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I am looking at an example problem in my textbook and its solution. Can someone look at this picture/ problem and its solution and tell me where they got the yield to maturity.

enter image description here

Ayoshna
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1 Answers1

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The yield maturity is the constant discount rate that makes the present value of the cash flows equal to the bond price.

Thus

$$114.49 = 100(1+y)^{-4} + 10\sum_{k=1}^{4} (1+y)^{-k} = 10\frac{(1+y)^{-1}[1-(1+y)^{-4}]}{1 - (1+y)^{-1}}+100(1+y)^{-4}\\=10\frac{1-(1+y)^{-4}}{y}+100(1+y)^{-4}$$

Solving numerically (using for example Newton's method) we find $y= 0.05834= 5.834\, \%.%$

RRL
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  • are you summing up just $(1+y)^{-k}$ for k from 1 to 4 or the entire equation $(1+y)^{-k} + 100(1+y)^{-4}$? – Ayoshna Nov 10 '14 at 08:27
  • @Ayoshna: Just $(1+y)^{-k}$. Those are the discount factors for the coupons. Clarified above. – RRL Nov 10 '14 at 08:31
  • Okay, I don't understand what's going on your summation. What happened to when k=2 and 3. How did you get $(1+y)^{-1}[1-(1+y)^{-4}]/ 1-(1+y)^{-4}$ – Ayoshna Nov 10 '14 at 08:47
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    Im summing a finite geometric series. Let $S = a + a^2 + a^3 + a^4$. Then $S - aS = a - a^5$ and $S = a(1-a^4)/(1-a)$ – RRL Nov 10 '14 at 08:52
  • Okay, I'm trying to follow this and do it for a homework problem however I keep getting the incorrect value. Is there an easier way to do this or on the BA II plus calculator? – Ayoshna Nov 10 '14 at 09:24
  • If you substitute $y = 0.05834$ into the formula you will get 114.49 -- did you check that? This is the standard formula for yield to maturity and its a nonlinear equation with no closed form solution - so solving requires iteration -- or a calculator programmed for YTM. I dont have a calculator. – RRL Nov 10 '14 at 09:30
  • Try bisection; guess $y0$ and $y1$ with $f(y0) < P$ and $f(y1) > P$ then set $y2 = (y0 + y1)/2$. Repeat with $y2$ and whichever of $y1$ or $y0$ brackets the solution. Repeat a few times. – RRL Nov 10 '14 at 09:32