Growth, decay and compound interest

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Simple vs Compound Interest

Simple interest is calculated only on the original principal each period.
Compound interest is calculated on the principal plus any interest already added.
Compound interest grows faster than simple interest over time because interest earns interest.
The extra growth from compounding is called anatocism in older texts.

The Compound Interest Formula

The final amount after compound interest is given by A = P\left(1 + \frac{r}{n}\right)^nt.
In this formula: A = final amount, P = principal, r = annual interest rate (as a decimal), n = number of compounding periods per year, t = time in years.
The total compound interest earned is I = A - P.
The multiplier for one period is 1 + \frac{r}{n}; raising it to the power nt applies it repeatedly.

Compounding Frequency

Compounding frequency is how many times per year interest is added to the principal.
Common frequencies: annually (n = 1), half-yearly (n = 2), quarterly (n = 4), monthly (n = 12), weekly (n = 52), daily (n = 365).
More frequent compounding gives a slightly higher final amount for the same annual rate.
For monthly compounding at an annual rate r, the monthly rate is \frac{r}{12}.

Growth and Decay Multipliers

For a percentage increase of p%, the multiplier is 1 + \frac{p}{100}.
For a percentage decrease of p%, the multiplier is 1 - \frac{p}{100}.
Repeated growth or decay over t years uses the multiplier raised to the power t.
Example: a 5% annual increase for 3 years gives a total multiplier of 1.05³.

Depreciation

Depreciation is a repeated percentage decrease in value.
If an item loses p% of its value each year, its value after t years is V = P(1 - \frac{p}{100})^t.
Depreciation is a form of exponential decay.
Example: a car worth £12000 losing 15% per year is worth 12000 \times 0.85^t after t years.

Population Growth and Decay

Populations can grow or decay by a fixed percentage each year.
For a population growing by r% per year, after t years the population is P(1 + \frac{r}{100})^t.
For a population decaying by r% per year, after t years the population is P(1 - \frac{r}{100})^t.
These are examples of exponential growth and exponential decay.

Solving Problems with Iteration

To find how many years it takes to reach a target value, substitute different values of t until the amount passes the target.
This trial-and-improvement method is called iteration.
For growth, the amount increases with t; for decay, it decreases with t.
Example: to find when £500 at 4% compound interest exceeds £600, test t = 1, 2, 3, … until A > 600.

The Rule of 72

The Rule of 72 estimates the number of years for an investment to double.
Years to double ≈ \frac{72}{\text{interest rate as a percentage}}.
Example: at 6% compound interest, money doubles in about 72 ÷ 6 = 12 years.
This rule was described by Luca Pacioli in 1494.

Annual Equivalent Rate (AER)

The annual equivalent rate (AER) shows the effective annual interest when compounding happens more than once a year.
AER allows fair comparison between accounts with different compounding frequencies.
AER is the total interest earned in one year divided by the principal.
Other names include effective annual rate, effective interest rate, and annual percentage yield.

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1.In the compound interest formula A = P\left(1 + \frac{r}{n}\right)^nt, what does n represent?
Easy
- AThe number of times interest is compounded per year
- BThe number of years the money is invested
- CThe annual interest rate as a decimal
- DThe final amount of money
2.Which multiplier should be used to increase an amount by 6% each year for 3 years?
Medium
- A1.06³
- B1.6³
- C1.06 × 3
- D0.06³
3.A car is worth £12,000. It depreciates by 15% each year. Which calculation gives its value after 2 years?
Easy
- A12000 × 0.85²
- B12000 × 0.15²
- C12000 × 1.15²
- D12000 × (1 − 0.15 × 2)
4.£500 is invested at 4% per year compound interest. What is the total amount after 3 years, to the nearest penny?
Medium
- A£562.43
- B£560.00
- C£562.00
- D£624.32
5.£800 is invested at a nominal annual rate of 6% compounded monthly. What is the amount after 2 years, to the nearest penny?
Medium
- A£901.60
- B£898.88
- C£896.00
- D£904.32
6.Which of the following statements about compound interest are true? (Select all that apply.)
Medium
- ACompound interest is interest earned on both the principal and previously accumulated interest.
- BSimple interest is always greater than compound interest for the same rate and time.
- CThe annual equivalent rate (AER) allows fair comparison of different compound interest products.
- DIncreasing the compounding frequency always decreases the final amount.
- ECompound interest can be calculated using a multiplier raised to a power.
7.Compound interest is calculated only on the original principal sum, not on any interest already earned.
Easy
True or false?
8.Match each term with its correct meaning.
Medium
- Principal
- Compounding frequency
- Annual equivalent rate (AER)
- The original sum of money invested or borrowed
- The number of times per year that interest is added
- The effective annual rate allowing fair comparison of products

複習筆記

## Growth, decay and compound interest **Compound interest** is interest earned on both the original **principal** and on interest already added. **Simple interest** pays interest only on the original principal each period. ### The multiplier method For a percentage change of r% per period, the **multiplier** is: - increase: 1 + r/100 - decrease (decay/depreciation): 1 − r/100 Repeated over t periods, raise the multiplier to the power t: \(A = P\left(1 + \frac{r}{n}\right)^{tn}\) where A = final amount, P = original principal sum, r = nominal annual interest rate, n = compounding frequency (1: annually, 12: monthly, 52: weekly, 365: daily), t = overall length of time (same units as r, usually years). Compound interest generated: I = A − P. ### Simple vs compound Simple interest adds the same amount each period; compound interest grows faster because each period's interest is added to the principal. ### Depreciation and decay For a value falling by r% per year, use multiplier (1 − r/100)^t. This models depreciation, population decay and similar decreases. ### Growth For a population or value rising by r% per year, use multiplier (1 + r/100)^t. ### Compounding frequency - Yearly: n = 1 - Half-yearly: n = 2 - Quarterly: n = 4 - Monthly: n = 12 - Weekly: n = 52 - Daily: n = 365 More frequent compounding gives a higher final amount for the same nominal rate. ### Annual equivalent rate (AER) The effective annual rate is the total accumulated interest payable by the end of one year divided by the principal sum. It lets you compare products fairly. ### Finding how many years (iteration) To find how long it takes to reach a target value, try values of t one at a time until the amount passes the target. This is **iteration**. ### Rule of 72 To estimate the number of years for an investment to double, divide 72 by the interest rate (as a number, e.g. 6 for 6%). ### Quick checks - Interest rate as a decimal: r/100. - Multiplier for 5% increase: 1.05. - Multiplier for 5% decrease: 0.95. - After t years at 5% compound: multiply by 1.05^t.

Compound interest: Interest accumulated from a principal sum and previously accumulated interest; the result of reinvesting or retaining interest that would otherwise be paid out.
Simple interest: Interest where previously accumulated interest is not added to the principal amount of the current period.
Principal: The original sum of money (P) on which interest is calculated.
Compounding frequency: The number of times per given unit of time the accumulated interest is capitalized, on a regular basis (n).
Annual equivalent rate (AER): The total accumulated interest that would be payable up to the end of one year, divided by the principal sum; used to compare financial products fairly.
Depreciation: A decrease in value over time, modelled with a multiplier less than 1, e.g. (1 − r/100)^t.
Rule of 72: To find the number of years for an investment at compound interest to double, divide the interest rate into 72.
Iteration: Repeating a calculation with different values (e.g. trying different numbers of years) to find how many years it takes to reach a target value.

字卡

What is compound interest?

Interest calculated on the original principal plus any interest already added. Each period's interest is added to the amount before the next period's interest is worked out.

How does simple interest differ from compound interest?

With simple interest, previously accumulated interest is never added to the principal. With compound interest, accumulated interest is added to the principal each period, so later interest is earned on it too.

What does the compounding frequency tell you?

It is the number of times per unit of time that accumulated interest is added to the principal. It can be yearly, half-yearly, quarterly, monthly, weekly or daily.

For monthly compounding with an annual rate, what is the compounding frequency?

n = 12, and time is measured in months.

State the compound interest formula and name each symbol.

A = P\left(1 + \frac{r}{n}\right)^tn where A = final amount, P = original principal, r = nominal annual interest rate, n = compounding frequency, t = time in years.

How do you find the total compound interest earned?

Subtract the principal from the final amount: I = A − P.

What is the multiplier for a 5% increase?

1.05 (since 100% + 5% = 105%, and 105 ÷ 100 = 1.05).

What is the multiplier for a 12% decrease?

0.88 (since 100% − 12% = 88%, and 88 ÷ 100 = 0.88).

How do you calculate repeated percentage change over several years?

Raise the multiplier to the power of the number of years, then multiply by the starting amount. For example, after 3 years at 5%: amount × 1.05³.

What is depreciation, and how is it calculated?

Depreciation is a repeated percentage decrease in value. Use a multiplier below 1 raised to the number of years, e.g. a 15% yearly fall gives value × 0.85^t.

How is compound interest used to model population growth?

Treat the yearly percentage increase as a multiplier above 1 and raise it to the number of years: population × (1 + rate)^t.

How is exponential decay modelled with a multiplier?

Use a multiplier between 0 and 1 raised to the power of the number of periods, e.g. a 20% yearly decrease gives amount × 0.8^t.

How can iteration find how many years it takes to reach a target value?

Apply the multiplier year by year, recording the amount after each year, until the amount first reaches or passes the target. The number of steps is the number of years.

What is the annual equivalent rate (AER)?

The total interest payable by the end of one year divided by the principal, expressed as an annual rate. It lets you compare products with different compounding frequencies fairly.

Why is the yearly compounded rate higher than a disclosed rate paid twice yearly?

Interest paid after six months is added to the principal, so the second half-year's interest is earned on a larger amount.

State the Rule of 72.

To estimate the number of years for an investment at compound interest to double, divide the interest rate into 72.

Which mathematician discovered the constant e while studying compound interest, and when?

Jacob Bernoulli, in 1683.

Who wrote Arithmeticall Questions (1613), a landmark book devoted to compound interest?

Richard Witt, a London mathematical practitioner. The book contained 124 worked examples and tables based on 10% interest.

What is continuous compounding?

Compounding at ever-increasing frequency until a limit is approached, so interest is added continuously rather than at set intervals. It is often used in valuing financial derivatives.

An amount of £500 grows by 4% each year. Write the expression for its value after 6 years.

500 × 1.04⁶.

A car worth £12000 depreciates by 18% each year. Write the expression for its value after 4 years.

12000 × 0.82⁴.

A population of 2500 increases by 3% per year. Write the expression for the population after 10 years.

2500 × 1.03¹⁰.

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