Today in Precalculus

• Turn in graded worksheet• Notes: Mathematics of Finance

(need a calculator)– Interest rates– Annual Percentage Yield

• Homework

Compound Interest

• Compound interest:

1kt

rA P

k

A: final amountP: principalr: interest ratek: number of payments per yeart: number of years

Example 1

P=$18,000

k = 2

t = 6

r= 0.075(2 6)

.07518,000 1

2A

=$27,998.18

Example 1b

P=$18,000

k = 4

t = ?

r= 0.075

A=$27,00027,000=18,000(1.01875)4t

1.5=1.018754t

log1.5 = 4tlog1.01875

4t=21.827

t = 5.457 years

4.075

27,000 18,000 14

t

log1.54

log1.01875t

Example 1cA=$31,500

P=$18,000

k = 12

t = 15

r= ?

(180)

1.75 112

r

(12 15)

31,500 18,000 112

r

180 1.75 112

r

180 1.75 112

r

18012 1.75 1 r r = .0374 So an interest rate of 3.74% is required.

ContinuouslyCompound Interest

• Continuously Compounded interest: A=Pert

A: final amountP: principalr: interest ratet: number of years

Example 2

P=20,000

t = 8

r = 0.053

A=20,000e(8•.053)

= $30,561.23

Example 2b

A = $100,000P = $20,000r = 0.053t = ?

100,000 = 20,000e.053t

5 = e0.053t

ln5 = 0.053t

t = 30.367 years

ln 5

0.053t

Example 2c

A = $40,000

P = $20,000

r = ?

t = 10

40,000 = 20,000e10r

2 = e10r

ln2=10rr =.0693An interest rate of 6.93% is needed

Annual Percentage Yield (APY)

• A common basis for comparing investments with different interest rates and methods of compounding.

• The percentage rate that, if compounded annually, would yield the same return as the given interest rate with the given compounding.

Example 1Bob invests with the local bank at 2.12% interest compounded monthly. What is the equivalent APY?

So the APY is 2.14%.

This means his investment compounded monthly at 2.12% earns the same interest as if it earned 2.14%, compounded once a year.

(12).0212

1 112

x

=.0214

Example 2

Which investment is more attractive, one that pays 8.5% compounded quarterly or another that pays 8.45% compounded monthly?

x=.0877

4.085

1 14

x

12.0845

1 112

x

x=.0879

So the investment with the 8.45% rate compounded monthly is the better investment

Example 3The interest charge by your credit card company is 18.75% APY. What is the monthly interest charge on a balance of $5,000?

(12)

1.1875 112

r

(12)

.1875 1 112

r

12 1.1875 112

r

12 1.1875 112

r

1212 1.1875 1 r

r = .1731 $5000x.1731 = $865.43

Homework

• Pg 341: 3-9 odd, 21-39odd, 41-46all