Price & yield. The key to understanding the return on all fixed income instruments is to view a bond as a series of discounted cashflows. At the start of the period, the investor pays out cash to purchase the bond. Over the course of the bond’s life, the investor will then receive several payments, usually one or two a year from interest payments, known as coupons and a final repayment at the end of the bond’s lifespan, known as redemption. In this respect, bonds differ fundamentally from equities, where the future cashflows are unknown.
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; the income yield, the simple yield and the yield to maturity or YTM.
Par/purchase price * coupon = running yield
Or 100/95 * 5% = 5.26% per annum.
At the time of writing this article, the UK Treasury % 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) does not take into account and 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 (and more accurate) 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.
£40 coupon payment (our running yield)
£1000 redemption payment ( a £30 profit)
Our return over the twelve month period is £70 on £970, or 7.2%. From the point of view of the private investor, 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:
Annual coupon/Market price + (parmarket price)/Market price /life x 100
Yield to Maturity (YTM)
With longer dated bonds, the same theory applies, but to gain a more accurate measure, we must discount each future cash flow according 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 reinvested 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. For readers who enjoy a challenge, it is:
Luckily for us, YTMs for Sterling bonds are published on this website! See the "Bond Prices and Yields" section for more information. What is more, we have developed an online yield calculator which you can access from the top navigation bar (just click on Yield Calculator).
Alternatively YTMs may be calculated by using the YIELDMAT function on Microsoft Excel or on a dedicated financial calculator such as a Hewlett Packard 12C or 17B. Online calculators provide another easy route to determining the value of a bond, with an excellent example to be found in the “Bonds” section of Yahoo Finance (http://bonds.yahoo.com/calculator.html).
Why does the price 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 and will have a considerable effect on the volatility of the asset over a range of different interest rate scenarios. Let’s take three UK Gilts as an example (calculations based from March 2010).
Bond

Duration

2% YTM

3% YTM

4% YTM

5% YTM

Treasury 5% 2012 
1.9 
105.8 
103.8 
101.9

100 
Treasury 4.75% 2015

4.89  114.22 
108.8 
103.7 
98.8 
Treasury 4.75% 2020 
7.9 
124.8 
115 
106 
98.05 
And how about corporate bonds? The relationship between price and yield for a corporate bond is exactly the same as a government bond, and the same yield calculators can be used for both. However, compared to a supersafe government bond, investors will demand an additional return for lending money to corporations due to the risk of default. This premium over the equivalent government bond yield is known as the spread.
Remember, corporate bond spreads reflect the market’s view of the creditworthness of the issuer. This opinion can change quickly, adding price volatility to this type of bond over and above that determined by interest rate fluctuations.
The yield curve. Both government and corporate bonds are issued in a variety of maturities, ranging from supershort 3month treasury bills and corporate paper through to 30 year or even undated or “perpetual” bonds with no final maturity.
Interest rates change over time, and bonds of different maturities will have different yields, reflecting the market’s expectations for future interest rates Generally, investors will require an incremental yield for longer dated securities (see illustrated chart of UK yield curve as of March 2010, right). This means that long bonds generally yield more than short bonds. This is known as a “positive yield curve” and is the usual state of play in the markets. If investors expect interest rates to rise in the future, the price of longer dated bonds will fall, pushing up yields at the long end of the curve. This is known as a “steepening” of the yield curve.
Alternatively, the perception of falling rates can lead to an “inverse” yield curve, where investors scramble to lock in fixed rates at the long end, pushing yields down below current money market rates. This is the situation reflected by the UK Gilt yield curve ilustrated (left), seen as of December 2006.