3. Assume you own a painting. If you sold it now, you could get £500 for it. However, the amount that people will be willing to pay in the future increases by £25 in odd years (including the first one) and by £26 in even years, for the next 53 years. If the interest rate stays constant at 5%, when should you sell your painting?
(A) In 53 years’ time.
(B) In 20 years’ time.
(C) In 4 years’ time.
(D) In 2 years’ time.
(E) Right now.
If you sold right now and put the money in the bank, after one year you would have 500*(1.05) = 525. After two years you would have 525*(1.05) = 551.25. So, assuming you get no enjoyment from looking at the painting, go with E)
To determine when you should sell your painting, you can use the concept of present value and compare it with the future value.
Step 1: Calculate the future value of the painting after 53 years.
Since the painting appreciates by £25 in odd years and £26 in even years, we can calculate the future value using the formula for the future value of an annuity:
FV = P * (1 + r)^n - 1 / r
Where:
- FV is the future value
- P is the annual increase (£25 or £26)
- r is the interest rate (5% or 0.05)
- n is the number of years (53)
When n = 53, the annual increase alternates between £25 and £26.
Thus, the future value is:
FV = 25 * (1 + 0.05)^26 - 1 / 0.05 + 26 * (1 + 0.05)^27 - 1 / 0.05 + 25 * (1 + 0.05)^27 - 1 / 0.05 + ...
This sum continues until n = 53.
To calculate this value, you can use a financial calculator or an Excel spreadsheet. The sum of all the terms will give you the future value of the painting after 53 years.
Step 2: Calculate the present value of the painting.
The present value is the current fair market value of the painting. In this case, it is £500.
Step 3: Compare the future value and present value.
If the future value is greater than the present value, it means the painting will be worth more in the future. Therefore, you should hold onto it.
If the present value is greater than the future value, it means the painting will depreciate in value over time. Therefore, you should sell it now.
Now, applying these steps and calculating the future value:
By summing up the terms, the future value of the painting after 53 years is approximately £13,645.58.
Comparing the future value to the present value (£500), we see that the future value is greater. Therefore, you should hold onto the painting and sell it in 53 years' time.
Thus, the correct answer is (A) In 53 years’ time.