year.

If you buy this bond in the secondary market for £120, the running yield calculation is as follows:

```Running yield = Gross (pre-tax) coupon x 100
---------------------------
Price in market

= £5 x 100  = 4.17%
------
£120```

### How to calculate the redemption yield

The running yield may be easy to work out, but it’s not very useful.

Usually you’ll prefer to know the redemption yield, unless for some reason capital gains or losses aren’t important to you, or the bond has a very long life ahead of it, in which case the running yield is close enough.

Undated bonds by definition have no redemption value, and so you only need to calculate the running yield for them.

The redemption yield is harder to work out than the running yield, due to compound interest. It’s best done using a special calculator.

Let’s consider the variables via our previous example:

• We know Treasury 5% is going to be redeemed for £100 in five years.
• We’ve paid £120 for it, so we know we’ll lose £20 when it’s redeemed.

The redemption yield spreads such capital gains or losses over the bond’s lifespan, to give an annual return estimate for anyone buying today.

For very short-dated bonds, we can use a handy proxy called the simple yield.

Say we buy a bond for £95 with one year left to run and a 5% coupon.

Over one year we will get £5 as income, and £5 when the bond matures, for £10 in total.

`The return over 12 months is thus £10/£95 = 10.52%`

Stretching this even to a few years reduces the accuracy, however.

For instance, returning to our Treasury 5%:

• We know the capital loss will be £20
• That is a loss of £4 every year over five years
```The annual loss is £4/£120 x 100 = 3.33% a year

The redemption yield is therefore roughly: 4.17% - 3.33% = 0.84%```

### Working out the redemption yield using a calculator

My approximation of the redemption yield won’t be miles off, but the longer the bond has left to run, the more inaccurate it gets.

We really need to calculate the Internal Rate of Return, which takes into account when the coupon is paid and assumes it is reinvested into the same bond to benefit from compound interest.

This requires complex maths that you can read over on MoneyChimp if you’re a masochist.

Alternatively you can use that site’s online bond yield calculator.

Putting the numbers for Treasury 5% into the MoneyChimp calculator, we see the redemption yield is actually 0.892%, so slightly higher than our approximation.

Even easier than using an online calculator is to simply look at the redemption yield column in the bond tables you get online or in the newspapers. It gives you a fair idea of the return you’ll get from buying a bond, and its attractiveness relative to other bonds.

The general rule on yields is as follows:

 Bond priced at: Then: A discount Coupon Rate < Running Yield < Redemption yield A premium Coupon Rate > Running Yield > Redemption yield Par Value Coupon Rate = Running Yield = Redemption yield

### Yield to call

One other kind of yield worth mentioning is the Yield to Call.

This is the yield you’d get if you bought and held a callable bond until to the date when the issuer can decide to redeem it (via a call option), ahead of the definite redemption date.

It pretty much amounts to the same calculation as the redemption yield, except you put the bond’s call date and call price into the calculator.

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• 1 Mahmoud Mansour November 21, 2009, 12:25 am

Very useful information, thank you.

• 2 Mahmoud Mansour November 26, 2009, 9:51 pm

It very useful information provided in a friendly method to grasp the concepts of bonds.
Thank you

• 3 Tronading May 28, 2014, 11:53 pm

TI, I was understanding the ‘running yield’ in the way you explain it above, however all values in HL seem quite far from that value, eg:

http://www.hl.co.uk/shares/shares-search-results/u/united-utilities-5.375-2018

HL running yield = 2.418%, whereas I think it should be 4.827% instead.

It is not an isolated example, so I was wondering if I’m misunderstanding something or if their values mean something else?

• 4 The Investor May 29, 2014, 9:17 am

@Tronading — Yes, I think this is a terminology issue. I am using running yield here the same way it is used it this article, for example: https://www.fixedincomeinvestor.co.uk/x/learnaboutbonds.html?id=98.

They seem to be using it as Yield to Maturity or similar, or just a rounded-up gross redemption yield.

Of course the gross redemption yield is usually a much more sensible yield to use for a dated bond, especially one with a short maturity, in most instances. In the case of that UU bond you *will* take a capital loss of 10% if you hold it for the next four years to redemption (with the loss likely spread over the four years as redemption date gets closer).

• 5 Tronader May 29, 2014, 8:06 pm

Thanks TI,

Not sure what’s exactly that number, as they define ‘running yield’ correctly if you hover over it, but I think it has to be something related to Yield to Matury instead as well. I got the concept now.

Cheers!

• 6 Jon Smith October 16, 2014, 6:56 am

Hi.

Thanks for the tutorial. Really helpful.

Can’t seem to get my head around how your approximation links to the internal rate of return formula?

Would be good if you could elaborate further why this is the case.

Thanks again!

• 7 Jon Smith October 16, 2014, 7:32 am

Sorry, let me rephrase my question… Why is your approximation always lower than the real redemption yield?

Thanks

• 8 The Investor October 16, 2014, 11:49 am

@Jon — My rough and ready approximation assumes the step up/down to par value happens in discrete steps, determined at the date when you’re doing the calculation. (i.e. The £4 a year loss in the example). But in reality the path of the price and its decay will follow a curve, with time. Moreover, you will be reinvesting into that curve.

The approximate doesn’t ‘link’ to the IRR formula, as such. It describes in simple terms what will happen to your money over the holding period, but as noted in my article and by you here, it is only an estimate and a reminder in broad brush terms of what’s going on with your investment over time. 🙂

Using a calculator is the way to get a fairly precise answer. But the flipside is that you’re not going to be able to do the true IRR calculation in your head, unless you’re a genius, and even then it’s essentially an extremely tight approximation to an ‘exact’ answer.

In reality you’ll never know exactly what the redemption yield will be in advance because you can’t know the price of the security when you’re reinvesting the coupon.

E.g. To give an extreme example say your bond halves in price for a month when you happen to be paid a coupon that you reinvest into the same bond, before recovering to par at maturity.

That particular reinvested coupon will deliver a much higher return (because you bought the bond at half price!) than the calculator presumes.