Given that the future cashflows are known quantities, the relationship between the price of a bond and the yield received by an investor is governed by mathematical formulae. We are going to look at three methods of analysing a bond’s yield:
1 income yield
2 simple yield
3 yield to maturity (YTM).
1. Income (or running) yield
Let’s take an example of a gilt, the UK Treasury 5% 2014. This bond pays a 5% coupon (divided into two semi-annual payment) and matures on the 7th September 2014. Thus, if we were able to buy the bond at the face value of 100% (or par), we know that we would receive an income 5% per annum on our investment until maturity.
But what would happen if we paid less than par for the bond? Let us assume that we purchase the bond for 95% of face value. Our income (or running) yield would be:
par/purchase price * coupon = running yield
Or
100/95 * 5% = 5.26% per annum
In this example, the UK Treasury 5% 2014 is trading at a price of 110. This premium to par has the effect of reducing the bond’s income yield as follows:
100/110 * 5 = 4.5% per annum
The income or running yield (sometimes also known as the flat yield or current yield) does not take into account any profit or loss made by holding the bond to redemption, and simply assumes that the investor will be able to sell the bond at the same price that he or she purchased it for.
For a more accurate measure of yield, we must turn to the yield to maturity – the standard calculation employed by market professionals (also sometimes known as the redemption yield).
Before we turn to the more complex yield to maturity, it is worth considering the simple yield. This is a good rough guide to the return available on a bond, and can often be worked out in one’s head.
2. Simple yield
Let us take a theoretical bond with one year left to run until redemption. The bond has a 4% coupon and we have purchased it in the market for 97%. Our return will consist of two factors, the running yield over the 12-month period and the profit made on maturity. Let us assume that we invest £1,000. Thus, for our initial investment of £970, we will receive the following:
1 £40 coupon payment (our running yield)
2 £30 profit on redemption (£1000 – £970)
Our return over the twelve month period is £70 on £970, or 7.2%. From the point of view of many investors, this type of calculation is perfectly adequate for assessing the return on a bond. Known as the simple yield, the formula can be expressed as follows:
simple yield = running yield + (annualised) profit on redemption
= (coupon/bond price) + (100 – bond price)/(bond price x years to redemption)
or, expressed mathematically:
simple yield = (C/P) + ((100-P)/P x t))
where,
C = coupon
P = current bond price
t = life to maturity of bond (years)
For longer-dated bonds, the same theory holds true. Let us take the Goldman Sachs 5.25% 15 December 2015. In June 2011 the bond had four-and-a-half years left to maturity and was trading at a price of 106.
Common sense dictates that a bond with a price standing at over par will yield less than the coupon. Let’s see what our simple yield approach throws out:
Again, we can start with the running yield. The 5.25% coupon will be diminished by the over par market price as follows:
running yield = 5.25% x 100/106 = 4.95%
now, the profit (or, in this case, loss) on redemption:
profit on redemption = (100-106)/106 = -5.66%
annualised over the four and a half years holding period:
profit on redemption (annualised) = -5.66%/4.5 = -1.26%
finally, adding the two (the running yield and profit to redemption) together gives:
simple yield = 4.95% -1.26% = 3.69%
Thus, our simple yield on the bond is 3.69%
The simple yield is a useful quick and dirty calculation, but it does not account for the time value of money for the cash received at different times over the life of the bond. This is particularly significant for longer-dated bonds where the premium or discount paid for a bond may be many years away from the final redemption payment. For this, we need to turn to a more complex calculation – the yield to maturity.
3. Yield to maturity (YTM)
With longer dated bonds, the same methodology as above (for the simple yield) applies; but to gain a more accurate measure we must discount each future cash flow according to when it will be paid. The formula used to calculate this is known as the yield to maturity (YTM) and is effectively the internal rate of return on the investment, allowing for each and every cash flow). The calculation assumes that the interest payments received on the bond can be re-invested at the same rate, although this may not be the case in real life.
The formula for this calculation is somewhat of a handful, and certainly not one for mental arithmetic. It can be expressed as:
price = coupon * 1/r [1 -1/(1+r)n ] + redemption/(1+r)n
where r is the YTM
Mathematicians will be interested to note that working out the YTM from the price is an iterative process.
Note – for more on yield calculation see the appendix.
Bond calculators
I would not recommend attempting to work out YTMs, however this calculation is the industry standard for the comparison of value in bonds. Luckily, there are many easy routes for establishing a bond’s YTM.
YTMs for sterling bonds are published on the www.fixedincomeinvestor.co.uk website. There is also an online yield calculator.
A yield calculator is a must for any serious bond investor. YTMs may be calculated by using the YIELD function in Microsoft Excel or on a dedicated financial calculator such as a Hewlett Packard 12C or 17B (eBay is often a good source of these old-model calculators). Online calculators provide another easy route to determining the value of a bond; an excellent example can be found in the Bonds section of Yahoo Finance: bonds.yahoo.com/calculator.html
Some readers may prefer to download one of the many excellent calculators available as an app on Apple iPhones and other hardware.
Duration
Why do the prices of some bonds move more than others?
Using the example of our theoretical 4% bond with 12 months left to run until maturity, a 1% shift in the yield demanded by investors will produce a change in price roughly equivalent to 1%. In the case of a longer dated bond, with many more years to run until redemption, the price move will be considerably more.
This relationship between a given change in yield and the resulting change in price is known as the duration of the bond. Duration is based on the weighted average of the cash flows, broadly speaking how long it will take you to get your money back. This will have a considerable