Question

8.2 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.

(20 marks total)

a. What is the yield to maturity on Bond #1?

b. What is the price of Bond #3?

c. 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 flow from these two portfolios would be identical, in amount and timing.

d. Based on the information in the given table,

i. What would it cost to buy 1000 units of Bond #3? (1 mark)

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

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? (1 mark)

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?

g. Fill in the missing information in the given table: (1 mark)

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 ($xxxx.xx)	-950.00
Time 1 cash flow	+1,000.00		+60.00	+70.00
Time 2 cash flow	0	+1,000.00	+1,060.00	+1070.00
Yield to maturity (xx.xx%			5.50%

Answer 1

Since, there are multiple parts to the question, I have answered the first four parts (Part a to Part d)

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Part a)

The yield to maturity on Bond 1 is calculated as below:

Current Price = Cash Flow Year 1/(1+Yield to Maturity)^1

Here, Current Price = $950 and Cash Flow Year 1 = $1,000

Using these values in the above formula, we get,

950 = 1000/(1+Yield to Maturity)^1

Rearranging values, we get,

Yield to Maturity on Bond 1 = (1,000/950) - 1 = 5.26%

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Part b)

The price of Bond 3 is determined as follows:

Price of Bond 3 = Cash Flow Year 1/(1+Yield to Maturity)^1 + Cash Flow Year 2/(1+Yield to Maturity)^2

Here, Cash Flow Year 1 = $60, Cash Flow Year 2 = $1,060 and Yield to Maturity = 5.50%

Using these values in the above formula, we get,

Price of Bond 3 = 60/(1+5.50%)^1 + (1,060)/(1+5.50%)^2 = $1,009.23

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Part c)

The future cash flows associated with the two portfolios are calculated as below:

Portfolio 1 - 60 Units of Bond 1 + 1,060 Units of Bond 2

Cash Flow Year 1 = Total Units of Bond 1*Cash Flow Year 1 from Bond 1 + Total Units of Bond 2*Cash Flow Year 1 from Bond 2 = 60*1,000 + 1,060*0 = $60,000

Cash Flow Year 2 = Total Units of Bond 1*Cash Flow Year 2 from Bond 1 + Total Units of Bond 2*Cash Flow Year 2 from Bond 2 = 60*0 + 1,060*1,000 = $1,060,000

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Portfolio 2 - 1,000 Units of Bond 3

Cash Flow Year 1 = Total Units of Bond 3*Cash Flow Year 1 = 1,000*60 = $60,000

Cash Flow Year 2 = Total Units of Bond 3*Cash Flow Year 2 = 1,000*1,060 = $1,060,000

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As can be seen from the above calculations that the the future cash flow from these two portfolios would be identical, in amount and timing. A cash flow of $60,000 in Year 1 and cash flow of $1,060,000 in Year 2 would occur with both the portfolios.

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Part d)

i)

The total cost to buy 1000 units of Bond 3 is calculated as follows:

Total Cost to Buy 1,000 Units of Bond 3 = Total Units of Bond 3*Price of Bond 3 = 1,000*1,009.23 = $1,009,231.60

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ii)

The total cost to buy 60 units of Bond 1 is determined as below:

Total Cost to Buy 60 Units of Bond 1 = Total Units of Bond 1*Price of Bond 1 = 60*950 = $57,000

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iii)

The value of $1,060 units of Bond 2 is inferred as below:

Value of 60 Units of Bond 1 + Value of $1,060 Units of Bond 2 = Value of 1,000 Units of Bond 3 [using law of one price]

Substituting values in the above equation, we get,

57,000 + Value of $1,060 Units of Bond 2 = 1,009,231.60

Rearranging Values, we get,

Value of $1,060 Units of Bond 2 = 1,009,231.60 - 57,000 = $952,231.60

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iv)

The value of one unit of Bond 2 is determined as follows:

Value of One Unit of Bond 2 = Value of $1,060 Units of Bond 2/1,060 Units = 952,231.60/1,060 = $898.33

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v)

The implied yield of Bond 2 is arrived as below:

Implied Yield of Bond 2 = Yield to Maturity of Bond 3 = 5.50%

The same can also be calculated as follows:

Value of One Unit of Bond 2 = Cash Flow Year 0/(1+Yield)^1 + Cash Flow Year 1/(1+Yield)^2

898.33 = 0/(1+Yield)^1 + 1,000/(1+Yield)^2

Rearranging values, we get,

Yield = (1,000/898.33)^(1/2) - 1 = 5.51% which is close to 5.50%.