Question

A bond that pays annual coupons has a par value of $1,000, an 8% coupon rate, 3 years left to maturity, and is currently priced at a YTM of 6.0%.

(a) Calculate duration and modified duration for the bond.

(b) If the YTM on the bond changes from its current 6.0% up to 8.0%, what price change (% and $) and new price ($) is predicted by the modified duration calculated in part a.?

(c) What is the size and direction of the pricing error in the duration-predicted price you calculated in part (b)?

Answer 1

(a) Duration is the ration between Present Value of Weighted Cashflows and Current Market Price

As current market price is not given, lets assume perfect market conditions and find the pv of bond with YTM 6%

A)resent Value of Cash flows = ((100)/(1.06))+((100)/(1.06²))+((1100)/(1.06³)) = 75.47+71.2+906.79 = $1053.46

B)Present Value of Weighted Cash flows = ((100*1)/(1.06))+((100*2)/(1.06²))+((1100*3)/(1.06³)) = 75.47+142.4+2720.24 = $2938.24

Macaulay Duration = B/A = 2938.24/1053.46 = 2.789 Years

Modified duration = Duration/(1+(YTM/n)) = 2.789/(1.06) = 2.6313

That is for every 1% change in interest rate the value of the bond (price) moves inversely 2.6313%

(b)

So, for 2% change in YTM the bond price will move -Modified Duration * Interest rate change * 100% = -5.2626%

in $ that would be a change of about ($1053.46 * 5.26%) = $55.41

And hence the new price should be closer to $998.05

(c)

The directional error generally doesn't occur, as for increase in yield, the price will go down.

However the size error does occur, as the interest rate change increases generally the variation in the price deviates highly from the prediction.