# Jill has $10,000 to invest at time t = 0, and two possible ways to invest it. Investment A has a... ## Question: Jill has$10,000 to invest at time t = 0, and two possible ways to invest it.

Investment A has a force of interest equal to {eq}\frac{0.8}{1+0.8t} {/eq} at time t.

Investment B provides a 5% effective annual interest rate.

Jill can invest any portion of her principal in either investment A or B, and can transfer any portion of her money between the two investments at any time.

What is the maximum amount Jill can accumulate by time t = 20?

## Investment:

Investment is the buying of an asset with the expectation of reaping maximum returns o the investment. The investments are made thinking about the long term goals for which the investments are proposed to be made. Investments can be made into securities, assets, bonds, mutual funds, etc.

When investing money the investor aims to aims the highest interest rate on the invested money. The force of interest is given for Investment A. We need to convert to a force of interest for Investment B. Since the effective rate does not vary for Investment B, the force of interest is constant and is equal to ln(1.05)which is approx .04879. Therefore as long as the force of interest for Investment A is greater than .04879, then we want to invest in A.

{eq}$0.08/\left( {1 + 0.08t} \right) \succ 0.04879$ {/eq}

We find that t < 8

So we invest in A for 8 years and then B for the remaining 12 years.

By inspection we see that;

{eq}$A\left( t \right) = 1 + 0.08t$ {/eq}

So the accumulated value at time is 20.

{eq}$10000\left( {1 + 0.08*8} \right) + 1.05*12$ {/eq}

That is aprox \$29,452 will be maximum amount of return. 