Question

Homework Problem 13_1

Q13_1 Suppose the current spot rate of a one year bond YTM₁ =5% and the current spot rate of a two year bond YTM₂ =5.5%. The bond buyer wants an option to prepay the bond next year at $952.3810 (yield of 5%). In other words, he wants the option to put or sell the bond at $952.3810. Assume the bond yield can go to 8% with probability .5 or come down to 4% with probability .5.

a) The Replicating Portfolio is a portfolio of 1-yr and 2-yr bonds that exactly replicates the payoff of the put option under all conditions, in this case the two states.

What is this portfolio of one-year bonds (number of units of par, long/short) and two-year bonds (number of units of par, long/short) that has same payoff of this Put option?

b) What is the fair value of this Put? and why?

c) What is the yield of this bond?

d) What is the OAS of the bond if the market price is $910?

Answer 1

Answer :

a) To calculate portfolio of one-year bonds (number of units of par, long/short) and two-year bonds (number of units of par, long/short) :

We assume that there is $ 100,000 funds available to invest in bonds.

calculation of portfolio of one-year bonds (number of units of par, long/short) :

YTM 1 = 5%

Coupon Rate : 5%

Initial purchase price of per Bond $ 1,000

Hence,

calculation of portfolio of one-year bonds (number of units of par, long/short) :

Year	Particulars	Amount	Discounting Factor @ 5%	Discounted Flow
1	Interest on Bond	= 1,000 x 5% = 50	0.9524	=0.9524 x 50 = 47.62
1	Investment of Bond	1,000	0.9524	=0.9524 x 1,000 = 952.40
	Total			1,000.02

Note : Value of portfolio of bond will be same as YTM and Yield of bond is same.

portfolio of one-year bonds (number of units of par, long/short) :

No. of Units = $ 100,000 / $ 1,000 = 100 bonds at 1,000 each

Put option Payoff : $ 952.3810

Value of Portfolio : 1,000 bonds x 1,000.02 = $ 100,002 /-

As put option payoff is less than present value of bond hence, It is recommended to take position for long.

calculate portfolio of two-year bonds (number of units of par, long/short) :

Year	Particulars	Amount	Discounting Factor @ 5.5%	Discounted Flow
1	Interest on Bond	= 1,000 x 5% = 50	0.9479	=0.9479 x 50 = 47.395
2	Interest on Bond	= 1,000 x 5% = 50	0.8985	=0.8985 x 50 = 44.925
2	Investment of Bond	1,000	0.8985	=0.8985 x 1,000 = 898.50
	Total			990.82

portfolio of two-year bonds (number of units of par, long/short) :

No. of Units = $ 100,000 / $ 1,000 = 100 bonds at 1,000 each

Put option Payoff : $ 952.3810

Value of Portfolio : 1,000 bonds x 990.82 = $ 99,082 /-

As put option payoff is less than present value of bond hence, It is recommended to take position for long.

b) Fair Value of this put :

Step 1 : calculation of bond Yield :

It is given that Assume the bond yield can go to 8% with probability .5 or come down to 4% with probability .5.

Bond Yield in such case : 8% x 0.5 + 4% x 0.5 = 6%

Step 2 : Calculation of Fair Value of Bond :

Year	Particulars	Amount	Discounting Factor @ 6%	Discounted Flow
1	Interest on Bond	= 1,000 x 5% = 50	0.9434	=0.9434 x 50 = 47.17
2	Interest on Bond	= 1,000 x 5% = 50	0.8899	=0.8899 x 50 = 44.495
2	Investment of Bond	1,000	0.8899	=0.8899 x 1,000 = 889.90
	Fair Value of bond			981.565

Fair Value of bond : $ 981.565

Reason : As per probability given the yield of bond is 6% and the Coupon rate is 5%. Hence, we discounted bond value with interest as expected yield is more than the bond yield.

c) Yield of a Bond :

In such Case Yield of a Bond : [ ($ 981.565 - $ 952.3810 ) / 952.3810 ] x 100 = 3.0643%

Here, we take $ 952.3810 as Bond buyer is ready to prepay bond at such price.

Alternatively, We can calculate yield at with buying price of bond:

= [ (1,000 - $ 981.565 ) / 1,000 ] x 100 =1.8435%

d) The OAS of the bond if the market price is $910

Current Market Yield : [ ( $ 1,000 - $ 910 ) / $ 1,000 ] x 100 = 9%

OAS : Option adjusted spread = 9% - 5% = 4%