The Solver function in Excel can also be used. There are advantages and disadvantages to using the IRR. The first is that we are solving for the interest rate rather than plugging one in.
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Second, it is a widely used measure, i. Now we will illustrate some of the problems in using the IRR. Suppose we have 2 bonds: Bond A and Bond B. Assume that they compound annually, rather than semiannually. Furthermore, they have the same investment horizon of 3 years.
It appears as if Bond A is better -- having a higher yield. But this is not necessarily the case. Suppose the term structure was not flat. In particular, suppose we are facing the following term structure. The one period forward interest rates are:. Forward rates are discussed in more detail later. It is clear from this example that Bond B is a superior investment to Bond A. If we vary the shape of the term structure, then the IRR rule will not always work. We have shown that the price of the bond is sensitive to the interest rate.
Another factor that has to be taken into account when ranking bonds is the timing of the cash flows. If Bond B's cash flows are concentrated in the far future, then its price will be very sensitive to changes in interest rates. Conversely, if Bond A's cash flows are concentrated in the near future, it will not be as sensitive to changes in the interest rate. So the time path of cash flows is very important. Graphically, the present values of Bond A and Bond B appear below. We have calculated what happens to bond price when there is an interest rate change. Consider the same example that we previously pursued:.
The holding period return also decreases.
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If this was a six year bond bought at par and held for one year, then the return on holding the bond is 8. You can imagine that bigger swings in the rates could cause negative holding period returns.
In this case, the capital loss exceeds the gain from collecting the coupon. The longer the maturity of the bond the more severe the price changes when the yield changes.
A measure of the sensitivity of the bond price movement relative to changes in interest rates is valuable. We will consider two measures: duration and elasticity. Both of these measures will give us local approximations, i. The logical way to measure the sensitivity of the bond price to changes in the interest rate is to take the first derivative of B with respect to r.
We can write the bond price formula:. If we adjust this measure by dividing by minus the bond price and the number of periods per year, and multiplying by one plus the market yield, we get a measure of duration first introduced by Macaulay in Duration is often called Macaulay Duration. Duration was invented as an alternate measure of the timing of the cash flows from bonds. The pitfall in using the maturity of a bond as a measure of timing is that it only takes into consideration the final payment of the principal -- not the coupon payments.
Macaulay suggested using the duration as an alternative measure that could account for all of the expected cash flows. Duration is a weighted average term to maturity where the cash flows are in terms of their present value. We can rewrite the above equation in a simpler format:.
Now lets consider examples of duration calculations. We will calculate the duration of Bond A and Bond B. Both bonds have a maturity of 10 years. Before we calculate the duration measure, we know that Bond B will have a shorter duration. The cash flows from 1 to 9 years are larger yet the principal is identical.
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Now lets work it out. As expected, the bond with the higher coupon rate has a shorter duration. This example illustrates two important properties of duration. First, the duration of a bond is less than its time to maturity except for zero coupons. Second, the duration of the bond decreases the greater the coupon rate.
This can be graphically illustrated:. Notice that the duration and maturity are identical for the zero coupon and the duration decreases with higher coupon rates. This is because more weight Present Value Weight is being given to the coupon payments. The final property is that, as market yields increase, the duration of the bond decreases. This should be intuitively obvious because when we are discounting cash flows a higher discount rate means a lower weight on cash flows in the far future.
Hence, the weighitng of the cash flows will be more heavily placed on the early cash flows -- decreasing the duration. The link between duration and volatility of prices is clear because of the first derivative that we used in obtaining the duration formula. There are two alternative measures that are worth investigating.
The first is called modified duration.
This is derived by dividing the duration measure by one plus the current market yield. The elasticity measure will be very close in practice to the modified duration measure.
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The closeness is dependent upon the size of the movement in interest rates. Hence, we see the link between duration, modified duration, and elasticity. Now lets consider another example. We will look at bonds of different maturity. For ease of calculation, we assume annual compounding. Note that the price elasticity measure is very close to the modified duration measure. Note also that the closeness of this measure depends upon the size of the interest rate move.
We would like to use the modified duration to approximate bond price movements for a given change in interest rates. The approximation will only be accurate for small changes in the interest rate. This is because the bond price is convex in the yields. We have seen this convexity when we looked at the plot of the bond price against various yields to maturity. Below is a figure that illustrates the error. Also, if stated interest rates are compounded semi-annually, then the same procedure is used to calculate duration in half years.
The duration in half years is converted to years by dividing by 2 and then transformed into modified duration by dividing by the effective periodic semi-annual rate. One might wonder why we divide by the semi-annual rate rather than 1 the annual percentage rate or 2 the true annualized rate. The answer, unfortunately, has to do with convention. Most top investment houses calculate the modified duration by dividing by the semi-annual rate - even if coupons are paid quarterly or monthly. In most of my examples, I have annual compounding and don't to worry about this convention.
Below is a numerical example using annual rates.
Related the yield to maturity on one-year zero-coupon bonds is 8
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