] Consider the following bond (assume: credit risk free, no embedded options, pa
ID: 2783439 • Letter: #
Question
] Consider the following bond (assume: credit risk free, no embedded options, pays interest semiannually): COUPON=9%, YTM=8%, TERM(5 YEARS), PAR=100.00
Compute the following
(a)The price value of a basis point;
(b)The exact Macaulay duration;
(c)The exact modified duration;
(d) An approximate value for the modified duration, obtained by changing the yield by ±20bp (i.e., use numerical approximation to compute the first derivative). How does it compare to the exact value obtained in part (c)?
(e) The exact convexity measure;
(f) An approximate value of the convexity measure (i.e., use numerical approximation to compute the second derivative), obtained by changing the yield by ±20bp. How does it compare to the exact value obtained in part (e)?
PLEASE SHOW ALL WORK FOR THUMBS UP
Explanation / Answer
Coupon: 9%
YTM: 8%
Term (yrs): 5
Par: 100.00
(a)The price value of a basis point;
Bo = INT [1 – (1+r) ^-n] + Par value (1+r)^-n
r
Where;
Bo= Bond price
INT-interest = 9%*100.00
= 9.00
r= Y.T.M = 8%/2 (semiannually)
r=4%
n= 5*2=10 (semiannually)
Par value = 100.00
Bo = 9.00[1 – (1.04^-10)/0.04] = 73.00
+ 100 (1.04^-10) = 67.56
Bo = 140.56
(a)The exact Macaulay duration;
Formula for Macaulay duration
D=
n Ct (t)
t = 1 (1+i) ^t
n Ct
t = 1 (1+i) ^t
Where:
t = the time period in which the coupon or principal payment occurs
Ct = the interest or principal payment that occurs in period t
i = the yield to maturity on the bond
Year
Cash flow
PV at 8%
PV of cash flows
PV as % of price
Year*PV as % of price
1
18
0.9259
16.67
0.1186
0.1186
2
18
0.8573
15.43
0.1098
0.2196
3
18
0.7938
14.29
0.1017
0.3051
4
18
0.7350
13.23
0.0941
0.3764
5
118
06806
80.31
0.5714
2.857
Bo= 140.56
1.00
3.9
Semiannually = 9
Annually = 9*2 =18
In the fifth year, the cash flows will be interest + par value = 100 + 18 = 118
Macaulay duration = 3.9 years
(c) The exact modified duration;
Modified duration = Macaulay duration/ [1+ (Y.T.M/n]
Where n-is the number of payments in a year
M.D = 3.9 years/ [1+8/2]
Modified duration = 3.75 years
d) Approximate value for the modified duration, obtained by changing the yield by ±20bp
P = D mod × i
Where
P=the change in price for the bond
D mod = the modified duration of the bond: The minus sign is because of the inverse relationship between yield changes and price changes
i = the yield change in basis points divided by 100:
-3.75 *+ 20/100
(-3.75) * (-0.20) = 0.75
-3.75* (0.20) = -0.75
How it compares to exact 3.75 years
This indicates that the bond price at 3.75 years should increase and decrease by approximately + 0.75 percent in response to the 25-basis-point change in YTM.
e) The exact convexity measure
Convexity = d2P=di2
P
Year
Cash flow
PV at 8%
PV of cash flows
t^2 + t
PV of cash flows * t^2 + t
1
18
0.9259
16.67
2
33.34
2
18
0.8573
15.43
6
92.58
3
18
0.7938
14.29
12
171.48
4
18
0.7350
13.23
20
264.6
5
118
06806
80.31
30
2,409.3
Bo= 140.56
2,971.3
Convexity =1/ (1.08) ^2 = 0.86
2,971.3*0.86 = 2,547.41
Convexity = 2,547.41/ 140.56 =18.12
f) in Convexity = 18.2 * + 20/100
Approximate = 18.2 +3.624
How it compares with the 18.2
The exact 18.2 convexity can either curve by increasing and decreasing at 3.624
Year
Cash flow
PV at 8%
PV of cash flows
PV as % of price
Year*PV as % of price
1
18
0.9259
16.67
0.1186
0.1186
2
18
0.8573
15.43
0.1098
0.2196
3
18
0.7938
14.29
0.1017
0.3051
4
18
0.7350
13.23
0.0941
0.3764
5
118
06806
80.31
0.5714
2.857
Bo= 140.56
1.00
3.9
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