Question

A 30 year corporate bond with a face amount of $1,000 and a 5.6% coupon (paid semi-annually) was issued 15 years ago. Given a 30 BPS, what is its effective duration? Assume the current yield to maturity is 6.0%. Show all work.

Answer 1

The formula for effective duration is

Effective duration = (P1 – P2) / (2 * P0 * Y)

Where,

P0 is the original price of the bond

P1 is the price of the bond if the yield were to decrease by Y percent

P2 is the price of the bond if the yield were to increase by Y percent

Y is the estimated change in yield used to calculate P1 and P2 = 30 BPS

The Bond’s price can be calculated the help of following formula

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

Where,

The par value or face value of the Bond = $1,000

Current price of the bond P0 =?

C = coupon payment or annual interest payment = 5.6% per annum, but it makes coupon payments on a semi-annual basis therefore coupon payment = 5.6%/2 of $1,000 = $28

n = number of payments = 30 (2*15 for semiannual payments of up to remaining maturity of 15 years)

i = yield to maturity or priced to yield (YTM) = 6.0% per annum or 6.0%/2 = 3.0% semiannual

Therefore,

P0 = $28 * [1 – 1 / (1+3.0%) ^30] /3.0% + $1,000 / (1+3.0%) ^30

= $548.81 + $411.99

= $960.80

The original price of the bond is $960.80

Now calculate P1:

Bond price P1 = C* [1- 1/ (1+i) ^n] /i + M / (1+i) ^n

Where,

The par value or face value of the Bond = $1,000

Price of the bond after decrease of 30 BPS; P1 =?

C = coupon payment or annual interest payment = 5.6% per annum, but it makes coupon payments on a semi-annual basis therefore coupon payment = 5.6%/2 of $1,000 = $28

n = number of payments = 30 (2*15 for semiannual payments of up to remaining maturity of 15 years)

i = yield to maturity or priced to yield (YTM) = (5.7% = 6.0% - 0.30%) per annum or 5.7%/2 = 2.85% semiannual

Therefore,

P1 = $28 * [1 – 1 / (1+ 2.85%) ^30] /2.85% + $1,000 / (1+2.85%) ^30

= $559.61 + $430.40

= $990.01

The Price of the bond after decrease of 30 BPS in yield is $990.01

Now calculate P2:

Bond price P2 = C* [1- 1/ (1+i) ^n] /i + M / (1+i) ^n

Where,

The par value or face value of the Bond = $1,000

Price of the bond after increase of 30 BPS; P2 =?

C = coupon payment or annual interest payment = 5.6% per annum, but it makes coupon payments on a semi-annual basis therefore coupon payment = 5.6%/2 of $1,000 = $28

n = number of payments = 30 (2*15 for semiannual payments of up to remaining maturity of 15 years)

i = yield to maturity or priced to yield (YTM) = (6.3% = 6.0% + 0.30%) per annum or 6.3%/2 = 3.15% semiannual

Therefore,

P2 = $28 * [1 – 1 / (1+ 3.15%) ^30] /3.15% + $1,000 / (1+3.15%) ^30

= $538.32 + $394.39

= $932.71

The Price of the bond after increase of 30 BPS in yield is $932.71

Therefore,

Effective duration = ($990.01 – $932.71) / (2 * $960.80 * 0.30%)

= 57.30/ (2 * $960.80 * 0.003)

= 9.94

Therefore effective duration is 9.94