Appreciation of real estate calculation

Question

Hannah's parents recently purchased a vacation home in Arizona for $270,000, which was 35% below what the value of the home was 5 years ago. Her parents feel that the real estate prices have bottomed in Arizona and look for an average appreciation of 2.5% a year.

a) What was the value of the home 5 years ago?

b) How much will the home appreciate to get back to its original value?

Part (a) is easy for me. But what about part (b), Should I use the formula $A = P(1 + \frac{r}{100})^n$ to calculate A (appreciation)?

Can someone please explain what part (b) is asking.

Answer 1

Part b is just asking what percentage it has to appreciate to get back to the value $5$ years ago. It is not $35\%$, which is the point of the problem. Take your result from $a$ and divide by $\$270,000$. The $2.5\%$ per year appreciation is not used in the problem. There might be a part $c$ that asks how many years are needed to get back to where it was.

Answer 2

Part of your problem is compounding. When your problem spans multiple years it is conventional to consider the way the return compounds over time.

Start with what the value was 5 years ago. As you say, that is easy. Solve for x here:

$$.65 x = 270000$$

tells you the value started out at 415385 five years ago.

You then need to know the ANNUAL rate at which it declined. This involves solving for r in this equation

$$415385/(1+r)^5 = 270000$$

which turns out to be .08997698705 per annum. Check:

$$270000*(1+.08997698705)^5 = 4155385$$

Of course, to return to the same original value in the time it took to decline, the appreciation rate must be the same on the climb out as decay rate was during the decent (remember, economics and math are different, either may not have been a smooth path, the rates in either direction may vary year-by-year). Now, as @RossMillikan says, what you MAY want to know is: How long, at your specified rate, does it take to recover? It turns out it is quite a long time because of the difference in the nearly 9% decay rate and your smaller specified recovery rate of 2.5%. Anyway, you solve for n in the following

$$270000*(1+.025)^n = 415385$$

which involves inverse functions but produces 17.44582168 years. Check:

270000*(1+.025)^17.44582168 = 415385

The problem is different if return is compounded more frequently than annually, say, monthly. But the structure of the solution is the same. Many of these questions for real estate are answered at mathestate.com