What sum of money invested at 7.5% pa simple interest for 2 years produces twice as much?

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.

Simple Interest: A = P(1+rt)

Ex1: If $1000 is invested now with simple interest of 8% per year. Find the new amount after two years.

Compound Interest:

a) simple : A = P(1+rt)

b) compounded annually, n = 1:

c) compounded semiannually, n =2:

d) compounded quarterly, n = 4:

e) compounded monthly, n =12:

f) compounded daily, n =365:

Continuous Compound Interest:

As the table shows, as n increases in size, the limiting value of A is the special number

e = 2.71828

If the interest is compounded continuously for t years at a rate of r per year, then the compounded amount is given by:

A = P. e rt

Ex3: Suppose that $5000 is deposited in a saving account at the rate of 6% per year. Find the total amount on deposit at the end of 4 years if the interest is compounded continuously. (compare the result with example 2)

Ex4: If $8000 is invested for 6 years at a rate 8% compounded continuously, find the new amount:

Equivalent Value:

• If the interest rate is compounded continuously at an annual interest rate r, then Effective interest rate: = er - 1

• If the interest rate is compounded n times per year at an annual interest rate r, then Effective interest rate = (1+r/n)n - 1

Ex6: An amount is invested at 7.5% per year compounded continuously, what is the effective annual rate?

Ex7: A bank offers an effective rate of 5.41%, what is the nominal rate?

Present Value:

If the interest rate is compounded continuously at an annual rate r, the present value of a A dollars payable t years from now is

P = A. e-rt

4.3: The Growth, Decline Model:

Growth: P(t) = Po . ekt

Decline: P(t) = Po . e-kt

• P(t): the new balance the new population the new price ...
• Po: the original balance the original population the original price
• k: the rate of change the growth, decline rate the interest rate
• t: the time (years, days...)

For compounded continuously, the time T it takes to double the price, population or balance using k as the rate of change, the growth rate or the interest rate is given by:

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Ex9: The growth rate in a certain country is 15% per year. Assuming exponential growth :

Ex10: If an amount of money was doubled in 10 years, find the interest rate of the bank.

Ex11: In 1965 the price of a math book was $16. In 1980 it was $40. Assuming the exponential model :

Ex12: How long does it take money to triple in value at 6.36% compounded daily?

Ex13: A couple want an initial balance to grow to $ 211,700 in 5 years. The interest rate is compounded continuously at 15%. What should be the initial balance?

Ex14: The population of a city was 250,000 in 1970 and 200,000 in 1980 (Decline). Assuming the population is decreasing according to exponential-decay, find the population in 1990.