Q.16I
Expert-verifiedPension funds pay lifetime annuities to recipients. If a firm remains in business indefinitely, the pension obligation will resemble perpetuity. Suppose, therefore, that you are managing a pension fund with responsibilities to make perpetual payments of $2 million per year to beneficiaries. The yield to maturity on all bonds is 16%.
a. If the duration of 5-year maturity bonds with coupon rates of 12% (paid annually) is four years and the time of 20-year maturity bonds with coupon rates of 6% (paid annually) is 11 years, how much of each of these coupon bonds (in market value) will you want to hold to both entire fund and immunize your obligation?
b. What will be the par value of your holdings in the 20-year coupon bond?
Answer
a. $6.7 m and $5.8 m
b. $14.25 m
Perpetual payment = 2m
YTM = 16%
Present value (PV) of obligation = Payment/ YTM% = 2m/ 16% = $12.5 m
Duration of Obligation = ΔP/P = 1.16/ 16% = 7.25 years
To find the weight (w) of five year maturity period (with duration 4 years-given)
= (w x 4) + [(1 – w) x 11] = 7.25
w = 0.5357
So to find maturity of 5 year bond = 0.5357 x $12.5 = $6.7 m and
the maturity of 20 year bond = (1- 0.5357) x $ 12.5 = 0.4643 x $12.5 = $5.8 m
The price of 20 year bond can be calculated by:
[60 x annuity factor (16%, 20) + [1000 x PV factor (16%, 20)]
= $407.12
This implies that the bond sells for 0.4071 times par value.
Market value = Par value x 0.4071
To find its par value, let’s substitute its values,
$5.8 m = Par value x 0.4071
= $14.25 m
Question: The following table contains spot rates and forward rates for three years. However, the labels got mixed up. Can you identify which row of the interest rates represents spot rates and which one the forward rates?
Year | 1 | 2 | 3 | |
Spot rate of Forward rates? | 10.00% | 12.00% | 14.00% | |
Spot rate of Forward rates? | 10.00% | 14.0364% | 18.1078% |
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