## (solution) Hello there, I am unsure where the assumption of 1000 units came

Hello there,

I am unsure where the assumption of 1000 units came from for this answer for 8.2 and 8.3

Thanks, Crystal

1. Fill in the missing items in the following table, using the Law of One Price. Assume all

these bonds have the same risk, the yield curve is flat, and any coupon payments are

paid annually.

Bond # 1

1-year

strip bond

?950 Time 0 cash flow

(i.e., Purchase

Price for the bond)

Time 1 cash flow

+1000

Time 2 cash flow

0

Yield

5.26% 2

2-year

strip bond

898.46 3

2-year 6%

coupon bond

1009.38 4

2-year 7%

coupon bond

1027.85 0

+1000

5.50% +60

+1060

5.50% +70

+1070

5.50% Before we start, we will try to fill in all the question marks.

From Bond 1, we can find the interest rate in year 1 (r 1)

1000/ (1+r1) = 950, this implies that the interest rate in year 1 is 5.26%

Yield is the same as the interest rate in the one year case.

From bond 3, we see that the yield is 5.5%, recall that yield is always expressed as an

annualized term, that means if you hold bond 3 for 2 years, your total yield would be

1.052 = 1.113. You will receive an 11.3% return for two years in total.

Now, suppose that bond 3 is worth P30 at time 0; I can create another bond that does not

pay anything at time 1, and pays 1060 at time 2. This bond should worth P 30 ?

60/1.0526 = P30 ? 57. Since this bond is identical to bond 3, we also know that its yield

is 5.5%. So if I buy the bond for P30 ? 57, hold it for two year and sell for 1060, I will

have gained 11.3% during this two year period, thus 1.113(P 30 ? 57) = 1060. This leads

to P30 = 1009.38

With this, we can now calculate r2

60/ r1 + 1060/ r1r2 = 1009.38, since r1 is known, we solve for r2 and get r2 = 5.74%. r2

here is the interest rate from year 1 to year 2 (aka one year forward rate from year 1)

Now, we can calculate the price of bond 2

P2 = 1000/r1r2 = 898.46

Yield = (r1r2)1/2 = 1.055 Lastly, for bond 4, P4 = 70/r1 + 1070/r1r2 = 1027.85

Yield = (r1r2)1/2 = 1.055

(Remark, you may calculate the bond prices using yield to maturity without using

forward rates, and you should get the exact same answers. Try this way yourself as an

exercise)

A. You are considering two investments from the bonds listed in the question above.

Show that the cash flows from the following two investments would be identical.

i.60 units of Bond #1 + 1060 units of Bond #2, and

ii.1000 units of Bond #3.

Assumption: I assume 1000 units of bond is one bond (i.e. 1000 units of bond 1 would

be worth \$950). The answer is the same re

For i) In year 1, you get \$60 (=\$1 per unit X 60 units) from bond 1, and 0 from bond 2. In

year 2, \$1,060 (=\$1 per unit X 1060 units) from bond 2.

For ii) In year one, you get \$60 and year 2 you get \$1060. Exactly the same.

B. How many units of Bond #1 and #2 would you need to replicate the cash flows of

1000 units of Bond #4?

To replicate bond 4, you want \$70 in the first year and \$1070 in the second year. Buying

70 units of bond 1 gives you \$70 in the first year and buying 1070 units of bond 2 gives

\$1070 in year 2.

C.

i. If the yield of Bond #3 is 5.5%, what would it cost to buy 1000 units of

Bond #3? 1009.38, solution is in the very beginning

ii. What would it cost to buy 60 units of Bond #1? \$57

iii. From part A. above, infer the value of 1060 units of Bond #2. 952.37

iv. What is the value of one unit of Bond #2? \$0.898 Yield of Bond #2? 5.5%

D. What's the value of 1000 units of Bond #4? \$1027.85 Yield? 5.5%

E. What have you learned about the Law of One Price from questions above? Law of one price tells you that all identical bonds must have the same price. This may

be rephrased slightly different, and we can say that given bonds that mature in the same

time (bonds themselves may differ in price and/or coupon payments), their yields to

maturity must be the same. If not, there exists arbitrage opportunities as you can short

the one with lower yield and buy the one with higher yield. As evident from part 1, bond

2, 3 and 4 have different prices and different coupon payments, but because they

mature in the same time, their YTM must be the same.

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