. Assume a constant annual interest rate of 5%. A bank has an asset base that de
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Question
. Assume a constant annual interest rate of 5%. A bank has an asset base that delivers cash flows for the next 5 years as shown in the table below. It wishes to structure its liabilities so as to match the value and duration of its assets. Two bonds (A and B) are available to the bank as liabilities and their respective cash flows are shown below.
a. Derive the quantities of the two bonds that the bank should use to meet its objectives.
b. Check that the value and duration of the bond portfolio match those of the asset cash flow stream.
Time12 3 4 5 Assets 400450 600200 300 Bond A8888108 Bond B 00 0 1000Explanation / Answer
Macaulay's Duration of Bond A = 1*CF1/(1+r) + 2*CF2/(1+r)^2 + ... + n*CFn/(1+r)^n / current bond price
=> Macaulay's Duration numerator = 1*8/1.05 + 2*8/1.05^2 + 3*8/1.05^3 + 4*8/1.05^4 + 5*108/1.05^5 = 492.29
PV of cash flow (denominator/ current price) = 8/1.05 + 8/1.05^2 + 8/1.05^3 + 8/1.05^4 + 108/1.05^5 = 112.99
Hence, Macaulays's duration = 492.29/112.99 = 4.36
Hence, Modified duration = Macaulays's duration / (1+r) = 4.36/1.05 = 4.15
For zero coupon bond B with maturity = 4 yeras, Macaulays's Duration = 4 (based on same formula, also simplified due to lack of coupon)
Hence, Modified duration = 4/1.05 = 3.81
Based on the same formula as for Bond A, Asset's Maculay's Duration can be calculated as below:
= (1*400/1.05 + 2*450/1.05^2 + 3*600/1.05^3 + 4*200/1.05^4 + 5*300/1.05^5)/
(400/1.05 + 450/1.05^2 + 600/1.05^3 + 200/1.05^4 + 300/1.05^5) = 4585.64/ 1707.02 = 2.69
Modified Duration of asset = 2.69/1.05 = 2.56 (based on same formula)
Now, Asset's Duration = 2.56 ; Bond 1 = 4.15 and Bond 2 = 3.81
Any combination of bond 1 and Bond 2 will not match the duration of asset as it is very low. Hence, We use bond 2 alone to match the asset with the liability (at least 2.56 is much closer to 3.81 compared to 4.15)
Hence, NPV of Assets = (400/1.05 + 450/1.05^2 + 600/1.05^3 + 200/1.05^4 + 300/1.05^5) = 1707.02
Bond 2 price = 100/1.05^4 = 82.27
No of Bond 2 Required = NPV of Assets / Price of Bond 2 = 1707.02/ 82.27 = 20.74 i.e. 21 Bond 2 is required.
a. Bond 1 = 0 and Bond 2 = 21
b. Value will match approximatly but there is mismatch in duration as assets duration is lower than both the bond's duration.
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