Section 5.1: Simple and Compound Interest. Simple Interest Simple Interest: Used to calculate interest on loans…often of one year or less. Formula: I.

Section 5.1: Section 5.1: Simple and Compound InterestSimple and Compound Interest

Simple InterestSimple InterestSimple Interest: Used to calculate interest on

loans…often of one year or less.

Formula: I = Prt I : interest earned (or owed) P : principal invested (or borrowed) r : annual interest rate t : time in years

Example 1Example 1

To buy furniture for a new apartment, Jennifer Wall borrowed $5,000 at 6.5% simple interest for 11 months.

a. How much interest will she pay?

Simple interest: I = Prt

I = ?

Example 1Example 1




I = ? P = $5,000 r = .065 t =11/12

Example 1Example 1




I = ? P = $5,000 r = .065 t =11/12

I = Prt = (5000)(0.065)(11/12) =

a. How much interest will she pay? a. How much interest will she pay? Simple interest: Simple interest: II = = PrtPrt

II = ? = ? PP = $5,000 = $5,000 rr = .065 = .065 tt =11/12 =11/12

II = = PrtPrt = (5000)(0.065)(11/12) = $______ = (5000)(0.065)(11/12) = $______

Example 1Example 1




I = ? P = $5,000 r = .065 t =11/12

I = Prt = (5000)(0.065)(11/12) = $297.92

Example 1Example 1


b. What is the total amount to be repaid?

Example 1Example 1



Amount to Repay = Principal + Interest

Example 1Example 1




= 5000 + 297.92

Example 1Example 1




= 5000 + 297.92 = $ 5,297.92

Example 1Example 1




= 5000 + 297.92 = $ 5,297.92

Notice here that we really have:

A = P + I … or A = P + Prt = P(1 + rt)

Example 1Example 1




= 5000 + 297.92 = $ 5,297.92


A = P + I … or A = P + Prt = P(1 + rt)

So, if you want a direct formula for A with simple interest, useA = P(1 + rt)

Example 1Example 1




= 5000 + 297.92 = $ 5,297.92


A = P + I … or A = P + Prt = P(1 + rt)

So, if you want a direct formula for A with simple interest, useA = P(1 + rt)

and, of course if you only want I, then useI = Prt

Alabama will beat Michigan Saturday Alabama will beat Michigan Saturday in Dallas.in Dallas.

1. Yes2. No

Find simple interestFind simple interest

$10,502 at 4.2% for 10 months

A. $370.66B. $367.57C. $404.33D. $330.81

Compound InterestCompound InterestCompound Interest: more commonly used than simple interest.With compound interest, the interest itself earns interest.

Formula:

tm

m

rPA

1


Formula:

Where A is the compound amount (includes principal and interest)

tm

m

rPA

1


Formula:

Where A is the compound amount (includes principal and interest) P is the initial investment

tm

m

rPA

1


Formula:

Where A is the compound amount (includes principal and interest) P is the initial investment r is the annual percentage rate

tm

m

rPA

1


Formula:

Where A is the compound amount (includes principal and interest) P is the initial investment r is the annual percentage rate m is the number of compounding periods per year:

tm

m

rPA

1


Formula:


Compounded annually, m = 1 Compounded semiannually, m = 2 Compounded quarterly, m = 4, etc.

tm

m

rPA

1


Formula:



t is the number of years

tm

m

rPA

1


Formula:



t is the number of years n = mt is the total # of compounding periods over all t years i = r/m is the interest rate per compounding period

ntm

iPm

rPA

11

Example 2 Example 2 Suppose that $22,000 is invested at 5.5% interest. Find the amount of money in the account after 5 years if the interest is compounded annually.


ntm

iPm

rPA

11


A = ?; P = 22,000; r = 0.055; m = 1; t = 5

ntm

iPm

rPA

11


A = ?; P = 22,000; r = 0.055; m = 1; t = 5

ntm

iPm

rPA

11

)5)(1(

1

055.0122000

A


A = ?; P = 22,000; r = 0.055; m = 1; t = 5

ntm

iPm

rPA

11

12.753,28$1

055.0122000

)5)(1(

A


A = ?; P = 22,000; r = 0.055; m = 1; t = 5

ntm

iPm

rPA

11

12.753,28$1

055.0122000

)5)(1(

A

Find the amount of interest earned.


A = ?; P = 22,000; r = 0.055; m = 1; t = 5

ntm

iPm

rPA

11

12.753,28$1

055.0122000

)5)(1(

A

Find the amount of interest earned.Compound Amount (A) = Principal (P) + Interest (I), so I = A – P


A = ?; P = 22,000; r = 0.055; m = 1; t = 5

ntm

iPm

rPA

11

12.753,28$1

055.0122000

)5)(1(

A

Find the amount of interest earned.Compound Amount (A) = Principal (P) + Interest (I), so I = A – P

= 28,753.12 – 22,000 = $ 6,753.12

Example 3 Example 3 If $22,000 is invested at 5.5% interest. Find the amount of money in the account after 5 years if interest is compounded monthly. (Round answer to nearest dollar.)


ntm

iPm

rPA

11


A = ?; P = 22,000; r = 0.055; m = 12; t = 5

ntm

iPm

rPA

11


A = ?; P = 22,000; r = 0.055; m = 12; t = 5

ntm

iPm

rPA

11

)5)(12(

12

055.0122000

A


A = ?; P = 22,000; r = 0.055; m = 12; t = 5

ntm

iPm

rPA

11

945,28$12

055.0122000

)5)(12(

A

to the nearest DOLLAR


A = ?; P = 22,000; r = 0.055; m = 12; t = 5

ntm

iPm

rPA

11

945,28$12

055.0122000

)5)(12(

A

to the nearest DOLLARFind the amount of interest earned.


A = ?; P = 22,000; r = 0.055; m = 12; t = 5

ntm

iPm

rPA

11

945,28$12

055.0122000

)5)(12(

A


Compound Amount (A) = Principal (P) + Interest (I), so I = A – P


A = ?; P = 22,000; r = 0.055; m = 12; t = 5

ntm

iPm

rPA

11

945,28$12

055.0122000

)5)(12(

A


Compound Amount (A) = Principal (P) + Interest (I), so I = A – P

= 28,945 – 22,000 = $ 6,945

Find the compound Find the compound amountamount

A. $10,444.87B. $10,433.47C. $10,350.00D. $9,695.56

$9000 At 3% compounded semiannually for 5 years

Example 4: Effective RateExample 4: Effective RateThe Effective Annual Rate (EAR) is the rate that

would be paid using simple interest in order to be equivalent to a stated compounded rate.

Example 4: Effective RateExample 4: Effective RateThe Effective Annual Rate (EAR) is the rate that

would be paid using simple interest in order to be equivalent to a stated compounded rate.

Financial institutions are usually required by law to provide the effective rate so that consumers can easily compare ‘apples to apples’.

Example 4: Effective RateExample 4: Effective Rate

Ex. Find the effective annual rate corresponding to a rate of 8% compounded quarterly.

The Effective Annual Rate (EAR) is the rate that would be paid using simple interest in order to be equivalent to a stated compounded rate.






This question is easy to answer if we notice a simplifying fact: The interest rate doesn’t change based on the principal or the amount of time.





This question is easy to answer if we notice a simplifying fact: The interest rate doesn’t change based on the principal or the amount of time.So, in our formulas, we can just calculate using $1 for 1 year.





This question is easy to answer if we notice a simplifying fact: The interest rate doesn’t change based on the principal or the amount of time.So, in our formulas, we can just calculate using $1 for 1 year.First see how much would be earned with compounding:






)1(4

4

08.111

tm

m

rPA






0824.14

02.1

)1(4

4

08.111

A

tm

m

rPA






0824.14

02.1

)1(4

4

08.111

A

tm

m

rPA

So $1 would turn into $1.0824 in 1 year.






0824.14

02.1

)1(4

4

08.111

A

tm

m

rPA


Now use A = $1.0824 in the simple interest formula & solve for r. (This will be the EAR.)






0824.14

02.1

)1(4

4

08.111

A

tm

m

rPA



A = P(1+rt)






0824.14

02.1

)1(4

4

08.111

A

tm

m

rPA



A = P(1+rt)

1.0824 = 1[1 + r(1)]






0824.14

02.1

)1(4

4

08.111

A

tm

m

rPA



A = P(1+rt)

1.0824 = 1[1 + r(1)]

1.0824 = 1 + r






0824.14

02.1

)1(4

4

08.111

A

tm

m

rPA



A = P(1+rt)

1.0824 = 1[1 + r(1)]

1.0824 = 1 + r

r = .0824






0824.14

02.1

)1(4

4

08.111

A

tm

m

rPA



A = P(1+rt)

1.0824 = 1[1 + r(1)]

1.0824 = 1 + r

r = .0824

So, the EAR is 8.24%

Example 4: Effective RateExample 4: Effective RateIf you would rather have a formula for EAR, here it is:

The effective rate corresponding to a stated rate of interest r compounded m times per year is

11

m

e m

rr

This formula gives the same answer that you would get if you just ‘figured it out’ as we did earlier. Try it yourself and see!

Example 5Example 5A family plans to retire in 15 years and expects to need

$300,000. Determine how much they must invest today at 12.3% compounded semiannually to accomplish their goal.



tm

m

rPA

1



tm

m

rPA

1

A = $300,000 P = ? r = 0.123 m = 2 t = 15



tm

m

rPA

1

A = $300,000 P = ? r = 0.123 m = 2 t = 15

)15)(2(

2

123.01300000

P



tm

m

rPA

1

A = $300,000 P = ? r = 0.123 m = 2 t = 15

P = $50,063.51

)15)(2(

2

123.01300000

P

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1. All or most2. A lot of it3. About half of it4. Not too much of it5. None or hardly any of it

Section 5.1: Simple and Compound Interest. Simple Interest Simple Interest: Used to calculate interest on loans…often of one year or less. Formula: I.

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