Question

An investor has $300,000 to invest, part at 12% and the remainder in a less risky

investment at 7%. If her investment goal is to have an annual income of $27,000, how
much should she put in each investment?

Answer 1

She have to invest at least $120,000 in the 12%-return asset and $180,000 in the 7%-asset to have an annual income of $27,000.

Explanation:

We will solve this as investing the minimum proportion in the risky investment (12% return), with the condition of having an annual income of $27,000.

The return can be calculated as:

`R=C\cdot(\alpha i_1+(1-\alpha)i_2)=300,000(0.12\alpha+0.07(1-\alpha))`

R: annual return, C: capital, α: proportion of the risky asset, i1: rate of return of the risky asset, i2: rate of return of the less risky asset.

Then, we can calculate the proportion of the risky asset as:

`R=27,000=300,000(0.12\alpha+0.07(1-\alpha))\\\\0.12\alpha+0.07-0.07\alpha=27,000/300,000\\\\0.05\alpha+0.07=0.09\\\\0.05\alpha=0.09-0.07=0.02\\\\\alpha=0.02/0.05=0.4`

The proportion of the risky asset is 40%.

The investments in each asset will be:

`I_1=0.4*300,000=120,000\\\\I_2=(1-0.4)*300,000=0.6*300,000=180,000`