An Interactive Applet powered by Sage and MathJax.

(By Prof. Gregory V. Bard. Updated by Ryan G. Hornberger.)

A widow has been left $\$$400,000 from her deceased husband. She is too old to work, and so must select from low-risk investments. She contacts a CFP (certified financial planner), who recommends a speculative-grade bond fund, which has relatively low but non-zero risk, and also an investment-grade bond fund, which has negligible risk. The funds have a forecasted rate of return of 6% and 3% respectively. The CFP instructs her to go over her bills and expenses to compute an income requirement.

The widow looks over her monthly bills and calculates that she requires $\$$17,160 investment income per year to pay her living expenses, because social security is not giving her enough to live on.

Of course, the first thing that the CFP computes is that

$$ (400,000)(0.03) = 12,000 $$and so because 12,000 is insufficient to meet the widow's needs, it is not the case that the widow can simply put all the money in the investment-grade bond fund. Accordingly, a diversified portfolio is necessary, and the CFP must solve a portfolio balancing problem.

When you click the "launch the applet" button below, you will be presented with two sliders. One controls how much is invested in each fund. Please click "launch the applet" now and try to find a solution. (After a while, you can read "Part Two" and change "show the graph" from "no" to "yes." This will show a visual representation of what is going on.)

**Note:** When you drag the slider back and forth, the applet won't recompute
the answers until you let go of the mouse button. There's also a 1-3 second delay,
depending on the speed of your internet connection.

Once you've explored the question numerically a bit---even if you didn't find the answer from Part One---you can now try exploring it graphically. Move the time sliders to some arbitrary values like 200,000 in each investment. Now, next to the words "Show the Graph?" is a choice between "no" and "yes." Change that to "yes."

This graph has several pieces to it:- The $x$-axis is the amount deposited in the investment-grade bond fund.
- The $y$-axis is the amount deposited in the speculative-grade bond fund.
- The units on both scales is "thousands of dollars."
- The green line, $x+y=400,000$ represents investing a total of 400,000 dollars between both investments. (Hint: green is the color of money.)
- The indigo line, $0.03x+0.06y=17,160$ represents getting an income of exactly $\$$17,160 which is our income goal. (Hint: "indigo" and "income" as well as "interest" all begin with the letter 'i'.)
- The big pink dot represents the investment mix or plan that your $x$-value and $y$-value currently are suggesting. (Hint: "pink" and "plan" both begin with the letter 'p'.)
- The two black lines are to help you figure out the $x$ and $y$ coordinates of your plan/point.
- The place where the green line and the indigo line cross is the actual answer. This is investment mix or plan that will exactly satisfy both the income requirement and invest exactly 400,000 among the two funds.

Just like in the graphical approach in part two, let's let $x$ be the amount invested in the investment-grade fund, and let $y$ be the amount invested in the speculative-grade fund.

- Since the widow only has $\$$400,000 we must require $ x + y = 400,000 $.
- Since the income of the two funds are 3% and 6%, the income is $ 0.03x + 0.06y $.
- Because the income requirement is $\$$17,160 we should have $ 0.03x + 0.06y = 17,160 $.
- We can manipulate the first equation to be $ y = 400,000 - x $.
- Then we insert that into the second equation to get: $$ \begin{array}{rcl} 0.03x + 0.06y & = & 17,160 \\ 0.03x + 0.06(400,000 - x) & = & 17,160 \\ 0.03x + 24,000 - 0.06x & = & 17,160 \\ 0.03x - 0.06x & = & 17,1600 - 24,000 \\ -0.03x & = & -6840 \\ x & = & (-6840)/(-0.03) \\ x & = & 228,000 \\ % y & = & 400,000 - x = 400,000 - 228,000 = 172,000 \\ \end{array} $$
- Finally we can solve for $y$ by plugging $x$ back into the first equation. $$ y = 400,000 - x = 400,000 - 228,000 = 172,000 $$

From my perspective, this is a really fun example. That's because the algebra to solve this problem is extremely easy. However, solving the problem numerically or graphically, as in "Part One" and "Part Two" above, was rather tedious. This shows the power of using algebra in financial mathematics.

If we look at the green line, it is easy to verify that all points *above* the green
line are going to representing investing more than $\$$400,000. For example, the point $(300,200)$
is above the green line, and represents investing $\$$500,000. Likewise, all the points below
the green line represent leaving some money uninvested. In the context of this problem, that
really doesn't make any sense. For example, the point $(100,200)$ is below the green line,
and represents investing $\$$300,000. Therefore, in order for a plan to invest all of the
widow's money---no more and no less---the point selected must lie along the green line. The green
line is the "happy medium" between the two problems of investing money that the widow doesn't
have, and leaving some money uninvested.

Similarly, the indigo line has an interpretation. The point $(100,100)$ is below the indigo line, and represents an income of $$ (0.03)(100,000) + (0.06)(100,000) = 9000 $$ which is not enough. The widow requires $\$$17,160. All points below the indigo line represent having too little income, which can be catastrophic. blah blah blarg.

then show two coordinate planes, one coordinate plane shows the "available funds line" and indicates two regions "under-invested" and "insufficient funds."

the second plane shows the "income requirement line" and indicates two regions "insufficient income" and "lower-risk solutions exist."

Last modified on June 8th, 2022.