Question

5. If the value of an asset increases as function: V(t) = Ke2*sqrt(t) (read as 2 times the square root of t), how long should this asset be held given a discount rate = r?

Answer 1

Solution:

You should only hold an asset if it is earning you more than the discount rate. I am going to go ahead here and assume that r is a continuously compounded discount rate.

What is the rate of return on the the given asset? If the returns are continuously compounded values, the continuously compounded return you are getting would be (dV/V)/dt. dV/V is the percentage change in the portfolio value or your returns. why is it dV and not delta V? Because it is continuously compounded returns we are looking for and hence the differentiation.

so, (dV/V)/dt is nothing but (dV/dt)/V.

dV/dt = K e^(2sqrt(t) 2 1/(2sqrt(t)

= Ke^(2sqrt(t)/sqrt(t)

=> (dV/dt)/V = 1/sqrt(t)

So, returning to the question, how long should you hold? You should hold as long as the returns you calculated above are greater than discount rate 'r'.

=> Hold as long as 1/sqrt(t) > r => t < 1/r^2. Hence, the asset should be held till "t = 1/r^2".