Price of a $10$-year $8.6\%$ annual coupon bond?

Question

How do you find the price of a 10-year 8.6% annual coupon bond with a 7% yield? Par is 100.

My thought was to calculate the present value for it:

$$100\cdot(1.086)^{10}+\frac{100}{(1.06)^{10}} = 279.03$$ However the answer is apprently $111.2$.

Answer 1

Because you did not show any of your work, I suspect you are not sure how to compute the present value of a sum of money you'll receive in the future.

Suppose the annual yield is $y$. If you now place $100$ in the bank and receive yield $y$, then in one year you have $100(1 + y)$. If you leave this amount of money on the bank, it grows to $100(1 + y)(1 + y) = 100(1+y)^2$ in two years. After $n$ years, you have $100(1 + y)^n$.

Consequently, if you now place $\frac{100}{(1 + y)^n}$ on the bank, it grows to $100$ in $n$ years. The amount $\frac{100}{(1 + y)^n}$ is the present value of $100$ received in $n$ years.

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Using your numbers, you receive $8.6$ at the end of years $1,2, \ldots, 9, 10$. Moreover, you also receive $100$ at the end of year $10$. You need to compute the sum of the present values of these amounts.

If things are still not clear, let me know, I'll expand the answer.

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An annual coupon bond does the following: untill maturity (i.e. the number of years the bond is valid), you receive coupons. In your case, you receive $8.6%$ of the par-value in every year. Hence, you receive $8.6$ in year 1, 8.6 in year 2,... and finally 8.6 in year 10. However, bonds are essentially loans, so you also receive back the principal which you lend. In your case this is 100. It is given that the yield is $7%$.

The present value of the 8.6 you receive at the end of year 1 equals $$\frac{8.6}{(1 + 0.07)}.$$ The present value of the $8.6$ you receive at the end of year 2 equals $$\frac{8.6}{(1 + 0.07)^2}.$$ You do this for every year. The only 'special' case, is the end of year 10: you receive a coupon (8.6) and the principal (100). In total, you receive 108.6 at the end of year 10. The present value equals $$\frac{108.6}{(1+0.07)^{10}}.$$

The only part left for you, is to sum all these terms (note that I left out the present values of the coupons received at the end of year 3 through 9. I leave it to you to compute those.)

at the end, you should obtain 111.2377 according to my computations in excel.