# Bonds and Yield Rate

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?

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.
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. • Interesting that you divide by$1.02$for a negative yield of$2\%$. It would seem that you should multiply by$0.98$. If you have a$-50\%$yield, wouldn't you expect to have half your money, not two thirds? For small percentages, the difference is very small, but four years out at$7\%$the difference is$.762$vs$.748$Jul 15, 2011 at 4:20 • If the interest rate is$i$, then a unit of currency invested today will be worth$(1+i)^t$at time$t.$When the unit of time is one period (whatever that is), various functions of$i$are useful enough to have standard notation. These are:$v:=\frac{1}{1+i}$(which is the value that must be deposited today to have$1$in one period) and$d:=\frac{i}{1+i}$(which is what must be deposited today to have the interest gained on$1$in one period). We have$v+d=1,$but$1-i$is hard to interpret. – Wes Jul 15, 2011 at 5:06 • @Wes: This is the IRR as well or the internal rate of interest. Jul 15, 2011 at 15:31 • @Damien: Yes, that is another name for it. By the way, I noticed that you said above that the price of the bond is$110.60,$but my value is higher. How did you get the$110.60$?I think that I need to fix something in my answer, but I cannot spot any errors. Are you sure that the interest rates are$2\%,6\%,7\%,7\%$? – Wes Jul 16, 2011 at 6:52 • @Wes: I think it is just a typo. Jul 16, 2011 at 15:10 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. • The answer says$6.88 \%$. Jul 14, 2011 at 21:16 • @Damien: then I am not understanding how the data is used. I get pretty close if I take the reduction in value after$4$years to be only$0.93$, not$0.93^4$and similarly for the other terms. Then I get$6.94\%$. But when I see annual yield for$4$years of$-7\%$I think it is telling me I lose that much each year. Jul 14, 2011 at 21:21 • The$10 \%$is not interest....it is a coupon rate. Jul 15, 2011 at 1:53 • @Damien: But I took credit for 10% paid annually at the end, which is how I think a coupon works. How is it different? I didn't compound it. Jul 15, 2011 at 2:00 • Ross, I think you're misunderstanding. If you receive$C$in$n$years time at an annually compounded effective interest rate of$r$, then the present value is what you'd need to invest now at rate$r$to receive$C$after$n$years, i.e.$P(1+r)^n = C$, so$P = C/(1+r)^n$. You also receive the face value of the bond (100) as well as the coupon (10) in the final year. So the equation to solve is$\sum_{i=1}^4 10/(1+x)^i + 100/(1+x)^4 = 110.785$, and the solution is$x=6.828\%\$, as found by Wolfram Alpha: is.gd/9rBczE Jul 15, 2011 at 7:55