Money and You: Course Notes

For example, suppose you invest $1,000. Sometime later, your investment is valued at $1,250. Then your ROI would be \(\displaystyle ROI = \frac{1,250 - 1,000}{1,000}\times 100 = 25\%.\) Your investment has grown by 25%.

The issue with ROI is that it measures only how one investment has grown over some unknown time period. It is then difficult to compare different ROI’s without setting some standards,

The gold standard for measuring how successful your savings and investment outcomes are is the rate of return. The rate of return is a measurement of how your investments have grown or decayed each year. For example, if your rate of return on your 401(k) were 8%, that means that, on average, your investments in your 401(k) have increased by 8%each year.

The issue with rate of return is that it is difficult to compute exactly by hand. There are some approximations for it, but the computations are still tedious. What is important is how to interpret the rate of return for an investment.

From a computational standpoint, you can, in a way, think of your rate of return as the “interest’’ you have earned by investing your money. It isn’t exactly interest because your investment values can increase or decrease each day, but long term one way to think of rate of return is a measurement of the “interest rate” you’ve earned per year, on average. Consider the following example:

Suppose that Katherine invests $1,000 in a project that guarantees a rate of return of 10%over the next five years. How much will her investment be worth after 5 years?

If we treat that rate of return like interest, we can use compounding to get the investment value after 5 years. After the first year, her investment would be worth \(1,000\times (1.1) = \$1,100.\) After the second year, her investment would be worth \(1,100\times (1.1) = \$1,210\text{.}\) After three, it would be \(1,210 \times (1.1) = \$1,331\text{.}\) After four, it would be \(1,331 \times (1.1) = \$1,464.10.\text{.}\) After five, it would be \(1,464.10 \times (1.1) = \$1,610.51\text{.}\)

We can shortcut this idea. Suppose you invest and amount \(P\) and that the rate of return is \(r\) (as a decimal). Then after a period of \(t\) years, the investment would be worth \(P\times (1+r)^{t}\text{.}\)

You can also go backward. Suppose that you can invest with a rate of return of \(r\) and that you need to have an amount \(A\) after \(t\) years. Then the amount you would need to invest now is \(\displaystyle \frac{A}{(1+r)^{t}}\text{.}\)

In general, rates of return are guaranteed for low-risk savings/investments such as savings accounts, CDs, and bonds. However, for higher-risk investments, rates of return are rarely guaranteed. (Otherwise, they wouldn’t be risky.)

First, to discuss what a “good’’ rate of return for an investment is, we need to understand why people invest money. People risk their money with investments so that they will (hopefully) come away richer afterwards. Become wealthier means having more “buying power.’’ However, this does not simply mean more money.

Consider this - who had more buying power?: Someone with $1,000 in 1900 or someone with $2,000 today? Considering that a house in 1900 would have cost around $3,000-$5,000 and that $2,000 wouldn’t even get you a five minute shopping spree at Sephora, that $1,000 in 1900 had way more buying power. Why? Inflation.

Inflation is a natural byproduct of a money-based economy. Over time, the costs of goods and services increase. In response, wages and salaries need to increase. The effect of inflation is that the value of currency decreases over time. Inflation is often measured as an annual compound rate. Average inflation rates are thrown around. A common one to use in the United States is 3%. Of course, year to year, this can vary quite a bit.

Consider this example. Suppose that 10 years ago an item cost $100. Average inflation has been 3%. How much would the item cost today?

We can calculate this similar to how we found ending values for investments: \(\displaystyle 100 \times (1.03)^{10}= \$134.39\text{.}\)

The reason we need to consider inflation when comparing investment returns can be seen in the following example.

Suppose Regina would like to increase her wealth. She invests $1,000. After five years, her rate of return was 4%. However, inflation was 5%. Did Regina increase her wealth?

At the end of five years, Regina’s investment becomes \(1,000\times (1.04)^{5} = \$1,216.65.\) However, an item that cost $1,000 would cost \(1,000\times (1.05)^{5} = \$1,276.28\text{.}\) So, she could no longer afford an item that she could afford five years prior. So, while Regina did earn money, she lost buying power. So, she actually lost wealth.

Here is what to take away from this exercise: If you earn a rate of return that is less than inflation, you are actually losing wealth. Of course, a small rate of return is better than no return at all.

So, what is a “good” rate of return for an investment? Hopefully, it is clear that a “good’’ rate of return is at least as large as normal inflation. However, you would also like to increase wealth on top of simply keeping up with inflation. There is no universally-accepted number that defines a “good’’ rate of return. However, many people say that 7%has historically been a solid rate of return that can be expected from fairly passive investments in financial markets. So, some people say that 7%of above are “good’’ rates of return.

Because people don’t tend to like whipping out a calculator everytime they talk about investments, some people use the Rule of 72 to think about how investments will grow under a certain rate of return.

The Rule of 72 allows someone to approximate how long it will take an investment to double with a given rate of return. If \(R\%\) is your investment rate of return as a percent, then the approximate time for an investment to double can be calculated as \(\frac{72}{R}\) years. For example, if your rate of return for an investment were 7.2%, then it would take approximately \(\frac{72}{7.2}= 10\) years for your money to double.

10 years may sound like a long time for money to double. The thing is that returns compound. This means that if your money doubles after 7 years, that doubled amount doubles after the next 7 years. That quadrupled amount doubles after the next 7 years. And so on.

Consider the following example.

Suppose you inherit $10,000 from a relative today. You want to invest the money in a mutual fund that earns a rate of return so that your money doubles every ten years. How much would you have when you retire in 50 years?

In ten years, you would have $20,000. After 20, that $20,000 doubles to $40,000. After 30 years, that doubles to $80,000. After 40 years, that doubles to $160,000. After 50 years, that doubles to $320,000.

The Rule of 72 can help you compare rates of return in a different way. For example, suppose you compare rates of return of 5%and 10%. Using the Rule of 72, under a 5%rate of return, it would take \(\frac{72}{5}= 14.4\) years for your investments to double. Under a 10%rate of return, it would take \(\frac{72}{10}= 7.2\) years for your investments to double.