Portfolio Optimization

Quantitative Finance Level 2

You have $100,000 to invest in bonds, and you have selected three types of bonds: federal bonds paying 5% interest, municipal bonds paying 6%, and a corporate bonds paying 9%. No more that $20,000 can be invested in corporate bonds, and the amount of corporate bonds cannot exceed the amount of federal bonds. How much should be invested in municipal bonds to maximize the interest?

Answer is in units of dollars.

The answer is 60000.

1 solution

Pranshu Gaba
Jun 29, 2015

Let $F, C$ and $M$ denote the amount of money invested in federal, corporate and municipal bonds respectively. Then we have the following conditions:

$F + C + M = 100{,}000 ~~~~~ C \leq 20{,}000 ~~~~~ C \leq F$

We want to maximize $( 5 F + 6 M + 9 C) \frac{T}{100}$ , where $T$ is the time to maturity. This equivalent to maximizing $5 F + 6 M + 9 C$ since $\frac{T}{100}$ is just a positive constant.

Since federal bonds give the least interest, $F$ should be minimized, i.e. $F = C$ . Now we rewrite the equation.

$2C + M = 100{,}000 ~~~~~ C \leq 20{,}000$

We want to maximize $14 C + 6 M$ , it is equivalent to maximizing $7C + 3M$ . Substituting $M = 100{,}000 - 2C$ , we want to maximize $300{,}000 + C$ . This means $C$ should be maximum, i.e. $C = 20{,}000$ . So $F$ is also $20{,}000$ and $M = 100{,}000 - 2C = \boxed{60{,}000}$ $_\square$

Moderator note:

Good approach. This is a basic linear programming question where the function we want to optimize is also linear.

$\text{Maximize}[\{0.09 c+0.05 f+0.06 m,c+f+m=100000\land c\leq 20000\land c\leq f\},\{c,f,m\}]\Longrightarrow \{6400.,\{c\to 20000.,f\to 20000.,m\to 60000.\}\}$

A Former Brilliant Member - 2 years, 3 months ago

Portfolio Optimization

The answer is 60000.

1 solution

Moderator note:

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