Question

The market value of the apple tree in year t is V(t) = 200t − 5t²

b) At what year is the apple tree most valuable? What is the value of the tree?

c) Assume that this tree can be harvested only one time. If the real interest is 3%, when should the tree be harvested? (Round years to the closest integer.)

d) If the real interest is 5%, when should the tree be harvested? Compare your results from c) and d) and comment on your findings. Is this the result as you expected?

e) What will be the value of the apple tree at the time of harvest? What is the value of your asset if you invest the proceedings from harvest into bank account that pays 3% annual rate for the rest of period you calculated in part b)?

Answer 1

The market value is given as .

(b) The apple tree is most valuable at where V is maximum. The maximum of V can be found by equating and checking that .

Now, or . Thus for , we have or . Also, or or , and hence and thus, V is maximum at t=20.

The value at t=20 is .

(c) If real interest rate is 3%, the maximum that can be obtained will be for is for where present value, is maximum. The maximum of it can be found as and checking that . Now, or or or .

For , we have .

or or

The will be positive for all t. Hence, solution will be quadratic solution of or , which will be or or or or or or or or (approx). Hence, 11 years (and 4months), is the time.

(d) If the interest rate is 5%, then , and the differentiation will be analogously,, and for , we have . By the means of same quadratic equation method, we will have or or or or . Hence, in this case, 9 years is the time.

The result is expected, because yet with more interest rate, less time is required to miximize the present value (PV), but this is due to V(t), as it increase, it also decrease. The decrease in V, if can overcome the increase in PV due to interest rate, will decrease the PV as a whole. At the maximum of PV, decrease in V overcomes the interest effect in PV.

(e) At it's maximum for 5% interest and t is 9 (approx), or . Also, for 3%, we have maximum of or . These will be the value of the apple tree at the time of the harvest.

We calculated 20 years in part (b). In case of part(c), we had 11 years, and difference is 9 years, and in case of part (d), we had 9 years and the difference is 11 years.

Thus, for part(c), if we take $2207.853 and put it for 9 years at 3% interest, and at the end, we will have dollars.

Also, for part (d), if we take $2164.103 and put it for 11 years at 3% interest, and at the end, we will have dollars.