Financial Math Answers

A shop keeper offers a discount of 12.5% on a digital camera should the customer pay cash. The camera normally retails for $1245. What is the discounted price if a customer is prepared to pay cash for the camera?

1. At time t=0 the following instruments are quoted in an arbitrage-free market

• a ZCB with maturity 2 years, notional 100 and priced 96.2 euros;
• a coupon bond with maturity 2 years, notional 100, paying annual coupons, with coupon rate 2.2% (on the notional) and priced 100.5 euros;
• an immediate postponed annuity paying 3 instalments of 40 euros, priced 116.2 euros Determine i(0, 1), i(0, 2) and i(0, 3).

1. Consider an arbitrage-free market, where the yield to maturity is described by (time is measured in years)
2. h(0, t) = 0.02 + 0.005t
   

• Compute v(0, 1), v(0, 2), v(0, 3), m(0, 1), δ(0, 3).
• In such market, compute the no-arbitrage price of a ZCB with maturity 3 years and notional 200.

Question 1

Siphesihle borrows money from the bank at a discount rate of 16,5% per year. He must pay the bank R30 000 in eight months from now. The amount of money he receives from the bank now is

[1] R33 707,87.

[2] R26 700,00.

[3] R27 027,03.

[4] R33 463,26.

[5] R33 300,00.

Question 2

Phathu needs R10 500 in ten months' time to buy herself a new lens for her camera. Two months ago she deposited R9 000 in a savings account at a simple interest rate of 11,5% per year. How much money will Phathu still need to buy the lens ten months from now?

[1] R465,00

[2] R229,50

[3] R637,50

[4] R408,67

[5] None of the above

Question 15

Tshepiso owes Thapelo R3 000, which is due ten months from now, and R25 000, which is due 32 months from now. Tshepiso asks Thapelo if she can discharge her obligations by two equal payments: one now and the other one 28 months from now. Thapelo agrees on condition that a 14,75% interest rate, compounded every two months, is applicable. The amount that Tshepiso will pay Thapelo 28 months from now is approximately

[1] R20 000.

[2] R14 000.

[3] R11 511.

[4] R11 907.

[5] R11 455

Questions 9 and 10 relate to the following situation: Three years ago Thokozile borrowed R7 500 from Alfred. The condition was that she would pay him back in seven years’ time at an interest rate of 11,21% per year, compounded semi-annually. Six months ago she also borrowed R25 000 from Alfred at 9,45% per year, compounded monthly. Thokozile would like to pay off her debt four years from now.

Question 9

The amount of money that Thokozile will have to pay Alfred four years from now is

[1] R36 607,98.

[2] R45 181,81.

[3] R55 336,49.

[4] R48 032,20.

[5] R54 278,92

Question 10

After seeing what she must pay Alfred, Thokozile decides to reschedule her debt as two equal payments: one payment now and one three years from now. Alfred agrees on condition that the new agreement, that will run from now, will be subjected to 10,67% interest, compounded quarterly. The amount that Thokozile will pay Alfred three years from now is

[1] R22 286,88.

[2] R25 103,93.

[3] R32 500,00.

[4] R21 171,35.

[5] none of the above

Question 11

Siya wants to buy a new state of the art computer for R35 000. He decides to save by depositing an amount of R500 once a month into an account earning 11,32% interest per year, compounded monthly. The approximate time it will take Siya to have R35 000 available is

[1] 70 months.

[2] 40 months.

[3] 115 months.

[4] 54 months.

[5] none of the above.

Question 12

The accumulated amount after eight years of monthly payments of R1 900 each into an account earning 9,7%

interest per year, compounded monthly, is

[1] R274 069,25.

[2] R182 400,00.

[3] R126 532,64.

[4] R395 077,74.

[5] none of the above.

Question 13

Vanessa decides to invest R140 000 into an account earning 13,5% interest per year, compounded quarterly. This new account allows her to withdraw an amount of money every quarter for ten years after which time the account will be exhausted. The amount of money that Vanessa can withdraw every quarter is

[1] R3 500,00.

[2] R1 704,28.

[3] R6 429,28.

[4] R8 594,82.

[5] none of the above.

Question 7

If money is worth 12% per annum, compounded monthly, how long will it take the principal P to double?

[1] 69,66 years

[2] 8,33 years

[3] 7,27 years

[4] 6,12 years

[5] None of the above

Question 8

A savings account pays interest at the rate of 5% per year, compounded semi-annually. The amount that should be deposited now so that R250 can be withdrawn at the end of every six months for the next ten years is

[1] R3 144,47.

[2] R6 386,16.

[3] R1 930,43.

[4] R3 897,29.

[5] none of the above.

Question 14

If 15% per year, interest is compounded every two months, then the equivalent weekly compounded rate is

[1] 14,464%.

[2] 14,837%.

[3] 14,484%.

[4] 14,816%.

[5] none of the above.

6. Consider the following utility function: 𝒖 = 𝟒𝟎𝟎 (𝟏 + 𝟏)−𝟏. 𝒙𝒚

a. Is this utility function homogenous? Briefly explain. [3]

b. Use implicit differentiation to find the 𝑀𝑅𝐶𝑆. [2]

c. Write down an expression for the indifference curve if 𝒖 = 𝟐𝟎. [2]