Learn about Bonds: Calculating the yield

Calculating the yield

Investors will generally buy a bond for two reasons. The first is to lock-in a known future income stream. The second is to attempt to benefit from rising bond prices. But what would cause the value of a bond to rise? As with all traded assets, it will be down to our old friends, supply and demand. There are two main variables affecting the price of bonds, the first being interest rates and the second the perceived credit quality or risk of default for the bond. Let’s consider the effect of the former on bond prices. As interest rates fall in the money markets, a bond paying a fixed rate of interest every year will become increasingly sought after by investors. Conversely, rising interest rates in the money markets, perhaps accompanied by inflation, will make the fixed income stream unattractive to investors and the market price of the asset will fall. This relationship between price and yield is the key to understanding the factors moving the fixed income markets.  

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 life-span, 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.

Income 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.

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.

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:

£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 + (par-market 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:

Price = Coupon * 1/r [1-1/(1+r)n] + Redemption/(1+r)n

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

Note that the higher the duration of the bond the greater the price move shown per change in yield. Duration is governed by the length of time to maturity and the size of the coupon, in effect, the average period of all cash flows. A long bond with a low coupon will have the greatest duration, a short bond with a high coupon will have the lowest duration. Investors looking to benefit from falling yields should look to add duration to their bond portfolios, defensive investors, or those envisioning a rising interest rate scenario will look to reduce.

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 super-safe 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.

Non-government bond spreads vary greatly, with the highest quality banks and supranational agencies (such as the World Bank) trading at only a tiny fraction over governments while the debt of smaller, risky or unfashionable companies may trade at a level returning several precent or more over a government bond of an equivalent maturity.

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 super-short 3-month treasury bills and corporate paper through to 30 year or even undated or “perpetual” bonds with no final maturity. Gilt YC March 2010

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.

 

 

  

Please note, the content on this section of the website is provided for educational purposes. Examples shown of prices, yields and credit ratings may have changed since the time of publication.  

Stockcube Research, March 2010