Chapter 6 Chapter 6

21
Convexity is a better technique to find the change in the value of the bond on
change in YTM. When we calculate change in bond value due to change in YTM
using the convexity, values are nearer to the values obtained by direct
calculations (By direct calculation we mean finding the value of bond as present
value of all cash flows from that bond). Changes in bond values due to changes in
YTM using convexity are
(i) non-linear (ii) upside capture, downside protection. (The increase in
value for a decrease in YTM will be greater than the decrease in value
due to an equal rise in YTM. Suppose the YTM declines by 1% and as
result the bond value increase by x%; Now suppose the YTM increases
by 1% and a result the bond value decreases by y% ; comparing X with
Y we find that X > Y.)
Convexity = ∑XW(1+X) / Current Bond value
Where, X represents the series of periods of cash inflows, and
W represents Present value (calculated on the basis of prevailing YTM) of cash
inflows corresponding to X periods
Using Convexity:
% Change in the price of the bond on change in YTM:
= [(-1) (Modified Duration)(Yield change in terms of percentage points)}
+ [(C/200) (Yield Chg in terms of percentage points) 2 ]
Q. o.30: 10 % bond with maturity 5 years. Face value : Rs.100. Current YTM = 10 %.
Find convexity. Using convexity, estimate the value of bond assuming YTM at 8%; what
if 12%.
Answer : Statement showing calculation of convexity:
X
W (Present values of XW(1+X)
cash flows)
1 9.09 9.09x2 = 18.18
2 8.26 16.52x3 = 49.56
3 7.51 22.53x4 = 90.12
4 6.83 27.32x5 = 136.60
5 68.31 341.55x6 = 2049.30
Total = 15 100 2343.76
Convexity = ∑XW(1+X) / Current Bond value = 2343.76/ 100 = 23.4376
Using Convexity: