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Quantitative Finance Questions and Solutions
Answered

## Question 1 (16 points)

Show the key steps leading to your solutions to quantitative questions

(For answers using the P(n) or F(n) formula, just quote the formula used and put down the corresponding n, r and A.)

A financial institution has entered into an interest rate swap with company X.  Under the terms of the swap, the financial institution pays fixed rate of 4% per annum and receives 12 month LIBOR (with 30/360 day basis) on a principal of \$10 million every year for six years.  Interest rate (with annual compounding) moved down to 3% per annum for all maturities after six months and stayed there since then.  Suppose further that the credit quality of company X started to deteriorate two years after and then defaulted right before it made the third payment (at the end of year 3).

1. What is the value of the swap to the financial institution then (i.e. at the end of year 3)?  (7 points)
2. What is the lossto the financial institution, if any, assuming the recovery ratio is 30%?  (3 points)
3. Company X entered into the swap three years ago in order to hedge the interest payments of a bond it issued back then.  In your opinion, what is likely to be the historical movement of the price of the bond in the past three years?  Please explain in details.  (6 points)

[Please put your answers to Q1 here]

You are a portfolio manager and have \$100 million to invest.  Per the guideline, you have to maintain the duration of your portfolio at around 4-5.  You have a view that the yield curve will move up and flatten modestly in the coming three months.

1. Construct the portfolio by choosing at most 2 bonds from the below table.  Explain your choice.  (10 points)

 Year to Coupon Modified Bond Maturity Duration A 2 4% 1.8 B 5 4.60% 4.1 C 10 5% 7.2

2. Three months have passed and the curve has moved in accordance to your view.  Do you need to adjust your portfolio now?  Explain.  (4 points)

[Please put your answers to Q2 here]

Consider the following fixed-rate level payment mortgage:

Maturity = 25 years; amount borrowed \$200,000; mortgage rate = 6% per annum

1. Calculate the monthly mortgage payment. (4 points)
2. Calculate i) the interest, ii) scheduled principal payment for the first month and iii) the outstanding balance of the mortgage loan at the beginning of the second month. (6 points)

[Please put your answers to Q3 here]

Susan is the treasurer of ABC Property Ltd.  The company needs to raise money to build a residential complex which takes 5 years to complete.  The company has planned to sell the complex after completion, but has a fall back plan of renting some of the flats out should the market condition is not to their favour to sell.  In general, the demand for property purchase is low during a high interest rates environment.

Currently, interest rates are relatively low and the yield curve is relatively flat.  The credit spread (i.e. the risk premium for ABC to issue bond) is all time low too.  Mary with a view that interest rates will stay low for a long period of time, is considering issuing a 10 year bond with one time put in year 5 in order to capture her view, as opposed to the traditional 5 year bond or 10 year bond.

1.    Discuss the pros and cons of her decision.  What will likely the consequence be should rates move up in 5 years’ time.  (10 points)
2.    ABC is a listed company whose board members as well as the shareholders are relatively conservative.  In view of this, her colleague is suggesting to issue a 10 year bond and callable in 5 years.  Mary has the concern of the cost as the coupon rate for traditional callable bond is likely to be higher than a straight 10 year bond.  Name two alternatives for Susan to alleviate her concern.  (6 points)
3.    What will Mary likely to do eventually?  Explain.  (4 points)

[Please put your answers to Q4 here]

The current market price of a two-year 25% coupon bond, paying annual coupons with \$100 face value is trading at \$121.97.  The current market price of a one year zero coupon bond with \$100 face value is \$89.28.

1. What must the price of a two-year zero coupon bond with a \$100 face value be in order to avoid arbitrage? (6 points)

(Hint:  Let P be the value of a two year zero coupon bond with \$100 face value if needed.)

2. Suppose the price of the two year zero coupon bond with a face value of \$100 is \$76.  Is there an arbitrage opportunity?  If yes, describe in details the transactions you will put together in order to capture the profits upfront.  (9 points)

3. Practically, what are the constraints you might face when executing the trade you proposed in b.  (5 points)