Question

Both Bond Sam and Bond Dave have 8 percent coupons, make semiannual payments, and are priced at par value. Bond Sam has 4 years to maturity, whereas Bond Dave has 15 years to maturity.

If interest rates suddenly rise by 5 percent, what is the percentage change in the price of Bond Sam?

If interest rates suddenly rise by 5 percent, what is the percentage change in the price of Bond Dave?

If rates were to suddenly fall by 5 percent instead, what would the percentage change in the price of Bond Sam be then?

If rates were to suddenly fall by 5 percent instead, what would the percentage change in the price of Bond Dave be then?

Answer 1

Assume that the price of bond P0 and its par value both are $1000 as the bond is priced at par

And market interest rate and coupon rate = 8% as it is selling at par

If interest rates suddenly rise by 5 percent, what is the percentage change in the price of Bond Sam?

We have following formula for calculation of bond’s price after change in interest rate

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

Where,

M = value at maturity, or par value = $1000

P0 = the current market price of bond =?

C = coupon payment = 8.0% of $1000 = $80 but semiannual coupon, therefore C = $80/2 = $40

Years remaining prior to maturity = 4 years; therefore number of payments n = 4 *2 = 8

YTM = interest rate, or yield to maturity = 8%+ 5% = 13% or 13%/2 = 6.5% semiannual

Now we have,

Bond price P0 = $40 * [1 – 1 / (1+6.5%) ^8] /6.5% + 1000 / (1+6.5%) ^8

= $243.55 + $604.23

= $847.78

The percentage change in the price of Bond Sam = ($847.78 -$1000)/$1000 =- 15.22%

If interest rates suddenly rise by 5 percent, what is the percentage change in the price of Bond Dave?

We have following formula for calculation of bond’s price after change in interest rate

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

Where,

M = value at maturity, or par value = $1000

P0 = the current market price of bond =?

C = coupon payment = 8.0% of $1000 = $80 but semiannual coupon, therefore C = $80/2 = $40

Years remaining prior to maturity = 15 years; therefore number of payments n = 15 *2 = 30

YTM = interest rate, or yield to maturity = 8%+ 5% = 13% or 13%/2 = 6.5% semiannual

Now we have,

Bond price P0 = $40 * [1 – 1 / (1+6.5%) ^30] /6.5% + 1000 / (1+6.5%) ^30

= $522.35 + $151.19

= $673.53

The percentage change in the price of Bond Dave = ($673.53 -$1000)/$1000 =- 32.65%

If rates were to suddenly fall by 5 percent instead, what would the percentage change in the price of Bond Sam be then?

We have following formula for calculation of bond’s price after change in interest rate

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

Where,

M = value at maturity, or par value = $1000

P0 = the current market price of bond =?

C = coupon payment = 8.0% of $1000 = $80 but semiannual coupon, therefore C = $80/2 = $40

Years remaining prior to maturity = 4 years; therefore number of payments n = 4 *2 = 8

YTM = interest rate, or yield to maturity = 8%- 5% = 3% or 3%/2 = 1.5% semiannual

Now we have,

Bond price P0 = $40 * [1 – 1 / (1+1.5%) ^8] /1.5% + 1000 / (1+1.5%) ^8

= $299.44 + $887.71

= $1,187.15

The percentage change in the price of Bond Sam = ($1,187.15-$1000)/$1000 = 18.71%

If rates were to suddenly fall by 5 percent instead, what would the percentage change in the price of Bond Dave be then?

We have following formula for calculation of bond’s price after change in interest rate

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

Where,

M = value at maturity, or par value = $1000

P0 = the current market price of bond =?

C = coupon payment = 8.0% of $1000 = $80 but semiannual coupon, therefore C = $80/2 = $40

Years remaining prior to maturity = 15 years; therefore number of payments n = 15 *2 = 30

YTM = interest rate, or yield to maturity = 8%- 5% = 3% or 3%/2 = 1.5% semiannual

Now we have,

Bond price P0 = $40 * [1 – 1 / (1+1.5%) ^30] /1.5% + 1000 / (1+1.5%) ^30

= $960.63 + $639.76

= $1,600.40

The percentage change in the price of Bond Dave = ($1600.40 -$1000)/$1000 = 60.40%