(1 point) Say you buy an house as an investment for (assume that you did not need a mortgage). You estimate that the house will increase in value continuously by
per year. At any time in the future you can sell the house and invest the money in a fund with a yearly interest rate of compounded quarterly.
If you want to maximize your return, after how many years should you sell the house? Report your answer to 1 decimal place.
years =

Ask by Guerrero Horton. in the United States

Mar 25,2025

Question

(1 point) Say you buy an house as an investment for (assume that you did not need a mortgage). You estimate that the house will increase in value continuously by
per year. At any time in the future you can sell the house and invest the money in a fund with a yearly interest rate of compounded quarterly.
If you want to maximize your return, after how many years should you sell the house? Report your answer to 1 decimal place.
years =

Ask by Guerrero Horton. in the United States

Mar 25,2025

UpStudy AI · Accepted Answer

1. Compute the continuous equivalent rate for the fund. An annual nominal rate of $$ 8.5\% $$ compounded quarterly gives a quarterly rate of
   $$
   \frac{0.085}{4} = 0.02125.
   $$
   The continuous equivalent rate is then
   $$
   r = 4\ln\Bigl(1+0.02125\Bigr).
   $$
   Evaluating,
   $$
   r \approx 4\ln(1.02125) \approx 4 (0.02103) \approx 0.08412.
   $$

2. The house increases in value continuously by a fixed amount of $$ \$62500 $$ per year. Therefore, if the initial value is $$ \$500000 $$, at time $$ t $$ years the value is
   $$
   V(t)=500000+62500t.
   $$
   Its instantaneous rate of appreciation relative to its current value is
   $$
   \frac{dV/dt}{V(t)}=\frac{62500}{500000+62500t}.
   $$

3. To maximize the return when transferring the money to the fund, sell the house at the time $$ t $$ when the house’s instantaneous rate of increase equals the continuous rate $$ r $$ of the fund. That is, set
   $$
   \frac{62500}{500000+62500t}=r.
   $$
   Solve for $$ t $$:
   $$
   500000+62500t=\frac{62500}{r}.
   $$
   Isolate $$ t $$:
   $$
   t=\frac{1}{r}-\frac{500000}{62500}.
   $$
   Note that
   $$
   \frac{500000}{62500}=8.
   $$

4. Substitute $$ r\approx 0.08412 $$:
   $$
   t\approx \frac{1}{0.08412} - 8.
   $$
   Calculate
   $$
   \frac{1}{0.08412}\approx 11.887,
   $$
   hence,
   $$
   t\approx 11.887 - 8 \approx 3.887.
   $$

5. Rounding to one decimal place gives the optimal time to sell the house as
   $$
   t\approx 3.9 \text{ years}.
   $$

years = $$ 3.9 $$
You should sell the house after approximately 3.9 years to maximize your return.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Mind Expander

preguntas relacionadas

Latest Calculus Questions