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    SCHOOL OF MATHEMATICAL SCIENCES                        Question 1 RM1,000 is invested at a market rate i

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SCHOOL OF MATHEMATICAL SCIENCES                     

 

Question 1

RM1,000 is invested at a market rate i. Inflation is at a constant 3.5% per annum. At the end of 15 years, the purchasing power of the investment has increased by 55%. Determine i.

 

Note that RM1,000 is not a must for this question. 

Question 2

Don invests RM1,000 at time 0. His nominal rate of interest is 8% in year 1, 7% in year 2 and 6% in year 3. The rate of inflation is 4.5% in year 1, 3.5% in year 2, and 3% in year 3. What is Don’s equivalent level annual real rate of interest/return over the 3-year period?

 

Note that RM1,000 is not a must for the question.

Question 3

The real rate of interest is 5%. The expected annual inflation rate over the next two years is 3%.

Determine the net present value of the following cashflows using the nominal interest.

 

Year	0	1	2
Cashflow	– 450	280	280

 

Question 4

The real interest rate is 7% and the inflation rate is 3%. Bob receives a payment of 1,000 at time 1, and subsequent payments increase by 100 for 5 more years. Determine the accumulated value of these payments in nominal interest at time 6 years.

Question 5 

An insurance company has an obligation to pay the medical costs for a claimant. Average annual claims costs today are RM5,000, and medical inflation is expected to be 7% per year. The claimant is expected to live an additional 20 years. Claims payments are made at yearly intervals, with the first claim payment to be made one year from today. Find the present value (using the real rate of interest) of the obligation if the annual (nominal rate of) interest rate is 5%.

Question 6

You are given the following term structure of interest rates.

 

Length of Investment	Interest Rate (or Spot Rate)
1	4%
2	5%
3	7%
4	8%

 

1. Compute the 1-year, 2-year, and 3-year forward rate, one year from now.
2. Compute the 1-year, 2-year, and 3-year forward rate, one year from now.
3. Compute the 1-year forward rate three years from now.

   Question 7

   You are given the following term structure of spot interest rates.

    

   Term (in Years)	Interest Rate (or Spot Rate)
   1	7.50%
   2	8.50%
   3	9.25%
   4	9.75%
   5	9.85%

Find the price of a RM1,000 two-year bond with annual 6% coupon.  

1. Compute the yield to maturity.

   Question 8

   The following are the current prices of RM1,000 zero-coupon bonds.

    

   Term to Maturity	Price (RM)
   1	952.38
   2	X
   3	853.26

    

   The 1-year forward rate one year from now is 7%. Determine X.

   1000  

   

                         ²

   Question 9

   The following are the current prices of RM1,000 zero-coupon bonds.

    

   Term to Maturity	Price (RM)
   1	943.40
   2	898.47
   3	847.62
   4	792.16

    

   Determine the 1-year forward rate 3 years from now.

   Question 10

   The following are the current prices of RM100 zero-coupon bonds redeemable at par.

    

   Term to Maturity	Price (RM)
   1	95.23
   2	89.84
   3	84.56
   4	79.21

    

   Question 11

   Suppose payments of 3,000, 5,000, 8,000, and 9,000 are to be made at times 1, 2, 4, and 5 respectively. Assume, an annual yield of 25%. Find the

    

   (a) average term-to-maturity using the method of equated time
   (b) Macaulay duration of the investment.

   Question 12
   Determine the Macaulay duration and modified duration for each of the following assets at an annual

    

   effective rate of interest of 6%

2. 1. A perpetuity immediate with level annual payments
   2. b. A 15-year zero coupon bond
   3. c. A 15-year 100 bond with 5% annual coupons maturing at par
      d. A 15-year 100 bond with 10% annual coupons maturing at par
      A 15-year mortgage with level annual payments
      Question 14

      Determine the convexity for each of the following assets at an annual effective rate of interest of 6%

       

   4. 1. A 15-year zero coupon bond. We can take the redemption value, C = 1.
      2. b. A 15-year 100 bond with 5% annual coupons maturing at par
      3. A 15-year 100 bond with 10% annual coupons maturing at par
      4. A 15-year mortgage with level annual payments
         Question 15

         A company's liabilities include loss reserves, which are liability reserves set aside to make future claim payments on policies which the company has already sold. You believe that these liabilities, totalling RM100 million on December 31, 2016, will be paid out according to the following schedule

          

         Calendar Year	Proportion of Reserves Paid Out
         2017	30%
         2018	35%
         2019	25%
         2020	10%

          

         Find the modified duration of the company's loss reserves. Assume that the annual interest rate is 8%, and that all losses paid during a given calendar year are paid at the midpoint of that calendar year.

         Question 15

         Consider the following bonds.

      5. • Bond A with price 2,000, modified duration 8 and convexity 64. ? Bond B with price 1,000, modified duration 6 and convexity 36; and
         • Bond C with price 800, modified duration 5 and convexity 25. Calculate the modified duration and convexity of the portfolio. 

           Question 16

           n

           ^{2 t}            1           ^{2 n          2                      2}

           Question 16
           Prove that ?t v ^? ?_?2?Ia?_n ? ^a_n ?1?(n?1) v ?_?. (Hint: t ?(t ?1) ?2t ?1)
            

           Question 17

           For a 25-year home mortgage with level payments and an interest rate of 9% convertible monthly, find the modified duration and the convexity of the payments.

           Question 18

           Prove that P(i ??i) ? P(i)?1?v?i? and P(i ? ?i) ? P(i)_?? ^?1?v?i ? ^c₂ (?i)^2?_??   . Use the results to

           estimate the new bond price (with annual coupon) if the yield rate increases 0.5% using both approximations. Given the yield rate i = 5%, initial price of bond = 1000, v ?10 and c ?100.

           Question 19

           A client deposits RM100,000 in a bank, with the bank agreeing to pay 7% effective for two years. The client indicates that half of the account balance will be withdrawn at the end of the first year. The bank can invest in either one-year or two-year zero-coupon bonds. The one-year bonds yield 9% and the two-year bonds yield 10%. Develop an investment program based on immunization.

Question 21

An insurance company has an obligation to pay RM1,000,000 at the end of 10 years. It has a zerocoupon bond that matures for RM413,947.55 in 5 years, and it has a zero-coupon bond that matures for RM864,580.82 in 20 years. The effective yield for assets and liability is 10%. Determine whether the company's position is fully immunized.

Question 22

An investor has a single liability of RM1,000,000 due in 15 years' time. The yield on zero coupon bonds of any term is currently 5% per annum, and the investor possesses cash equal to the present value of his liability. He wishes to invest in 10-year and 20-year zero coupon bonds in such a way that he will make a profit on any immediate change in the force of interest. How much of each security should he buy, and how large a profit will he make if the rate of interest per annum immediately becomes 0.01, 0.02, 0.03, 0.04, 0.06, 0.07, or 0.08?

Question 23

An insurance company accepts an obligation to pay RM10,000 at the end of each year for 2 years. The insurance company purchases a combination of the following two bonds at a total cost of X in order to exactly match its obligation.

 

(Bond I)  A 1-year 4% annual coupon bond with a yield rate of 5% (Bond II) A 2-year 6% annual coupon bond with a yield rate of 5%.

 

Calculate X.

Question 24

A company must pay liabilities of 1,000 and 2,000 at the end of years 1 and 2, respectively. The only investments available to the company are the following two zero coupon bonds.

 

Question 25

John wants to absolutely match a debt that he owes under which he must make payment of 1,000 in one year and 2,000 in two years. He can purchase the following bonds.

(Bond A) A 2-year bond with annual coupons of 100 and a maturity value of 1,000 (Bond B) A 1-year bond with annual coupons of 80 and a maturity value of 1,000 Calculate the amount of Bond B that John should purchase.