Question

In: Finance

Pension funds pay lifetime annuities to recipients. If a firm expects to remain in business indefinitely,...

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)

Solutions

Expert Solution

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


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