Question

The Law of One Price implies that financial instruments with the same risk and the same cash flows at the same time should have the same price.

You are given the following table containing incomplete information on four different bonds. Assume that all these bonds have the same risk, and any coupon payments are paid annually. Once calculations are done please fill the table

Bond #	1	2	3	4
	1 year strip bond	2 year strip bond	2 year 6% coupon bond	2 year 7% coupon bond
Purchase price	-950.00
Time 1 cash flow	+1000.00	0	+60.00	+70.00
Time 2 cash flow	0	$1000.00	+1060.00	+1070.00
Yield to maturity			5.50%

1. What is the yield to maturity on Bond #1?                                              
2. What is the price of Bond #3?
3. You are considering two investments from the bonds listed in the table.    Portfolio 1: 60 units of Bond #1 + 1060 units of Bond #2 . Portfolio 2: 1000 units of Bond #3. Show that the future cash flows from these two portfolios would be identical, in amount and timing.
4. Based on the information in the given table,       

i.    What would it cost to buy 1000 units of Bond #3?                             

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

iii. From part c. above and your answers in part d.i and ii, infer the value of 1060 units of Bond #2.                                                                              

iv. What is the value of one unit of Bond #2?                                         

v. What is the implied yield of Bond #2?

e. How many units of Bond #1 and #2 would you need to replicate the future cash flows of 1000 units of Bond #4?

f. using your answer to part e above, determine the following                       

i.    What’s the value of 1000 units of Bond #4?                                      

ii.    What’s the yield of Bond 4?

Answer 1

a) Bond 1

Yield is the expected return from an asset. We need to find the rate which discounts 1000 Future Year1 Cash flow to PV of 950

i.e 1000 = 950 (1+R)^1

R = 5.26%

b) We disount the cash flows of Bond B to find the price of bond 3

DF for each year = 1/ (1+ R)^N

Year	Cash Flows	DF	PV
1	60	0.947867299	56.87203791
2	1060	0.898452416	952.3595607
			1009.231599

Value of Bond B = 1009.23

c)Portfolio 1 = 60 units bond 1 + 1060 units of Bond 2

Portfolio 2 = 1000 units of Bond 3

Hence we compare the cash flows by multiplying each cash flow of respective bond with number of units

For eg for Bond 1 , for Year 1 = 1000* 60 units

For eg for Bond 3 , for Year 1 = 60* 1000 units

Year	Bond 1	Bond 2	Total Portfolio 1	Bond 3 / Portfolio 2
1	60000	0	60000	60000
2	0	1060000	1060000	1060000

Hence we observe that, cash flows for year 1 is 60,000 and Year 2 is 1060,000 for each portfolio

d)

1) Cost to buy 1000 units bond 3 = 1009.23*1000 = 1009230 (Price calculated in part b solution)

2) Cost to buy 60 units of Bond 1 = 60* 950 = 57,000

3) We know that, The Law of One Price implies that financial instruments with the same risk and the same cash flows at the same time should have the same price. Hence cost of portfolio 1 should be equal to portfolio 2

i.e  60 units of Bond #1 + 1060 units of Bond #2 = 1000 units of Bond #3

57000 + 1060 * Price Bond 2 = 1009230

Price Bond 2 = 898.33

Value of1060 units Bond 2 = 952230

4) Price Bond B = 898.33

5) To replicate Bond 4 using bond 1 and 2 , we shall have following strategy so that cash flows are same for 2 years

Portfolio 1 = 70 units Bond1 and 1070 units of Bond 2

Portfolio 2 = 1000 units of Bond 4

e) 1000 units of Bond 4 = 70 units bond 1 + 1070 units Bond 2

= 70 * 950 + 1070* 898.33

= 1027713

Price Per Bond 4 = 1027.70