# Find yield-to-maturity of a bond

Given: A one-year zero-coupon bond has an annual yield of $$6.25\%$$. A two-year zero-coupon bond has an annual yield of $$7.00\%$$. A three-year zero-coupon bond has an annual yield of $$7.50\%$$. A three- year $$12\%$$ annual coupon bond has a face value of $$1,000.$$

Find: Yield-to-maturity of this three-year $$12\%$$ annual coupon bond.

My solution: $$PV = \cfrac{120}{1.0625}+\cfrac{120}{1.07^2}+\cfrac{1120}{1.075^3}=1119.31,$$ $$\ \ \ \ AV(t=3)=3*120+1000=1360.$$

$$1119.31(1+i)^3=1360 \implies i=6.71\%$$

Correct solution: $$PV = \cfrac{120}{1.0625}+\cfrac{120}{1.07^2}+\cfrac{1120}{1.075^3}=1119.31$$

$$1119.31 = 120\require{enclose}a_{\enclose{actuarial}{3}i}+1000v_i^3 \implies i = 7.42\%.$$

The disparity between my solution and my textbook's shows me I am really missing some intuition on how bonds work. If you can provide any insights that would be great. Thanks.

• If $AV$ stands for "accumulated value", then you're computing it incorrectly. You are just adding up the cash flows, but you have to take interest into account. We don't know the yield rate yet, so we're not in a position to compute the accumulated value. Mar 25 '20 at 22:51
• The problem doesn't make much sense to me. The yield is the interest rate that makes the present value of the cash flows equal to the price. This calculation makes sense to me only if the holder of the $12\%$ bond is willing to accept the three zero-coupons bonds in payment, which is not stated anywhere. Is there something more to the problem statement than what you've stated? Mar 25 '20 at 23:00
• @saulspatz Thanks for commenting. Yes AV stands for accumulated value. The problem reads: "A one-year zero-coupon bond has an annual yield of 6.25%. A two-year zero-coupon bonds has an annual yield of 7.00%. A three-year zero-coupon bond has an annual yield of 7.50%. A three-year 12% annual coupon bond has a face value of \$1,000. Find the yield to maturity on this three-year 12% annual coupon bond." Your first comment makes me think I'm misunderstanding bond payments. If I'm paid 120 thrice and 1000 once, then I end up with 1360, right? Why would I get interest after receiving the payments? Mar 26 '20 at 1:49
• You don't get interest after receiving the payments, but to compare the values of two streams of payments, you have take interest into account, whether the payments are in the future or the past. "Accumulated value" is just the present value, computed as of some future date. Mar 26 '20 at 2:11
• @saulspatz I thought that's what I did in the last step of my solution? At t=3, I have 1360. Mar 26 '20 at 2:21

You haven't taken into account when then payments are received. You are accumulating the $$1191.31$$ with interest for $$3$$ years, but you aren't adding interest to the bond payments at all. You would need to say, $$1191.31(i+1)^3=120(1+i)^2+120(1+i)+1120$$
You say in a comment that this seems to imply that you receive interest on the bond payments after receiving them. In a way it does, but not from the bond issuer. It assumes that you reinvest, the bond proceeds at the yield rate. Just as $$\120$$ a year in the future is worth less than $$\120$$, $$\120$$ a year in the past is worth more than $$\120$$.