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