Question 1: The Ministry is obligated to make a payment of 100000 Galleons (gold

Question

Question 1: The Ministry is obligated to make a payment of 100000 Galleons (gold coins), in exactly five years. A financial advisor, Kingsley Shacklebolt, wishes to fund this liability with a combination of two-year, zero-coupon bonds and seven-year, zero-coupon bonds, purchased today. The current yield rate for bonds is 5%. How much should they purchase each type of the bond in order to have Redington immunization? Also, show that the resulting portfolio provides the liability amount even if the rates decrease to 4%, or rise to 6%.

Explanation / Answer

In this problem, we need to immunize our portfolio such that we end up receiving 100,000 after 5 years exactly. Since we we only have the option of investing in zero coupon bonds wwhich have the duration as 2 years and seven years, our objective is to calculate the value of investment required in each portfolio such that our duration matches with the portfolio's duration.

The present value of 100,000 5 years from now is 78,352.62 which is calculated as per below:

100,000 / (1+0.05)^5

where 0.05 is the current yield and 5 is the number of years

Let A be the total amount invested in a 2 year bond and B be the total amount invested in a 7 year bond.

A + B = 78,352.62

The duration of a zero coupon bond is the time to maturity. So duration for A is 2 and duration for B is 7.

Let Wa be the weight of 2 year Bonds in our total investment. Weight of Investment in 7 years bonds is Wb = (1-Wa) as Wa + Wb = 1

Since we need to make sure that our total duration matches with the liability time (5):

Wa * 2 + (1-Wa) * 7 = 5

Solving the above, we get Wa = 0.4 and Wb = 0.6

A = Wa * Total Investment = 0.4 * 78,352.62 = 31,341.05

B = Wb * Total Investment = 0.6 * 78,352.62 = 47,011.57

Since we've immunized our portfolio with respect to duration, there won't be any effect of rates increase or decrease on our liability amount due after 5 years.

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