Financial Math Answers

1. Consider the following discrete time one-period market model. The savings
account is $1 at time 0 and $β at time 1. The stock price is given by S0 = 1
and S1 = ξ where ξ is a random variable taking two possible values u and d,
each with positive probability. Moreover, assume that d < β < u.
(a) Define what is meant by an equivalent martingale measure (EMM). Find,
with proof, the EMM of this model. Does this model have arbitrage opportunities?
(b) Consider a contract which pays D1 = S^2 1 at time 1. Prove that the time 0
price of this contract is given by D0 = u + d − ud/β
(c) If we assume d = β < u instead, would the EMM from part (a) still be a
valid probability measure? Is it still a valid EMM?
(d) In the case of d = β < u, find an arbitrage strategy

Say that you (or your parents) are purchasing a used car for $19,850. The sales tax is 7.5%, the down payment is $1,000.00, and you have an average credit rating. If your first payment is $425.98, how much of the payment goes toward the principal?

A lottery offers two options for the prize.

Option A: $1000 a week for life.

Option B: $600 000 in one lump sum.

The current expected rate of return for large investment is 3%/a, compounded monthly.

a. Which option would the winner choose if s/he expects to live for another 50 years?
b. At what point in time is Option A better than Option B?

Mary would like to save $10 000 at the end of 5 years for a future down payment on a car.

How much should she deposit at the end of each week in a savings account that pays 1.2%/a, compounded monthly, to meet her goal?

Write a brief reflection on the advantages and disadvantages to long-term saving for a purchase, compared to borrowing a large sum of money and paying it off over time.

4.1. You invest R 16 000.00 at an interest rate of 15% per annum. After how many years will your
investment grow to a value of R 40 000.00 if the interest earned is:
4.1.1. simple? (3 marks)
4.1.2. compounded? (5 marks)
4.2. You consider buying a business for R 900 000.
The business is expected to run for 5 years and the following net returns are expected.
YEAR 1 : R 150 000
YEAR 2 : R 200 000
YEAR 3 : R 290 000
YEAR 4 : R 320 000
YEAR 5 : R 220 000
Should you buy the business? A project of this type is expected to return at least 12% per annum. (17 marks)

Before you enter any values, make a guess as to which of the following will result in the investment of the greatest value. In other words, will you make more money if…
…you invest $2000 per year for twenty years at 2% interest OR …you invest $2500 per year for twenty years at 1% interest?
…you invest $5000 per year for ten years at 3% interest OR …you invest $2000 per year for twenty years at 4% interest?
…you invest $2000 per year for five years at 4% interest OR …you invest $2000 per year for four years at 8% interest?
…you invest $2000 per year for twenty years at 5% interest OR …you invest $2000 per year for twenty–one years at 4% interest?
Using the spreadsheet (or any other means), determine the future value of each of the investments described above and record these. For each choice, which option will make you more money?
Were the results what you expected? Comment on how your predictions compared to the outcome in each case.

Determine an interest rate less than 15%, a period of investment greater than two years, and a regular payment that will result in the total amount of interest you earn being equal to the total amount of money you put in? (for example, under what conditions will you have a future value of $1 000 000, having earned $500 000 interest?)

How many years of periodic, equal payments of $1,000 would be required to be made into an account earing 9% compounded annually, at the end of each compounding period to have on deposit $29,361 to pay future debt of the business?

Let’s say that you want to make $1 000 000 through regular investments. How much would you need to contribute each year if:
The interest rate was 2% and you invested for twenty years?
The interest rate was 4% and you invested for ten years?
The interest rate was 4% and you invested for twenty years?
The interest rate was 10% and you invested for twenty years?