l>Chapter 4 Handout

## Simple Interest: A = P(1+rt)

P: the principal, the amount investedA: the new balancet: the timer: the rate, (in decimal form)Ex1: If \$1000 is invested now with simple interest of 8% per year. Find the new amount after two years.P = \$1000, t = 2 years, r = 0.08. A = 1000(1+0.08(2)) = 1000(1.16) = 1160

## Compound Interest: P: the principal, amount investedA: the new balancet: the timer: the rate, (in decimal form)n: the number of times it is compounded.

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Ex2: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:P =\$5000, r = 6% , t = 4 yearsa) simple : A = P(1+rt) A = 5000(1+(0.06)(4)) = 5000(1.24) = \$6200b) compounded annually, n = 1: A = 5000(1 + 0.06/1)(1)(4) = 5000(1.06)(4) = \$6312.38c) compounded semiannually, n =2: A = 5000(1 + 0.06/2)(2)(4) = 5000(1.03)(8) = \$6333.85d) compounded quarterly, n = 4: A = 5000(1 + 0.06/4)(4)(4) = 5000(1.015)(16) = \$6344.93e) compounded monthly, n =12: A = 5000(1 + 0.06/12)(12)(4) = 5000(1.005)(48) = \$6352.44f) compounded daily, n =365: A = 5000(1 + 0.06/365)(365)(4) = 5000(1.00016)(1460) = \$6356.12

## Continuous Compound Interest:

Continuous compounding means compound every instant, consider investment of 1\$ for 1 year at 100% interest rate. If the interest rate is compounded n times per year, the compounded amount as we saw before is given by: A = P(1+ r/n)ntthe following table shows the compound interest that results as the number of compounding periods increases:P = \$1; r = 100% = 1; t = 1 yearCompoundedNumber of periods per yearCompound amountannually1(1+1/1)1 = \$2monthly12(1+1/12)12 = \$2.6130daily360(1+1/360)360 = \$2.7145hourly8640(1+1/8640)8640 = \$2.71812each minute518,400(1+1/518,400)518,400= \$2.71827As the table shows, as n increases in size, the limiting value of A is the special number e = 2.71828If 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 rtEx3: 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)P =\$5000, r = 6% , t = 4 yearsA = 5000.e(0.06)(4) = 5000.(1.27125) = \$6356.24Ex4: If \$8000 is invested for 6 years at a rate 8% compounded continuously, find the new amount:P = \$8000, r = 0.08, t = 6 years.A = 8000.e(0.08)(6) = 8000.(1.6160740) = \$12,928.60

## Equivalent Value:

When a bank offers you an annual interest rate of 6% compounded continuously, they are really paying you more than 6%. Because of compounding, the 6% is in fact a yield of 6.18% for the year. To see this, consider investing \$1 at 6% per year compounded continuously for 1 year. The total return is:A = Pert = 1.e(0.06)(1) = \$1.0618If we subtract from \$1.618 the \$1 we invested, the return is \$0.618, which is 6.18% of the amount invested.

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The 6% annual interest rate of this example is called the nominal rate The 6.18% is called the effective rate.If the interest rate is compounded continuously at an annual interest rate r, then Effective interest rate: = er - 1If the interest rate is compounded n times per year at an annual interest rate r, thenEffective interest rate = (1+r/n)n - 1Ex5: Which yield better return: a) 9% compounded daily or b) 9.1% compounded monthly?a) effective rate = (1+0.09/365)365 - 1 = 0.094162b) effective rate = (1+0.091/12)12 - 1 = 0.094893the second rate is better.Ex6: An amount is invested at 7.5% per year compounded continuously, what is the effective annual rate?the effective rate = er - 1 = e 0.075 - 1 = 1.0079 - 1 = 0.0779 = 7.79%Ex7: A bank offers an effective rate of 5.41%, what is the nominal rate?er - 1 = 0.0541er = 1.0541r = ln 1.0541 then r = 0.0527 or 5.27%

## Present Value:

If the interest rate is compounded n times per year at an annual rate r, the present value of a A dollars payable t years from now is:

## 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

Ex8: how much should you invest now at annual rate of 8% so that your balance 20 years from now will be \$10,000 if the interest is compounded a) quarterly: P = 10,000.(1+0.08/4)-(4)(20)= \$ 2,051.10b) continuously: P = 10,000.e-(0.08)(20) = \$2.018.97