Question

In: Finance

Consider a 8% coupon bond making annual coupon payments with 4 years until maturity and a...

Consider a 8% coupon bond making annual coupon payments with 4 years until maturity and a yield to maturity of 10%.

  1. What is the modified duration of this bond?
  1. If the market yield increases by 75 basis points, what is the actual percentage change in the bond’s price? [Actual, not approximation]
  1. Given that this bond’s convexity is 14.13, what price would you predict using the duration-with-convexity approximation for this bond at this new yield?
  1. What is the percentage error?

Solutions

Expert Solution

a) Let us Consider the par value of the Bond to be 100

First we need to the find the macaulay duration of the bond

Year (t) Cash flow(CF) PV factor CF* PV (CF* PV)/ Total [(CF* PV)/ Total] * t
1 8 0.909091 7.272727 0.07765 0.0776503
2 8 0.826446 6.61157 0.070591 0.1411824
3 8 0.751315 6.010518 0.064174 0.1925214
4 108 0.683013 73.76545 0.787588 3.1503503
Total 93.66 3.5617044

The macaulay duration is 3.5617

The modified duration = (macaulay duration) / ( 1 + YTM /n ) , where n is the number of coupons per period

= 3.56 / (1 + 1.1)

= 3.2379

Alternatively we can also calculate using MDURATION func in excel

b) The PV of the bond when YTM =10% is 93.66027

Year CF PV factor PV of CF
1 8 0.909091 7.272727
2 8 0.826446 6.61157
3 8 0.751315 6.010518
4 108 0.683013 73.76545
Total 93.66027

The PV of the bond when YTM =10.75% is 91.42253

Year CF PV factor PV of CF
1 8 0.902935 7.223476
2 8 0.815291 6.522326
3 8 0.736154 5.889234
4 108 0.664699 71.7875
Total 91.42253

Hence the actual price change % = (93.66027 - 91.42253)/ 93.66027 * 100 %

= 2.3892 %

The Approximate % decrease in bond price = (The change in yield* the modified duration) * 100%

= -( 0.0075 * 3.2379 ) * 100%

= -2.428 %

c)

Change in price accounting for convexity = Duration effect+Convexity effect

=(-Modifed Duration * ΔYield) + [0.5 * Convexity * (ΔYield)2 ]

=( - 3.2379 * 0.0075) + [ 0.5 * 14.13 * 0.00752 ]

= -0.0238868 or -2.38868  %

d)

Percentage error = (Approx - actual) / actual * 100%

Now, actual price after YTM increase = 91.42253

Approx price  after YTM increase = (1-0.0238868 )* 93.66027

= 91.42302

Hence , % error =( 91.42302 - 91.42253) / 91.42253 *100%

= 0.0005359%


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