Question

Consider a 10-year bond with a face value of $100 that pays an annual coupon of 8%. Assume spot rates are flat at 5%.

a.Find the bond’s price and modified duration.

b.Suppose that its yields increase by 10bps. Calculate the change in the bond’s price using your bond pricing formula and then using the duration approximation. How big is the difference?

c.Suppose now that its yields increase by 200bps. Repeat your calculations for part b.

Answer 1

Bond maturity = 10 years

Face value = $100

Coupon rate = 8%

coupon frequency = annual

Coupon = coupon rate * face value = 8% *100 = $8

Discount rate = 5%

a)

Bond price is given by the below formula

Where C_t is the cash flow at time t

r is the discount rate

n is the bond maturity

The final cash flow value will be coupon + face value

Bond price is calculated as shown below

Bond price = $123.1652

Duration is given by the below formula

Duration is calculated as shown below

Macaulay Duration = 7.5419

For annual coupon frequency, Modified Duration = Macaulay Duration / (1+r)

Modified Duration = 7.5419 /(1+5%) = 7.1828

b)

Change in yield = 10 bps = 10/100 % = 0.1%

new discount rate = 5% + 0.1% = 5.1%

Bond price is calculated as shown below

Bond Price = 122.2847

Change in bond price = Modified Duration * change in yield = 7.1828*0.1 = 0.7183

Bond price using modified duration = Old bond price - Change in bond price = 123.1652 - 0.7183 = 122.4469

Difference in bond prices = 122.2847 - 122.4469 = -0.1622

c)

Change in yield = 200 bps = 200/100 % = 2%

new discount rate = 5% + 2% = 7%

Bond price is calculated as shown below

Bond Price = 107.0236

Change in bond price = Modified Duration * change in yield = 7.1828*2 = 14.3656

Bond price using modified duration = Old bond price - Change in bond price = 123.1652 - 14.3656 = 108.7996

Difference in bond prices = 107.0236 - 108.7996 = -1.776

The difference is bigger now