There is a close relationship between math and finance. In fact, at an early age, you probably learned some basic arithmetic using pennies, nickels, dimes, quarters, and dollars. Once you get to a point where you want to invest some money, be it conservatively or aggressively, it’s helpful to know some basic financial terms and the math behind them.

Let’s get started!

## Compound Interest

Compound interest is interest that is paid on the initial principal as well as the accumulated interest from previous periods.

From an investing standpoint, compound interest helps your savings account grow faster. (On the other hand, if you have a loan where the interest is compounded, this can make paying off the loan a bit more difficult!)

Formula for Compound InterestA(t) = P(1 + (r/n))^{nt}where:A(t) =the amount after t yearsP=the principal; the original amount investedr =the interest rate, expressed as a decimalt =time, in yearsn=the number of times per year that the interest is compounded |

(Source: math.libretexts.org)

### Example

Fiona invests $1,000 in each of two different banks. Bank ABC compounds interest quarterly and offers a 3% interest rate. Bank XYZ compounds interest weekly and offers a 2.8% interest rate. Fiona wants to know, how much money will she have in each bank at the end of 5 years?

**Solution**

Use the formula for compound interest to determine Fiona’s account balance in each of the banks at the end of 5 years.

**Bank ABC:**

Let’s identify our key variables. The principal *P* is $1,000, the rate *r *expressed as a decimal is 0.03, the time *t* is 5, and *n* is 4 since the interest is compounded quarterly. Plugging in these values to our formula, we obtain:

*A*(*t*) =1,000(1 + 0.03/4)^{(4)(5)}

=1,000(1.0075)^{(20)}

= 1,161.18

At the end of five years, Fiona’s account balance at Bank ABC will be $1,161.18.

**Bank XYZ:**

Again, let’s identify our variables. The principal *P* is $1,000, *r *is 0.028, *t* is 5, and *n* is 52 since the interest is compounded weekly. Plugging in these values to our formula, we obtain:

*A*(*t*) =1,000(1 + 0.028/52)^{(52)(5)}

=1,000(1.000538)^{(260)}

= 1,150.09

At the end of five years, Fiona’s account balance at Bank XYZ will be $1,150.09.

Bank ABC offered a higher interest rate but compounded the interest less frequently than Bank XYZ. Even though Bank XYZ compounded the interest every week, it still resulted in a lower amount after 5 years because of the lower interest rate. ◾

## Annuities

In our example above, Fiona invested $1,000 in each of two different banks and left the money there for 2 years without making any additional deposits. In practice, this is not very common.

Frequently, people make additional deposits into an account, often at regular intervals such as monthly, quarterly, or annually. An **ordinary** **annuity** is a series of regular payments made at the end of a month on a regular basis. (Source: investopedia.com)

Let’s look at annuities a bit further and see how they relate to math. At some point in your math career, you learned about finding the sum geometric series. In a geometric series, each term is found by multiplying the preceding term by a common ratio. You also learned how to find the sum of a geometric series:

The sum of a geometric series is:a(x ^{n+1} – 1) / (x – 1)wheren is the number of termsa is the first termx is the common ratio. |

We can use this formula to derive another formula that will give us the **future value** of an annuity.

### Example

At the end of each month, James makes a deposit of $200 into an account that pays 6% monthly. At the end of 5 years, how much money will be in this account?

**Solution**

Over 5 years, there will be 60 payments made into the account. Since the first payment is made at the end of the first month, it will earn interest for 59 months.

The first payment accumulates to:

200(1+0.06/12)^{59}

The second payment will be deposited at the end of the second month, so it will earn interest for 58 months.

The second payment accumulates to:

200(1+0.06/12)^{58}

Similarly, the third payment accumulates to:

200(1+0.06/12)^{57}

This process continues until the final deposit is made at the end of the 60th month.

The second to last deposit which is made at the end of the 59th month, only earns interest for that final month.

The second to last payment accumulates to:

200(1+0.06/12)^{1}

The last payment won’t accumulate any interest, so the last term in the series is the value 200, the amount of the deposit.

Let’s put all of this together and write out our expressions as a geometric series:

200(1+0.06/12)^{59}+200(1+0.06/12)^{58}+200(1+0.06/12)^{57}+ … +200(1+0.06/12)^{1}+200

We’ll write the series in reverse order so that the exponents are increasing rather than decreasing.

200+200(1+0.06/12)^{1}+200(1+0.06/12)^{2}+ … +200(1+0.06/12)^{59}

This is a geometric series with *a* = 200, *x *= 1+0.0612, and *n* = 59. Using the sum of a geometric series formula, we obtain:

a(x^{n+1} – 1) / (x – 1)

200((1 + 0.06/12)^{60} – 1) / ((1 + 0.06/12 – 1)

200((1.005)^{60} – 1) / (1.005 – 1)

$13,954

So, the** future value** of the annuity is $13,954. Hopefully, you noticed the pattern with this calculation. We use this idea to derive the formula for the future value of an annuity.

Future Value = (m(1 + r/n)^{nt}-1) / (r/n)wherem is the amount of the regular depositn is the number of times per year the interest is compounded r is the interest rate expressed as a decimalt is the number of years. |

(Source: libretexts.org)

**Note:** The annuity we just discussed is *not* the same as the financial investment known as an annuity, which provides a guaranteed income throughout a person’s retirement.

## Price to Earnings Ratio

There are two key components that influence how well a stock performs: the company’s profitability and how the investors value the company.

While a company earnings report can accurately provide information on a company’s profitability, it’s harder to gauge how investors view the company and its overall performance. This is where the price-to-earnings ratio (P/E) can help us.

P/E ratio =(market price of stock)/(earnings per share) |

The price-to-earnings ratio is the market price of a stock divided by the earnings per share. This simple ratio can help potential investors determine if the stock is overvalued or undervalued. If you compare P/E ratios among similar companies, you could see how a particular stock stacks up against similar stocks.

The average P/E ratio for the S&P 500 has historically been between 13 and 15. Suppose you’re interested in purchasing stock from Company ABC and it has a P/E ratio of 26.

Company ABC is trading at 26 times the earnings. The high number indicates that investors think the company will outperform the market. It’s a bit more complicated than that because sometimes a high P/E ratio causes cautious investors to think that the company’s stock is overvalued. (Source: schwab.com)

## The Rule of 72

While computers and calculators do an amazing job of calculating complex formulas with a great deal of accuracy, sometimes it’s helpful to be able to mentally estimate financial formulas. This is where the rule of 72 comes in handy.

Suppose you want to know how long it will take for you to double an investment based on a given annual rate of return. The rule of 72 says that if we divide 72 by the expected rate of return, that value will be the number of years it will take to double our investment.

Rule of 72number of years to double investment: 72/r where r is the rate of return, written as a percent. |

Let’s look at an example.

### Example

Kaleb wants to make a long term investment of $10,000. One investment option has an annual rate of return or 5%. How long will it take for Kaleb to double his investment?

**Solution**

Using the rule of 72, we get:

Number of years to double the investment: 72/5= 14.4, rounded down to 14.

So, it will take about 14 years for Kaleb’s investment to equal $20,000.

## All about dividends

A **dividend** is a portion of the company profits that is returned to the shareholders. Dividends are a way the company can reward investors for owning shares and it gives the company a way to distribute excess cash they don’t intend to reinvest in the company.

**What is dividend yield? **The dividend yield is equal to the dividend per share divided by the price per share. This value is usually expressed as a decimal.

Dividend Yield = (Annual Dividends per Share)/(Price per Share) |

The dividend yield for a particular company is constantly changing because the price of the stock is constantly changing and the amount of dividend that a company pays out can change over time. (Source: investopedia.com)

### Example

Mary wants to invest $100,000 in a dividend paying healthcare stock and is trying to decide between two companies. Company A trades at $50 and pays an annual dividend of $2 per share. Company B trades at $25 and also pays an annual dividend of $2 per year. If Mary’s goal is to increase the value of her portfolio over time, which Company should she invest in?

**Solution**

To compare the stocks, let’s calculate the dividend yield of each.

Dividend Yield of Company A = 250= 4%

Dividend Yield of Company B = 225= 8%

What does the dividend yield tell us about the two companies?

If a company’s dividend yield has steadily increased over time, that can mean that the company has been increasing their dividend or the share price has been decreasing, or both. For the scenario above, assuming the two companies are equivalent in other ways, Mary would likely choose to invest in Company B since its dividend yield is twice that of Company A. ◾

We’ve looked at a few ways that math can help us with finances – there are so many more ways that I didn’t list here!!! Math is a powerful tool in so many areas of life. It’s always great to know that there are real life examples out there where math is used constantly!

You might also be interested to learn how actuaries use math in their careers.

You can also learn about careers for math majors here.

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**About the author:**

Jean-Marie Gard is an independent math teacher and tutor based in Massachusetts. You can get in touch with Jean-Marie at https://testpreptoday.com/.