Question

A bond pays a coupon rate of 5% annually and matures in 10 years. The principal is $10,000 and current market price is $8,500.

1. Suppose the yield increases by 0.05% (0.0005, i.e. 5 bps). What is the new bond price? What is the actual change in price?

2. What is the change in price predicted by modified duration formula? Is this change larger or smaller compared to the actual price change in (c)? Why?

3. How would incorporating convexity help improve duration based approximation in (2)?

Answer 1

A bond pays a coupon rate of 5% annually and matures in 10 years. The principal is $10,000 and current market price is $8,500.

We will calculate the yield of this bond using the RATE function in excel. Inputs for the RATE function are:

Yield, y = RATE(Period, payment, PV, FV)

Period = 10 years

Payment = 5% x 10,000 = $ 500

PV = - current market price = -8,500

FV = future value = face value = $ 10,000

Hence, y = RATE(10, 500, -8500, 10000) = 7.15%

Suppose the yield increases by 0.05% (0.0005, i.e. 5 bps). What is the new bond price? What is the actual change in price?

Yield now = y + 0.05% = 7.15% + 0.05% = 7.20%

We will use the PV function of excel to get the New bond price.

New bond Price = -PV(rate, period, payment, FV) = PV(7.20%, 10, 500,10000) = $ 8,469.00

Actual change in price = 8,469 - 8,500 = - $ 31.00

What is the change in price predicted by modified duration formula? Is this change larger or smaller compared to the actual price change in (c)? Why?

We can calculate the modified duration of the bond using MDURATION function of excel. Inputs for MDURATION are:

Modified duration = MDURATION(Settlement date, maturity date, %coupon, %yield, frequency, basis)

Settlement date = 1/1/2019

Maturity date = 1/1/2029 (10 years to settlement date)

% coupon = 5%

% yield = 7.5%

frequency = 1

basis = 0 (default)

Hence, Modified duration = MDURATION(Settlement date, maturity date, %coupon, %yield, frequency, basis) = MDURATION(1/1/2019, 1/1/2029, 5%, 7.15%, 1, 0) = 7.393199716

%age change in price due to 0.5% increase in yield = - modified duration x %age change in yield = - 7.393199716 x 0.05% = -0.3697%
Hence, change in price = -0.3697% x 8,500 = - $ 31.42

This change is larger compared to the actual change.

This is because we have just considered the impact of duration. This assumes that bond price is a linear function of yield, which is not the case actually.

How would incorporating convexity help improve duration based approximation in (2)?

The concept of convexity of the bond takes care of this gap. Convexity is the double derivative of the price of the bond with respect to yield and thus goes a long way in explaining the non linear behavior of Price with respect to yield. Thus convexity helps bridge the gap in the change in price actually and as predicted by the modified duration.