Which of the following bonds would be cheapest to deliver given a T-note futures

Question

Which of the following bonds would be cheapest to deliver given a T-note futures price of 120.6773? (Assume that all bonds have semiannual coupon payments based on a par value of $100.)

9.5-year bond with 5% coupons and a yield of 3.5%

6.5-year bond with 3.5% coupons and a yield of 2.5%

9.5-year bond with 5% coupons and a yield of 3.5%

9-year bond with 3% coupons and a yield of 1.5%

Please explain the steps in details. No excel work without explanation

Which of the following bonds would be cheapest to deliver given a T-note futures price of 120.6773? (Assume that all bonds have semiannual coupon payments based on a par value of $100.)

Selected Answer: b.

9.5-year bond with 5% coupons and a yield of 3.5%

Answers: a.

6.5-year bond with 3.5% coupons and a yield of 2.5%

b.

9.5-year bond with 5% coupons and a yield of 3.5%

c.

9-year bond with 3% coupons and a yield of 1.5%

Please explain the steps in details. No excel work without explanation

Explanation / Answer

First let’s calculate the bond price with the help of following formula

Bond price P0 = C* [1- 1/ (1+i) ^n] /i + M / (1+i) ^n

Market price of the bond, P0 =?

C = coupon payment or annual interest payment = 3.5% per annum but it makes coupon payments on semiannual basis therefore coupon payment = 3.5%/2 of $100 = $1.75

n = number of payments or time remaining for the maturity of bond = 13 (6.5*2 for semiannual payments)

i = yield to maturity (YTM) = 2.5% per annum or 1.25% per semiannual

M = value at maturity, or par value = $ 100

Therefore,

Price of bond P0 = $1.75 * [1 – 1 / (1+1.25%) ^13] /1.25% + $100 / (1+1.25%) ^13

= $20.88 + $85.09

= $105.9651

C = coupon payment or annual interest payment = 5% per annum but it makes coupon payments on semiannual basis therefore coupon payment = 5%/2 of $100 = $2.50

n = number of payments or time remaining for the maturity of bond = 19 (9.5*2 for semiannual payments)

i = yield to maturity (YTM) = 3.5% per annum or 1.75% per semiannual

M = value at maturity, or par value = $ 100

Therefore,

Price of bond P0 = $2.50 * [1 – 1 / (1+1.75%) ^19] /1.75% + $100 / (1+1.75%) ^19

= $40.12 + $71.92

= $112.0345

C = coupon payment or annual interest payment = 3% per annum but it makes coupon payments on semiannual basis therefore coupon payment = 3%/2 of $100 = $1.50

n = number of payments or time remaining for the maturity of bond = 18 (9*2 for semiannual payments)

i = yield to maturity (YTM) = 1.5% per annum or 0.75% per semiannual

M = value at maturity, or par value = $ 100

Therefore,

Price of bond P0 = $1.50 * [1 – 1 / (1+0.75%) ^18] /0.75% + $100 / (1+0.75%) ^18

= $25.17 + $87.42

= $112.5844

The bonds would be cheapest to deliver given a T-note futures price of $120.6773

Cheapest to deliver bond is with minimum difference between T-note futures price and bond’s price

Therefore,

$120.6773 - $112.0345 = $8.6428

$120.6773 - $105.9651 = $14.7122

$120.6773 - $112.5844 = $8.0929

Therefore cheapest to deliver bond the bond in option c. with bond’s price of $112.5844

Therefore correct answer is option c. 9-year bond with 3% coupons and a yield of 1.5%

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