Bonds and Yield Rate

Question

I came across the following problem on bonds:

Suppose we are given the following term structure of annual effective yield rates for zero coupon bonds: $(1, 2 \%)$, $(2, 6 \%)$, $(3, 7 \%)$, and $(4, 7 \%)$ where the ordered pairs are of the form $(\text{time to maturity}, \text{yield rate})$.

Find the yield to maturity for a $4$ year bond with face and redemption amount $100$ and annual coupons at rate $10 \%$.

Now the price of the bond is the present value of the coupons plus the present value of the redemption amount. This comes to be $110.60$. Using this price, how do I get the yield to maturity using the above information?

Answer 1

For the present value of the bond, I get $$\frac{10}{1.02}+\frac{10}{1.06^2}+\frac{10}{1.07^3}+\frac{110}{1.07^4},$$ which is roughly $110.7853381.$ The following keystrokes for a TI BA II Plus will give the answer that you mentioned above: Set N equal to $4,$ PV equal to $110.7853381,$ PMT equal to $-10,$ and FV equal to $-100.$ Hitting CPT I/Y gives the result.

Edited (to give more details):

We pay some amount today, and in return for that get a $\$10$ "coupon" at the end of each of the next four years, along with $\$100$ at the end of the fourth year. (The $\$10$ is $10\%$ of the face value of the bond, which is $\$100$ in this question.) The given information about interest rates says that $\$1$ today will be worth $1\cdot 1.02$ dollars in one year, $1\cdot 1.06^2$ dollars in two years, and so on. Thus dividing by $1.02,$ $1.06^2,$ and so on gives the values today of the future payments. The question is asking for the constant interest rate $i$ for which $$\frac{10}{1+i}+\frac{10}{(1+i)^2}+\frac{10}{(1+i)^3}+\frac{110}{(1+i)^4}=110.7853381,$$ and there are different ways to approximate $i$--a financial calculator such as the BA II Plus is probably the quickest.

Answer 2

Not my specialty, but here goes. The payoff of the four year bond with no interest will be worth $0.93^4=.7480$ in today's dollars. Your $10\%$ interest per year is worth $.1*.98+.1*.94^2+.1*.93^3+.1*.93^4=.3416$, so at the end you have a stream worth $1.0896$. As it takes $4$ years to get that, the yield is $1$ less than the fourth root, giving $2.17\%$ annual interest. Please forgive the use of equal signs for what should be approximately equal.