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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?

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2 Answers 2

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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.

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  • $\begingroup$ 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 $.748$ $\endgroup$ Jul 15, 2011 at 4:20
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    $\begingroup$ 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. $\endgroup$
    – Wes
    Jul 15, 2011 at 5:06
  • $\begingroup$ @Wes: This is the IRR as well or the internal rate of interest. $\endgroup$
    – Damien
    Jul 15, 2011 at 15:31
  • $\begingroup$ @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 $2\%,$ $6\%,$ $7\%,$ $7\%$? $\endgroup$
    – Wes
    Jul 16, 2011 at 6:52
  • $\begingroup$ @Wes: I think it is just a typo. $\endgroup$
    – Damien
    Jul 16, 2011 at 15:10
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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.

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  • $\begingroup$ The answer says $6.88 \%$. $\endgroup$
    – Damien
    Jul 14, 2011 at 21:16
  • $\begingroup$ @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. $\endgroup$ Jul 14, 2011 at 21:21
  • $\begingroup$ The $10 \%$ is not interest....it is a coupon rate. $\endgroup$
    – Damien
    Jul 15, 2011 at 1:53
  • $\begingroup$ @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. $\endgroup$ Jul 15, 2011 at 2:00
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    $\begingroup$ 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 $\endgroup$ Jul 15, 2011 at 7:55

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