How many years will it take for a sum of money invested at 5% pa in simple interest to increase its value by 40 %?

The sooner you start to save, the more you'll earn with compound interest.

How compound interest works

Compound interest is the interest you get on:

• the money you initially deposited, called the principal
• the interest you've already earned

For example, if you have a savings account, you'll earn interest on your initial savings and on the interest you've already earned. You get interest on your interest.

This is different to simple interest. Simple interest is paid only on the principal at the end of the period. A term deposit usually earns simple interest.

Save more with compound interest

The power of compounding helps you to save more money. The longer you save, the more interest you earn. So start as soon as you can and save regularly. You'll earn a lot more than if you try to catch up later.

For example, if you put $10,000 into a savings account with 3% interest compounded monthly:

• After five years, you'd have $11,616. You'd earn $1,616 in interest.
• After 10 years you'd have $13,494. You'd earn $3,494 in interest.
• After 20 years you'd have $18,208. You'd earn $8,208 in interest.

Compound interest formula

To calculate compound interest, use the formula:

A = P x (1 + r)n

A = ending balance
P = starting balance (or principal)
r = interest rate per period as a decimal (for example, 2% becomes 0.02)
n = the number of time periods

How to calculate compound interest

To calculate how much $2,000 will earn over two years at an interest rate of 5% per year, compounded monthly:

1. Divide the annual interest rate of 5% by 12 (as interest compounds monthly) = 0.0042

2. Calculate the number of time periods (n) in months you'll be earning interest for (2 years x 12 months per year) = 24

3. Use the compound interest formula

A = $2,000 x (1+ 0.0042)24
A = $2,000 x 1.106
A = $2,211.64

Lorenzo and Sophia compare the compounding effect

Lorenzo and Sophia both decide to invest $10,000 at a 5% interest rate for five years. Sophia earns interest monthly, and Lorenzo earns interest at the end of the five-year term.

After five years:

• Sophia has $12,834.
• Lorenzo has $12,500.

Sophia and Lorenzo both started with the same amount. But Sophia gets $334 more interest than Lorenzo because of the compounding effect. Because Sophia is paid interest each month, the following month she earns interest on interest.

Compound Interest: The future value (FV) of an investment of present value (PV) dollars earning interest at an annual rate of r compounded m times per year for a period of t years is:

FV = PV(1 + r/m)mtor

FV = PV(1 + i)n

where i = r/m is the interest per compounding period and n = mt is the number of compounding periods.

One may solve for the present value PV to obtain:

PV = FV/(1 + r/m)mt

Numerical Example: For 4-year investment of $20,000 earning 8.5% per year, with interest re-invested each month, the future value is

FV = PV(1 + r/m)mt   = 20,000(1 + 0.085/12)(12)(4)   = $28,065.30

Notice that the interest earned is $28,065.30 - $20,000 = $8,065.30 -- considerably more than the corresponding simple interest.

Effective Interest Rate: If money is invested at an annual rate r, compounded m times per year, the effective interest rate is:

reff = (1 + r/m)m - 1.

This is the interest rate that would give the same yield if compounded only once per year. In this context r is also called the nominal rate, and is often denoted as rnom.

Numerical Example: A CD paying 9.8% compounded monthly has a nominal rate of rnom = 0.098, and an effective rate of:

r eff =(1 + rnom /m)m   =   (1 + 0.098/12)12 - 1   =  0.1025.

Thus, we get an effective interest rate of 10.25%, since the compounding makes the CD paying 9.8% compounded monthly really pay 10.25% interest over the course of the year.

Mortgage Payments Components: Let where P = principal, r = interest rate per period, n = number of periods, k = number of payments, R = monthly payment, and D = debt balance after K payments, then

R = P � r / [1 - (1 + r)-n]

D = P � (1 + r)k - R � [(1 + r)k - 1)/r]

Accelerating Mortgage Payments Components: Suppose one decides to pay more than the monthly payment, the question is how many months will it take until the mortgage is paid off? The answer is, the rounded-up, where:

n = log[x / (x � P � r)] / log (1 + r)

Future Value (FV) of an Annuity Components: Ler where R = payment, r = rate of interest, and n = number of payments, then

FV = [ R(1 + r)n - 1 ] / r

Future Value for an Increasing Annuity: It is an increasing annuity is an investment that is earning interest, and into which regular payments of a fixed amount are made. Suppose one makes a payment of R at the end of each compounding period into an investment with a present value of PV, paying interest at an annual rate of r compounded m times per year, then the future value after t years will be

FV = PV(1 + i)n + [ R ( (1 + i)n - 1 ) ] / i where i = r/m is the interest paid each period and n = m � t is the total number of periods.

Numerical Example: You deposit $100 per month into an account that now contains $5,000 and earns 5% interest per year compounded monthly. After 10 years, the amount of money in the account is:

FV = PV(1 + i)n + [ R(1 + i)n - 1 ] / i =
5,000(1+0.05/12)120 + [100(1+0.05/12)120 - 1 ] / (0.05/12) = $23,763.28

Value of a Bond:

V is the sum of the value of the dividends and the final payment.

You may like to perform some sensitivity analysis for the "what-if" scenarios by entering different numerical value(s), to make your "good" strategic decision.

Replace the existing numerical example, with your own case-information, and then click one the Calculate.

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