Question

Answer the following short questions. Show all your calculations and explain every step. There are several ways of solving the questions. I expect each of you to solve them in your own way.

1. What is the length of time required for an amount of money to quadruple in value at an interest rate of 8% per year?
2. What is the equal annual amount deposit needed in years 1 through 5 (i.e., in years 1, 2, 3, 4, and 5) to provide for a withdrawal of 1100 TL in year 6 and subsequent withdrawals that increase in 400 TL per year from year 7 to year 35 at an interest rate of 8% per year?
3. The maintenance cost for a certain machine is 1500 TL per year for the first 5 years (from n=1 to n=5), 2200 TL in year 6, and from year 7 until year 35 the maintenance cost per year increases 6% with respect to the amount of the previous year. At an interest rate of 8% per year, what is the equivalent equal payment series?

Draw the cash flow diagrams. Always use factor notation.

Answer 1

(a) In case the compounded annual rate is 8% per year, and n is the time required, as per the formulae: -

On solving,

Hence, it takes approximately 18 years for the money to quadruple at 8% rate of interest.

(b) For the purpose, let us assume that equal deposits through the years 1 to 5 is X. The trick is to calculate the PV of the deposits and withdrawals at t = 0 and equate them for deposits to be sufficient for withdrawals later.

Calculating the PV of the deposits,

Henceforth, at r = 8%,

On solving,

Calculating the PV of the withdrawals,

Henceforth, at r = 8%,

On equating,

On solving,

Hence, the required deposit to be made every year from year 1 to year 5 is 9,164.74 TL.

(c)

For the purpose, let us assume that equal deposits through the years 1 to 35 is X.

Calculating the PV of the costs,

Henceforth, at r = 8%,

On solving,

In the case of the other scenario where the payments are not equal, calculating the PV of costs at r = 8%,

On solving, we obtain: -

On equating,

On solving,

Hence, the required cost every year from year 1 to year 35 is 3,271.06 TL.