aim: how does money matter? the lowdown on interest rates

Aim: Money Matters – Effective Rate & APR Course: Math Literacy

Aim: How does money matter? The lowdown on interest rates.

Do Now:

Annie deposits $1000 in a local bank at 8% interest for 1 year. How much does Annie have in her account after 1 year?

Annie deposits $1000 in a local bank at 8% interest for 1 year compounded quarterly.How much does Annie have in her account after 1 year?


Simple & Exponential Money

Exponential growthCompound Interest

Simple annual interest A = P(1 + r)t

A P(1 r

n)nt

Exponential growthContinuous compounding

A Pe rt

A – ending amount also referred to an Future Value or Maturity Value

P – beginning amount also referred to an Present Value

I = A – P

Interest earned is the difference between Future Value and Present Value

Simple Interest I = Prt A = P + I


Effective Rate/Annual Percentage Yield

When compounding takes place we often would like to know the Effective Rate. For interest rate i per compounding period, a principal of $1 grows to (1 + i)n in n periods.

= $1 1 1the rate of simple interest thatwould realize exactly as much interest over the sa

Effectiv

me perio

e

d of time

Raten

i r

in

interest rate for compounded period

Also called

(APY) 1 1

the effective interest rate per yea

Annual Percentage Yield

r

nr

n


Model Problem

With a nominal annual rate of 6% compounded monthly, what is the APY?

APY = 1 1n

r

n

120.06

APY = 1 112

12APY = 1.005 1 0.06167 6.17%

(Effective Rate)


Model Problem

Suppose that the monthly statement from the fund reports a beginning balance (P) of $7373.93 and a closing balance (A) of $7382.59 for 28 days (n). What is the APY assuming daily compounding?

1n

A P i

1nA

iP

1

1n

Ai

P

1

1n

Ai

P

1 287382.59

17373.93

i

0.0000419194i

Compounded daily for 1 year (365 days)

365 = 1 0.0000419194 1A

= 1.01542 1.54%A APY


Model Problem

A credit union offers a certificate of deposit at an annual interest rate of 3%, compounded monthly. Find the effective rate. Round to nearest 100th of a percent.

APY = 1 1n

r

n

120.03

APY = 1 112

(Effective Rate)

= 3.04%

I = A – P

12 10.03100(1 )

12A

A P(1 r

n)nt

A 103.04- 100

3.04


Annual Percentage Rate (APR)

Annual Percentage Rate (APR) – the rate paid on a loan when that rate is based on the actual amount owed for the length of time that it is owed. It can be found for interest rate r , with N payments by using the formula

The cost of credit!

1969 Truth in Lending Act interest calculated only on the amount you owed at particular time not on original amount

2

1

nrAPR

n


Model Problem

You’re considering buying a computer for a price of $1399, to be paid in monthly installments over 3 years at a rate of 15%. What is the APR?

n = 3 years @ 12 months per year = 36

r = 0.15

0.292

29.2%!!!

2

1

nrAPR

n


Model Problem

A Blazer with a price of $18,436 is advertised at a monthly payment of $384.00 for 60 months. What is the APR (to nearest 10th of percent?

n = 60 months r = ?

I = A – P A = 384(60) = 23040 -184364604I = Prt

4604 = 18436(r)5 r = 0.0499457583

0.0989.8%

2

1

nrAPR

n


Model Problem

You purchase a refrigerator for $675. You pay 20% down and agree to repay the balance in 12 equal monthly payments. the finance charge on the balance is 9% simple interest.

a) Find the finance chargeb) Estimate the annual percentage rate.

I = Prt

16.6%

2

1

nrAPR

n

675(.20) = 135 – down payment675 – 135 = 540 – amount to be financed

a) I = 540(0.09)(1) = 48.60 – interest to be paid

2(12)(0.09) 2.16b) 0.166

12 1 13APR


Open-Ended Credit-Comparing APRs

Credit cards are convenient, but can lead to trouble! Compare APRs to find the best deal.

Find the APR for a rate of 1½% per month

Find the APR for a rate of 0.057553% per day

21.1% APR

aim: how does money matter? the lowdown on interest rates

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