Find the price of the bond using its book value

Question

A n year 1000 par-value bond with 8% annual coupons has an annual effective yieled of i, 1+i >0 . The book value of the bond at the end of the third year is 990.92 and the book value of the bond at the end of the fifth year is 995.10. Find the price of the bond

My thoughts:

My initial instinct was to divide the two book values to find v^n. However, this does not seem to work. The main reason I cannot solve for the price is because the values of n and the yield rate are missing. Substituting into the formula for the book values also does not seem to help. Any thoughts?

Answer 1

You cannot "analytically" find the yield of a bond given the price. You have to use numerical methods... which means guess and check, and adjust your guess.

The bond is marked at as discount. The yield is greater than 8%. The amount of the discount decays by about 5 points over the 3 years between 2 years from now and 5 years from now. And will still be at a 5 point discount. While this decay is not exactly linear, it is close enough.

So we make a guess that the bond will have 3 years of life, 5 years from now. And 6 years of life 2 years from now. Then we attempt to price the bond. See if we can find a yield that give us something close to the correct book values, and if necessary adjust the times to maturity from there.

I get $988.6

Answer 2

Let be the book value of the bond at the end of the third year $B_3=990.92$ and the book value of the bond at the end of the fifth year $B_5=995.10$.

We have $$ BV_5=B_3(1+i)^2-80s_{\overline{2}|i} =B_3(1+i)^2-80 (1+i)-80 $$ that is $$ 990.92 (i+1)^2-80 (i+1)-1075.1 = 0 $$ and solving the quadratic equation in $1+i$ we find the values $1+i=-1.00203$ and $1+i=1.08276$ and from the condition $1+i>0$ we choose $$i=1.08276-1=0.08276$$

So the price is $$ P=80a_{\overline{3}|i}+B_3 v^3\approx 985.77 $$