In: Finance
Pension funds pay lifetime annuities to recipients. If a firm expects to remain in business indefinitely, its pension obligation will resemble a perpetuity. Suppose, therefore, that you are managing a pension fund with obligations to make perpetual payments of $2 million per year to beneficiaries. You consider using the following two bonds to immunize your obligation. Assume the yield to maturity on all bonds is 10%.
Bond |
Par |
Maturity |
Coupon |
Yield to Maturity |
A |
1,000 |
5-year |
10%, Annual Payment |
10% |
B |
1,000 |
25-year |
4.5%, Annual Payment |
10% |
a) Suppose that the market interest rate increases from 10% to 12%, how much bond A’s price would change (in dollar value) by applying the duration rule? (3pts)
b) You also calculate the duration of bond B and find it to be 11.5 years. How much of each of these two bonds (in market value) will you want to hold to both fully fund and immunize your obligation? Assume market interest rate remains at 10%. (3pts)
Answer to the question:
a) Calculation of change in Bond A’s price due to change in interest rate:
Year |
Cash Flow |
PV @ YTM10% |
Year*PV of cash Flow |
1 |
100 |
90.91 |
90.91 |
2 |
100 |
82.64 |
165.28 |
3 |
100 |
75.13 |
225.39 |
4 |
100 |
68.30 |
273.20 |
5 |
1100 |
683.01 |
3415.05 |
1000 |
4169.83 |
Duration of the bond = ∑WX
∑W
=4169.83/1000
= 4.17
Convexity of the bond = Duration / (1+r)
= 4.17 / (1.10)
= 3.7909
Convexity is the % of price change due to change in each 1% change in interest rate.
Hence in the given case bond A’s price would change by 2*3.7909=7.5818%
Therefore bond A’s price would change by $75.818 per share (1000*7.5818%)
b) For calculating the proportion of each fund we want to hold, first of all we have to calculate the DL i.e. Duration of the liability
Therefore DL = (1+r) / r
= (1.10) / 0.10
= 11 years
Duration of 5 year bond D1 = 4.17 years
Duration of 25 year bond D2 = 11.50 years
Weight of 5 year bond = W
Weight of 25 year bond = (1-W)
Hence DL = W*D1 + (1-W)*D2
11= W*4.17 + (1-W)*11.50
= 4.17W + 11.50 -11.50 W
7.33 W = 0.50
Therefore W = 0.50/7.33
= 0.0682
Hence 1-W = 0.9318
Hence amount to be invested in 5 year bonds = $200,000*0.0682
= $13640
And amount to be invested in 25 year bonds = $200,000*0.9318
= $186360