## Chapter 4 Class Handout

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

**P:**the principal, the amount invested

**A:**the new balance

**t:**the time

**r:**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 invested

**A:**the new balance

**t:**the time

**r:**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)nt**

**the 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 = 2.71828**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)P =$5000, r = 6% , t = 4 yearsA = 5000.e(0.06)(4) = 5000.(1.27125) = $6356.24

**Ex4:**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.See more: Does John Frieda Test On Animals, Is John Frieda Cruelty Free And Vegan

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 - 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**

**Ex5:**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