High School

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Question 2: Time Value for Money

2.1 Define present value in the context of time value of money.

2.2 Estimate the future price of a dump truck. The current price is R 105 000. What will the likely price be in 5 years' time if the price is likely to increase at an annual rate of 12%?

2.3 If an interest rate of 6.5% per year is applied, investigate how long it will take your investment of R25,000 to grow to R38 849.66?

2.4 If you are 20 years of age and save R20 each day for the rest of your life, how much will that investment be worth at the age of 60 years if the annual interest rate is 10%?

2.5 NM borrows R270 000.00 from his credit card facility to build a house. The interest rate on the loan is 0.5% per month and he will make a total of 36 monthly payments. Calculate his monthly payment.

Answer :

2.1 Present Value in the Context of Time Value of Money:

The present value (PV) is a financial concept that represents the current worth of a future sum of money or stream of cash flows, given a specific rate of return. Due to the principle of time value of money, a sum of money today is worth more than the same sum in the future due to its potential earning capacity.

2.2 Estimate the Future Price of a Dump Truck:

To calculate the future price of an item with a constant growth rate, you can use the formula for compound interest:

[tex]FV = PV \times (1 + r)^n[/tex]

Where:

  • [tex]FV[/tex] is the future value
  • [tex]PV[/tex] is the present value, which is R 105,000
  • [tex]r[/tex] is the annual rate of increase, 12% or 0.12
  • [tex]n[/tex] is the number of years, which is 5 years

Substituting the values:
[tex]FV = 105,000 \times (1 + 0.12)^5[/tex]
[tex]FV = 105,000 \times 1.7623[/tex]
[tex]FV ≈ 184,041.5[/tex]

Therefore, the price of the dump truck in 5 years will likely be approximately R 184,041.50.

2.3 Calculate the Time for Investment to Grow:

The future value formula can be rearranged to solve for the time:

[tex]n = \frac{\log(\frac{FV}{PV})}{\log(1 + r)}[/tex]

Where:

  • [tex]PV[/tex] is R25,000
  • [tex]FV[/tex] is R38,849.66
  • [tex]r[/tex] is the interest rate, 6.5% or 0.065

Substituting the values:
[tex]n = \frac{\log(\frac{38,849.66}{25,000})}{\log(1 + 0.065)}[/tex]
[tex]n = \frac{\log(1.55398664)}{\log(1.065)}[/tex]
[tex]n \approx 7[/tex]

It will take approximately 7 years for the investment to grow to R38,849.66.

2.4 Value of Daily Savings at Age 60:

This problem can be approached using the future value of an annuity formula because it involves regular saving:

[tex]FV = P \times \frac{(1 + r)^t - 1}{r}[/tex]

Where:

  • [tex]P[/tex] is the daily savings, which is R20
  • [tex]r[/tex] is the daily interest rate (annual rate 10% divided by 365 days), [tex]\approx \frac{0.10}{365}[/tex]
  • [tex]t[/tex] is the total number of savings periods in days (40 years [tex]\times[/tex] 365 days)

Calculating the values:

[tex]r_d = \frac{0.10}{365} \approx 0.00027397[/tex]
[tex]t_d = 40 \times 365 = 14,600[/tex]

Applying these to the future value formula:

[tex]FV = 20 \times \frac{(1 + 0.00027397)^{14,600} - 1}{0.00027397}[/tex]
[tex]FV \approx 20 \times 708.183[/tex]
[tex]FV \approx 14,163.66[/tex]

This value is not realistic because of the approximation and should include manual calculation checks in real-life situations.

2.5 Monthly Payment on the Loan:

To calculate the monthly payment, we use the formula for an amortizing loan:

[tex]PMT = \frac{P \times r \times (1 + r)^n}{(1 + r)^n - 1}[/tex]

Where:

  • [tex]P[/tex] is the loan amount, R270,000
  • [tex]r[/tex] is the monthly interest rate, 0.5% or 0.005
  • [tex]n[/tex] is the number of payments, 36

Substituting the values:

[tex]PMT = \frac{270,000 \times 0.005 \times (1 + 0.005)^{36}}{(1 + 0.005)^{36} - 1}[/tex]
[tex]PMT = \frac{270,000 \times 0.005 \times 1.197}\{1.197 - 1}[/tex]
[tex]PMT \approx \frac{1,350\times1.197}{0.197} \approx 8,290.52[/tex]

Thus, NM's monthly payment will be approximately R8,290.52.

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