Consider an investor with a portfolio totaling 500,000 that...

Consider an investor with a portfolio totaling $\$ 500,000$ that is invested in certificates of deposit, municipal bonds, bluechip stocks, and growth or speculative stocks. How much is invested in each type of investment? The certificates of deposit pay $2 \%$ annually, and the municipal bonds pay $4 \%$ annually. Over a five-year period, the investor expects the blue-chip stocks to return $10 \%$ annually and the growth stocks to return $14 \%$ annually. The investor wants a combined annual return of $6 \%$ and also wants to have only one-fourth of the portfolio invested in stocks.

Consider an investor with a portfolio totaling $\$ 500,000$ that is invested in certificates of deposit, municipal bonds, bluechip stocks, and growth or speculative stocks. How much is invested in each type of investment?
The certificates of deposit pay $2 \%$ annually, and the municipal bonds pay $4 \%$ annually. Over a five-year period, the investor expects the blue-chip stocks to return $10 \%$ annually and the growth stocks to return $14 \%$ annually. The investor wants a combined annual return of $6 \%$ and also wants to have only one-fourth of the portfolio invested in stocks.

Key Concepts

Portfolio Allocation

Portfolio allocation involves distributing total investment funds among different asset classes to optimize returns and manage risk. It requires understanding the trade-offs between risk and return for each asset type, and aligning the distribution with the investor’s specific goals and constraints.

Weighted Average Return

Weighted average return is the calculation of an overall portfolio return based on the proportion of each asset class’s investment return in the portfolio. This concept is key in determining how each asset’s individual return contributes to the total return given the percentage of the portfolio that is allocated to it.

System of Linear Equations

A system of linear equations is used to represent multiple linear relationships between variables simultaneously. In the context of portfolio allocation, constraints such as total investment amount, desired overall return, and allocation percentages are modeled as linear equations, which can then be solved to find the amounts invested in each asset class.

Transcript

00:01 So in this problem, we have an investor with a total portfolio of $500 ,000, and they're invested in certificates of deposit, i'll call them cds, municipal bonds, i'll call them munis, blue chip stocks, blue chips, and growth or speculative stocks.

00:33 Okay? now, we want to know how much is invested in each type of investment.

00:39 And we're given that the cds pay 2 % annually.

00:45 Municipal bonds pay 4 % annually.

00:51 Over five years, the blue chips return 10 % annually.

00:57 And the growth stocks return 14 % annually.

01:04 And since we want to know how much of each one, we'll assign some variables to them.

01:10 Z and a, where these are the amounts in each investment.

01:32 Now, first we know that all of these add up to 500 ,000.

01:37 So we have 500 ,000 equals x plus y plus z plus a.

01:52 Okay.

01:54 Next, we're told you want to combine any return of 5%.

01:58 So that means i have 500 ,000 times .05 is .02 times x by the return on the cds plus 0 .4y plus 0 .10z plus 0 .10z plus 0 .14a.

02:44 All right.

02:46 And then, lastly, we're told that the only one's a fourth of the portfolio invested in stocks.

02:59 And remember, these two are both stocks.

03:03 So, what does that mean? that means if i take 500 ,000 times a fourth, that gives me z plus a.

03:16 And that's the only information i'm given.

03:18 Now the problem i have is i've got four unknowns and only three equations.

03:25 And so i've got to do something here to kind of simplify this into three equations and three unknowns.

03:38 So i can solve this, right? so here's what i want to do.

03:42 I'm going to say that the amount invested in stocks is the variable s.

03:48 Right this is the amount in stocks okay and so that means now that this part is s right and that means then i have some return times s well if i can assume he has the same amount in both z and a minute then what well then the return on that will be 12%.

04:35 So now i can rewrite my equations.

04:38 500 ,000 is x plus y plus s.

04:45 Call that equation 1.

04:47 And then 500 ,000 times 0 .05.

04:51 Well, that is 25 ,000 equals 0 .02x plus point 0 .4%.

05:15 This is supposed to be.

05:17 0 .04.

05:26 0 .04 0 .04 plus, we said if it's equal, then 0 .12 s.

05:37 Okay.

05:39 And we know from the last, see that's equation 2, we know from this equation that 500 ,000 divided by 4, is 125 ,000.

05:57 Is my s.

06:00 125 ,000 is my s.