What is the Duration of the Portfolio of Liabilities?


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What is the total value of the liability portfolio?


Question 1

You work as a senior fixed-income portfolio manager at a pension fund. Your fund has two business segments, namely a pension segment and a financial products segment. On December 31, 2013, your fund has the following liability positions:

i)The pension segment has issued a zero-coupon bond with face value $3,186,000 that matures in 20 years.

ii)The financial products segment has an annuity liability in which it needs to pay out $150,000 (annually) for the next 10 years.

The pension fund statutes require that only AAA-rated assets can be purchased. There are two such assets available for purchase in the markets: a 10-year Treasury bond paying 15% annual coupons, and a 25-year Treasury bond paying 3% annual coupons, both with face values of $1,000. Suppose the current market interest is 4% and, for simplicity, it is at the moment expected to say the same for the next 25 years.

a)Construct an immunized portfolio.

i)What is the total value of the liability portfolio? (rounded to full dollar amount)

ii)What is the duration of the portfolio of liabilities?

(Hint: you need to take the weighted average of the durations of each liabilities. The weight of each liability corresponds to its value in the liability portfolio.)

iii.What is the duration of the 10-year 15% coupon T-bond? Of the 25-year 3%- coupon T-bond?

iv)What is the combined face value of the 10-year T-bonds that you will need to purchase so to have an immunized portfolio? What is the combined face value of the 25-year T-bonds that you will need to purchase?

b) Suppose that after 1 year, the market interest rate falls to 2% (and, for simplicity) is expected to remain there for the next 25 years.

i)Is the portfolio still fully funded? (If overfunded, write down a positive dollar amount; for underfunding write it as a negative dollar amount; if fully funded, write $0)

ii)Is the portfolio still immunized? Provide the duration of the liability portfolio and the asset portfolio.

iii)Rebalance the portfolio. What do you need to buy and sell? How much of each? What are the net revenues after rebalancing? (put negative number if net costs)

c) Provide an Excel chart that shows the interest rate risk of your liability as of year 0. Specifically, provide a chart that shows the value of the asset portfolio and the value of the liabilities along the ordinate (in $) as the market interest rates change along the abscissa between 1% and 10% (increasing by 1%) for year 0. (This is similar to the chart shown on slide 49 in class 2.)

Question 2: Moonflight, Inc.

Moonflight Inc. expects its business to pick up in about 10 years but to not generate much cash flow before that. To finance its investments, it issues a long-term bond on January 1, 2014 with face value $1,000, with 30 years maturity, and with changing coupons over time. Specifically, the following table shows the coupon payments

         Time                                         Coupon

Jan 1, 2014 – Dec 31, 2023     4% each 6 months

Jan 1, 2024- Dec 31, 2033     3% every quarter

Jan 1, 2034 – Dec 31, 2043    14% once per year


Since the government is interested in spurring innovation in the aerospace industry, Moonflight’s management was successfully secured a government-backed guarantee for the bond, such that the bond is risk-free. Suppose that the yield curve predicts the following market interest rates over the next 30 years:


        Time                                        Forward rates predicted by yield curve*

Jan 1, 2014 – Dec 31, 2021                    5%

Jan 1, 2022 – Dec 31, 2027                    6%

Jan 1, 2028 – Dec 31, 2037                    7%

Jan 1, 2038 – Dec 31,2048                     8%

(Tip: use Excel to clearly outline payments, coupon rate changes and interest rate changes.)

a)Compute the expected bond price.

b) After the bond has been issued, a Court of Appeals decides that the government guarantee is not legal as it violates international WTO obligations. The news changes the price at which the bond trades in the secondary markets at $ 1,400. Compute the YTM (bond equivalent, annual).

c) Create a chart with Excel that shows “YTM (%) (bond-equivalent, annual)” from 1%-15% (increasing by 1%) along the abscissa and “bond price ($)” along the ordinate.


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