Internal Rate

At this point, we come against a complexity when calculating the yield to maturity of coupon paying bonds because of the multiple cashflows the bond pays. The complexity lies on the fact that interest rates are quoted on an annual basis whereas the yield to maturity is the Internal Rate of Return per cashflow period. According to the market convention, the results must be doubled and the IRR per cashflow period must be annualised to give the Effective Yield to Maturity. The yield to maturity computed on the basis of this market convention is called the Bond-equivalent yield.

Let us see now how the yield to maturity is related with the price of bonds. Conventionally, there is a negative relationship between the bond price and interest rates. The price of a bond falls when interest rates rise and conversely the price of a bond goes up when there is a decrease of interest rates. Now, if the yield to maturity is increasing then the price of the bond will be decreasing and vice-versa. How the yield to maturity affects the price of a bond depends on the yield level, the coupon rate and the time to maturity. Supposedly, we have one zero-coupon bond and two coupon-paying bonds.

The price level of these bonds will be determined by the slope of the term structure of interest rates. If the only difference between bonds is their maturity date, then the present value of the bond with the shorter time to maturity reacts less strongly to identical changes in the yield to maturity. Accordingly, the bond with the larger time to maturity fluctuates more throughout its life, indicating greater price risk. At the point where the price functions intersect, the coupon amount and the yield to maturity are identical for each bond.

YTM theory is based on three major assumptions, which present its weaknesses as a yield measure as it is unlikely that all assumptions are fulfilled in reality. The investor will realize the yield to maturity at the time of purchase only if the bond is held until the maturity date and the coupon payments can be reinvested at the computed yield to maturity. The risk the investor faces is that the future re-investment rates will be less than the yield to maturity at the time the bond is purchased.

This risk is referred to as reinvestment risk. The maturity and coupon payments of a bond are the two main characteristics that determine the importance of the interest-on-interest component and therefore the degree of reinvestment risk. For a given yield to maturity and a given coupon rate, the longer the maturity, the more dependent the bond’s total return is on the interest-on-interest component in order to realize the yield to maturity at the time of purchase. “In other words, the longer the maturity, the greater the re-investment risk.

The implication is that the yield-to-maturity measure for long-term coupon bonds tells little about the potential yield that an investor may realize if the bond is held to maturity”. 2 Turning to the coupon rate, for a given maturity and a given yield to maturity, the higher the coupon rate, the more dependent the bond’s return will be on the re-investment of the coupon payments in order to produce the yield to maturity anticipated at the time of purchase. When maturity and yield to maturity remain constant, premium bonds are more dependent on the interest-on-interest components than are bonds selling at par.

For a zero- coupon bond that is not the case as the yield earned on a zero-coupon bond held to maturity is equal to the promised yield to maturity. It seems here that the major drawback of the yield to maturity measure is its deficiency in assuming that all coupon interest can be reinvested at the yield to maturity. Furthermore, YTM is influenced by the size of the cashflows. In particular, with a positively sloped term structure of interest rates, high coupon bonds have a lower yield to maturity than an otherwise identical low coupon bond.

On the other hand, when a negatively sloped term structure exists, a higher coupon bond has a higher yield to maturity. In addition, when the term structure of interest rates is upward sloping, higher (lower) coupon bonds have a lower (higher) yield to maturity. Conversely, with a downward sloping term structure of interest rates, higher (lower) coupon bonds have a lower (higher) yield to maturity. The above critique on the assumptions of the yield to maturity indicates conceptual weaknesses as a summary measure of a bond’s relative attractiveness.

All assumptions are questionable in practice as most investors very rarely hold their bond positions until maturity, re-investment of the future coupon payments is based on the then prevailing market interest rates and not on the current yield to maturity. Finally, the assumption of flat term structure of nominal rates seems rather unrealistic. Due to the failure of the assumptions and the Coupon effect, we must conclude that YTM is not an optimal measure to evaluate bonds. As mentioned earlier, there are several bond yield measures, apart from the yield to maturity measure, quoted by dealers and used by portfolio managers.

All of them, however, fail to take things further and have shortcomings similar to those of the yield to maturity measure. REALISED COMPOUND YIELD The most sophisticated method of measuring yield is the Realised Compound Yield measure. Realised Compound Yield is the compound rate of growth in total value during the holding period expressed as an annualized rate of interest. It takes account for the investor’s desired holding period, the slope of the term structure of interest rates and the assumptions the investors make about the interest rates at which the coupons can be reinvested.

It is the most appropriate measure for comparing alternative bonds if the investors wish to hold a bond for less than the term to maturity. It can also be used if a bond is to be held until maturity date. The realized compound yield includes the influence of interest-on-interest in the terminal value of the investment. The actual yield calculated crucially depends upon the assumed reinvestment rate. It assumes futures rates of reinvestment and, provided that expected reinvestment rates are consistent between instruments, meaningful comparisons can be made.

Further assumptions involve the level of bond yields at the time of sale and reinvestment of redemption proceeds, depending on the life of the bond and how it lies in accordance with the desired holding period. To calculate the Realised Compound yield we derive the total coupon payments and interest upon reinvestment. The method uses the forward yield curve to reflect expectations on future interest rates. The price of the bond and whether it is equal to the redemption price lies upon the holding period and whether that is shorter or longer than the life of the bond.

The final step to the realized compound yield calculation is the derivation of the terminal value of the investment. The formula used for this calculation is: With TV being the terminal value of the investment, P the original price of the bond at which it was purchased and k the number of coupon paying periods. The following example shows the importance of forward rates. It considers two 4-year bonds. One pays 8% pa, is valued at 65. 32 with a yield to maturity of 24. 5%, and the other pays 4% pa, and is valued at 56. 19 with a yield to maturity of 22. 7%.