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 effect on the volatility of the bond over a range of different interest rate scenarios. Let’s take three UK gilts as an example (calculated in March 2010).
The following table shows three bonds of different maturities. Note how the longer-dated bonds have longer duration. As mentioned in the paragraph above, duration is a measure of a bond’s price volatility over a shift in yield. The table shows how these bond prices move over yield shifts between 2% and 3%.
Table 5.1: Duration example
Note that the higher the duration of the bond the greater the price move shown per change in yield.
Duration, which is expressed in units of years, is determined 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 duration.
Tip
A zero coupon bond will have a duration equivalent to its maturity.
Convexity
Duration is not set in stone. Obviously, it will shorten with the bond’s life, but a drop in price will also reduce the duration.
Why?
Because as the price falls, the fixed coupons are now a greater in proportion to the purchase price, thus shrinking the average life. The relationship between price, yield and the duration of a bond can be plotted on a chart and is known as convexity, due to the shape of the resulting curve (see illustration, below).
Figure 5.1: convexity
The subject of convexity is also applicable to bond portfolios, and is of some importance to the institutional fund manager who wishes to model how their portfolio might behave in different interest rate scenarios. For the purposes of the private investor, the subject is of rather less importance.
