Investment Strategy Problem

Quantitative Finance Level 2

A company reserved $30 million for investments. The money is to be divided among treasury notes, stocks, and bonds. It is required that at least $3 million be invested in each type of investment, and at at least $15 million be invested in treasury notes and bonds, while the amount invested in bonds cannot exceed twice the amount invested in treasury notes.

It is known that the annual yield for treasury notes is 7%, for stocks 9%, and 8% for bonds. What is the maximum return attainable by the given requirements in millions of dollars?

The answer is 2.5.

2 solutions

Roger Erisman
May 21, 2015

Note that stock have biggest yield so want to maximize the amount in stocks. Minimum for each is 3 million but T notes and bonds must add up to at least 15. Therefore set stocks = 15 million. Bonds cannot be greater than twice T notes, but they earn more interest so want to maximize bonds.

T notes Bonds

3 12 Not possible Bonds>2*Tnotes

4 11 Not possible same as above

5 10 Possible

6 9 Possible but interest will be lower than 5, 10

Interest at T note = 5, Stock = 15, Bonds = 10 is 2.5 million.

30 = TN + B + S TN, B, S > = 3 mln

15 mln = TN + B
B< 2TN B=15-TN 15-TN<2TN 15 < 3TN 3TN > 15
TN > 5

Let’s assume: TN = 5, then 15 = 5 + B B = 10

Then, 10 + 5 + S = 30 S = 15

Stock + Treasury Notes + Bonds = 5 + 10 + 15.

If we sum up the total return on interests: 5 x 0,07 + 10 x 0,08 + 15 x 0,09 = 2,5

Alessio Nardin - 1 year, 9 months ago

A Former Brilliant Member
Feb 26, 2019

$\text{Maximize}[\{0.08 b+0.09 s+0.07 t,b+s+t=30\land t\geq 3\land s\geq 3\land b\geq 3\land b+t\geq 15\land b\leq 2 t\},\{t,s,b\}] \Longrightarrow \{2.5,\{t\to 5.,s\to 15.,b\to 10.\}\}$

Investment Strategy Problem

The answer is 2.5.

2 solutions

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