Question

1. You have a bond that pays a $3 coupon in 1 year and a $3 coupon in two years and a $3 coupon in three years. It matures in three years with principal repayment of $100 (paid at the same time as the final coupon). Suppose 1-year, 2-year, and 3-year zero coupon bond prices are 0.99, 0.97, and 0.92, respectively. Which is closest to the fair market price of the bond that you have?

$96.00

$98.00

$100.00

$102.00

$104.00

2. You would like would like to swap the uneven payments (note that the payments are $3 in year 1, $3 in year 2, and $103 in year 3) associated with the bond described in (1) for payments of equal amounts in 1 year, in 2 years, and in 3 years. Assuming you can find a swap counterparty who offers you a fair deal (and that interest rates are unchanged from those described in (1)), what is closest to the swap rate amounts will you receive in each year as a result of the swap?  

$32.00

$34.00

$36.00

$38.00

$40.00

3. The price of a 1-year zero coupon bond is 0.97. The price of a 2-year zero coupon bond is 0.93. The price of a 3-year zero coupon bond is 0.85. The price of a 4-year zero coupon bond is 0.78. You are a borrower who anticipates needing to borrow $100,000 for one year at the end of year 3 and would like to guarantee the rate on your upcoming loan (i.e., after 3 years you will need a $100,000 loan which will last for one year). Which is closest to the interest rate you will be able to guarantee if there are no transactions costs and you get a fair deal in the interest rate forward market?

1%

2.5%

4%

5.5%

7%

Answer 1

Soln : Zero coupon bond prices given as 1 year = 0.99, 2 year = 0.97 and 3 year = 0.92

Let r_i be the rate for 1, 2 and 3 year

We can say that 1/(1+r₁) = 0.99, similarly, 1/(1+r₂)² = 0.97 and 1/(1+r₃)³ = 0.92

Now, for the bond given , giving $3 in each of the year 1, 2, 3with maturity value = 100

Price of the bond = 3/(1+r₁) + 3 /(1+r₂)² + 103/(1+r₃)³ = 3*0.99 +3*0.97 + 103*0.92 = $100.64

Option C is the closest to fair market value i.e. $100

2) Let S be the swap amounts received in each of the year

We can say that S*0.99 +S*0.97 +S*0.92 = 100.64

On solving we get S = 100.64/2.88 = 34.944

Option B is the correct ans i.e. $ 34 is the closest which is received as swap amount each year.

3) With the zero coupon bond given the interest rate/spot rate for each of the year

1st year, r₁ = 1/0.97 - 1 = 3.093%

2nd year , r₂ =(1/0.93)^0.5 -1 = 3.695%

3rd year , r₃ =(1/0.85)^0.33 -1 = 5.57%

4th year , r₄ = (1/0.78)^0.75 -1 = 6.41%

Now, $100,000 to be borrowed at the end of year 3 for one year, Let the rate of the same be R

As per interest rate parity we can say that

(1+r₃)³* (1+R) = (1+r₄)⁴

or (1+5.57%)³ *(1+R) = (1+6.41%)⁴

On solving we get R = 8.97%, It is closest to the last option i.e.7%