1.4 Financial applications of geometric sequences

Money placed in a savings account earns interest, and that interest then earns interest of its own. Each year the balance is multiplied by the same factor, so the balances form a geometric sequence. Cars, computers and machinery lose value in the same way, by a fixed percentage of their value each year. Geometric sequences therefore describe how savings grow, how loans build up and how assets lose value, and they are the basis of the calculations that banks, businesses and governments make every day.

Simple and compound interest

When money is invested, the bank pays interest as a percentage of the amount in the account. The amount invested at the start is the present value $PV$, and the amount it has grown to at a later date is the future value $FV$. There are two ways to calculate the interest.

With simple interest, the interest is always a percentage of the original amount, so the same interest is added every year and the balances form an arithmetic sequence (see 1.2). With compound interest, the interest each year is a percentage of the current balance, which includes the interest already earned, so the balance is multiplied by the same factor every year and the balances form a geometric sequence.

The difference between the two methods is small at first but grows quickly. Simple interest gives straight-line growth; compound interest gives exponential growth, because each year's interest is larger than the last.

The compound interest formula

Interest added once a year

If the interest rate is $r\%$ per year, then each year the balance is multiplied by $1 + \frac{r}{100}$. After $n$ years the present value has been multiplied by this factor $n$ times.

The balances form a geometric sequence with common ratio $1 + \frac{r}{100}$. The power is $n$, not $n - 1$, because the present value is the balance at time zero and the value after $n$ years has had $n$ multiplications applied.

Interest added several times a year

Banks often add interest more often than once a year. The interest rate is still quoted per year (the nominal annual rate), but it is shared equally between the compounding periods. With compounding periods of one month, for example, an annual rate of $6\%$ means $\frac{6}{12} = 0.5\%$ is added every month, and in $n$ years there are $12n$ additions.

With $k$ periods per year, the rate per period is $\frac{r}{k}\%$ and there are $kn$ periods in $n$ years. This gives the general compound interest formula.

The effect of compounding frequency

For the same nominal annual rate, more frequent compounding gives a slightly larger future value, because interest starts earning interest sooner. The table shows 10 000 dollars invested for one year at a nominal rate of $6\%$.

The effective annual rate is the actual percentage increase over a whole year, $\left(1 + \frac{r}{100k}\right)^{k} - 1$, written as a percentage. The gains from more frequent compounding get smaller and smaller: the values approach a limit of $10\,618.37$ dollars. That limit, continuous compounding, leads to the number $e$, introduced in 1.5.

Finding the present value, the rate or the time

The compound interest formula links five quantities: $FV$, $PV$, $r$, $k$ and $n$. When all but one are known, the remaining one can be found. The present value comes from rearranging, the rate from taking a root, and the time from logarithms (see 1.7) or a GDC.

Finding the present value

Rearranging the formula gives

which answers questions such as "how much must be invested now to reach a target later?"

Finding the interest rate

When $PV$, $FV$ and $n$ are known, divide to isolate the power, then take the $n$th root.

Finding the time

When the time is unknown it appears in the exponent. A GDC solver or table finds it directly; with logarithms, $a^x = b$ gives $x = \frac{\log b}{\log a}$. The answer usually needs interpreting: interest is only added at the end of each compounding period, so the time is rounded up to a whole number of periods.

The GDC finance solver

A graphic display calculator (GDC) has a finance solver, often called the TVM (time value of money) solver. It works with the same formula, and it can find any one of the variables when the others are entered. The names vary slightly between calculators, but the variables are the same.

The solver uses a sign convention for the direction of money. Money that leaves you, such as a deposit into an account, is entered as negative; money that comes back to you is positive. So for an investment, $PV$ is negative and $FV$ is positive. For a loan, the borrower receives $PV$ (positive) and repays $FV$ (negative).

Depreciation

Most assets lose value over time. A new car is worth less after a year of use, and a computer bought today will be worth very little in five years. This loss of value is depreciation. When the value falls by the same percentage each year, the values form a geometric sequence with common ratio between 0 and 1.

The formula has the same structure as compound interest, with the factor $1 - \frac{r}{100}$ in place of $1 + \frac{r}{100}$. A depreciation rate of $15\%$ gives a ratio of $0.85$, so each year the asset keeps $85\%$ of its value from the start of that year. The value approaches zero but never reaches it.

The same equation can be used the other way round. When the starting and final values are known, the annual rate of depreciation is found by taking a root, exactly as for an interest rate.

Comparing investments

To decide between investments, calculate what each one is worth at the same future date, or compare their effective annual rates. A higher nominal rate does not always win: the compounding frequency, the length of time and any fees all affect the outcome.

Inflation and the real value of money

Over time, the prices of most goods and services rise. This general rise in prices is inflation, measured as an annual percentage. A sum of money that stays the same buys less each year, so its real value falls. Inflation is itself a form of compound growth: prices are multiplied by the same factor each year.

Statistical agencies measure inflation with a price index, such as a consumer price index (CPI), which tracks the cost of a typical basket of goods and services. The table shows the annual average of the US consumer price index for three recent years.

Source: US Bureau of Labor Statistics, CPI-U annual averages (1982–84 = 100), 2021–2023.

The price rises from 2021 to 2023 compound: $\frac{304.702}{270.970} = 1.1245$, so prices rose by about $12.4\%$ over the two years, not $8.0\% + 4.1\% = 12.1\%$. To keep the same buying power, a salary of 60 000 dollars in 2021 would have needed to rise to $60\,000 \times 1.1245 \approx 67\,470$ dollars by 2023.

Real value after inflation

To find the real value of a future amount in today's money, divide it by the factor by which prices have risen. If inflation is $i\%$ per year for $n$ years, prices are multiplied by $\left(1 + \frac{i}{100}\right)^n$, so

An investment only increases in real value if its interest rate is higher than the inflation rate.

Exam-style questions on financial applications

Financial questions usually appear on the calculator paper, where the finance solver can be used, but simple compound interest and depreciation calculations with friendly numbers can appear without a calculator.

On the calculator paper the same ideas are combined in longer questions, often with a finance solver, an effective rate and an inflation adjustment in one context.

Self-check

• To do:

  I can explain why simple interest gives an arithmetic sequence and compound interest a geometric one.

• To do:

  I can use $FV = PV \times \left(1 + \frac{r}{100k}\right)^{kn}$ for any compounding frequency.

• To do:

  I can find the present value, the rate or the time, and round the time correctly in context.

• To do:

  I can use a GDC finance solver, entering the values with the correct signs and writing them down as working.

• To do:

  I can calculate depreciation with $V_n = V_0\left(1 - \frac{r}{100}\right)^n$

• To do:

  I can compare investments using future values or effective annual rates.

• To do:

  I can find the real value of money after inflation and explain why it differs from the face value.

The following table shows some values of $A(n)$, correct to two decimal places.

1. (a)

   Find the value of $p$, the value of $q$ and the value of $r$.

   [2]2 marks

   Markscheme (a)

   attempt to substitute at least one of $n = 12, 365, 8760$ into $A(n)$ (M1)

   $p = 1051.16$, $q = 1051.27$, $r = 1051.27$ A1

2. (b)

   Use the table to write down an estimate of $\displaystyle\lim_{n \to \infty} A(n)$, correct to two decimal places.

   [1]1 mark

   Markscheme (b)

   $1051.27$ A1

3. (c)

   A second account pays a nominal annual interest rate of $5.13\%$, compounded once per year. Kofi claims that if the first account compounded interest often enough, it would give him more money after one year than the second account.

   Use your answer to part (b) to determine whether Kofi is correct.

   [2]2 marks

   Markscheme (c)

   second account gives $1000 \times 1.0513 = 1051.30$ (dollars) A1

   $A(n)$ approaches $1051.27 < 1051.30$

Note: Do not award R1 without a comparison of $1051.30$ with their limit.