1. calculating simple interest

Calculating Interest and Exponential Growth

1. Calculating Simple Interest

• A dollar today is worth more than a dollar tomorrow

• Because of this cost, money earns interest over time

• If you are borrowing, you will pay interest

• If you are lending/investing, you will earn interest

• Simple Interest

• interest on an investment that is calculated once per period, usually annually

• on the amount of the capital alone

• interest that is not compounded



• Principal is the initial amount invested or borrowed (the loan amount or how much you save)

• Simple Interest Formula:

• P = Principal

• r = Annual Interest Rate

• t = Number of periods (usually years) the money is being borrowed

• Simple Interest = Principal times interest times years

• Simple Interest = P(r)(t)

• Total Owed = P + P(r)(t)



• Ex 1:

Mr. Vasu invests $5,000. His annual interest rate is 4.5% and he invests his money for 5 years. What is the total in his account after this time?

• P =

• r =

• t =

• Total = P + P(r)(t)

$5,000

0.0455

5000 + 5000(0.045)(5)

5000 + 1125 = $6,125



• Ex 2: Trayvond saves $10,000 to pay for a car. His earns 6% on his investment and invests his money for 7 years. What is the total in his account after this time?

• P =

• r =

• t =

• Total = P + P(r)(t)

$10,000

0.067

10000 + 10000(0.06)(7)

10000 + 4200 = $14,200


2. Calculating Compound Interest

• Constant Multiplication Factor and Interest Rate

• The constant multiplication factor = (1 + r)

• r = annual interest rate (as a decimal)

• Annual interest rate and growth rate are the same thing

• Ex 1: If you earn 6%, what is the constant multiplication factor: (1 + 0.06) = (1.06)

• Ex 2: If the CMF is 1.5, what is the growth rate?

1.5 = 1 + r; r=0.50, which is 50%



• Ex 3: Mr. Vasu invests $10,000 in an account that earns 6% annual interest that compounds annually. How much will he have in 2 years:

Year 0 Year 1 Year 2

$10,000

10,000(1.06)

= 10,600

10,600(1.06)

= $11,236

Mr. Vasu has $11,236 after two years.



• Ex 3: Mr. Vasu invests $10,000 in an account that earns 6% annual interest that compounds annually. How much will he have in 2 years:

Year 0 Year 1 Year 2

$10,000

10,000(1.06)

= 10,600

10,600 (1.06)

= $11,236

$10,000 10,000(1.06)

=10,000(1.06)1

=10,600

10,000(1.06)(1.06)

=10,000(1.06)2

=11,236

• Ex 4: Mr. Vasu invests $10,000 in an account that earns 6% annual interest that compounds annually. How much will he have in 7 years?

10,000(1.06)7 = $15,036.30Mr. Vasu has $15,036.30 after seven years.



• Compound Interest Formula

(Exponential Growth Function)

A = P(1 + r)t

A = Future Value or Final/Ending Value

P = Principal/Initial Value and Y-Intercept

r = Annual Interest Rate/Growth Rate

t = Years



• Ex 5: Aaliyah invests $6,000 and earns 5% per year.

• Write an exponential growth equation for how much money Aaliyah has after t years?

A = ?P = 6,000r = 0.05t = ?

A = 6000(1.05)t

• How much will she have after six years if interest is compounded annually?

t = 6 years

A = 6000(1.05)6

A = $8,040.57



• Ex 6: Ganiu invests $24,000 for ten years at 4.5%.

• How much does he have in his account after the ten years?

A = ?P = 24,000r = 0.045t = 10

A = 24000(1.045)10

A = $37,271.27Ganiu has $37,271.27 after 10 years.

• How much did he earn in interest alone?

$37,271.27 – 24,000 =

Ganiu earned $13,271.27 in interest.


3. Analyzing

Compound Interest Formula

• Ex 7: The following function represents how much money Lashawn has in her account after t years:A(t) = 6,500(1.17)t

• What is the y-intercept?

The coefficient is 6,500, so the y-intercept is 6,500.

• What is the constant multiplication factor?

The base is 1.17, so the CMF is 1.17.

• How much money does Lashawn invest at the beginning into her account?

The y-intercept is where t=0, the initial value. So, she started with $6,500.

• What is the annual interest rate?

CMF = (1+r) = 1.17, so r = 0.17 or 17%

• How much Lashawn have after twelve years?

A(t) = 6,500(1.17)12 = $42,770.44.


3. Analyzing

Compound Interest Formula

• Ex 8: The following function represents the number people living the Chinese city of Kunming:

C(t) = 50,000(2)t

• What is the y-intercept?

Coefficient is 50,000, so the y-intercept is 50,000.

• What is the constant multiplication factor?

The base is 2, so the CMF is 2.

• How many people were initially in Kunming?

The y-intercept is where t=0, the initial value. So, the initial population was 50,000 people.

• What is the annual growth rate in population?

CMF = (1+r) = 2, so r = 1 or 100% growth

• How many people in Kunming after 10 years?

C(t) = 50,000(2)10 = 51,200,000 people



w Periodic Compounding

Semiannual

Quarterly

Monthly

Daily

• Compound Interest Formula

with Periodic Compounding

A = P(1 + r/n)nt

A = Future Value or Final/Ending Value

:

P = Principal/Initial Value and Y-Intercept

r = Annual Interest Rate/Growth Rate

t = Years

n = Periods per Year (1, 2, 4, 12, 365)


• Ex 9: Henok invests $6,000 and earns 5% per year. How much will he have after six years

A(t) = 6000(1 + .05/n)6n • if interest is compounded annually (n=1)?

A = 6000(1.05)6

A = $8,040.57

• if interest is compounded semi-annually (n=2)?

A = 6000(1 + 0.05/2)(2●6)

A = 6000(1.025)12

A = $8,069.33

• if interest is compounded quarterly (n=4)?

A = 6000(1 + 0.05/4)(4●6)

A = 6000(1.0125)24

A = $8,084.11



Semiannual

Quarterly

Monthly

Daily


• Ex 9: Henok invests $6,000 and earns 5% per year. How much will he have after six years

A(t) = 6000(1 + .05/n)6n • if interest is compounded monthly (n=12)?

A = 6000(1 + 0.05/12)(12*6)

A = $8,094.11

• if interest is compounded daily (n=365)?

A = 6000(1 + 0.05/365)(365●6)

A = $8,098.99



Semiannual

Quarterly

Monthly

Daily

Annually Semi-Annually

Quarterly Monthly Daily

n = 1 n = 2 n = 4 n = 12 n = 365

$8,040.57 $8,069.33 $8,084.11 $8,094.11 $8,098.99

Henok’s investment gets bigger if interest compounds more frequently


Ex 10: Homer invests $1,000 at 10% for nine years

P = 1,000 r = 0.10 t = 95. Simple vs. Compound Interest

Linear vs. Exponential Functions

Simple Interest

Asimple = P + Prt

A = 1000 + 1000(0.10)(9)

Asimple = $1,900

Year A(t)

0 1,000

1 1,100

2 1,200

3 1,300

4 1,400

5 1,500

9 1,900

Compound Interest (annual)

Acompound = P(1+r)t

A = 1000(1.10)9

Acompound = $2,357.95

Year A(t)

0 1,000 1000(1.1)0

1 1,100 1000(1.1)1

2 1,210 1000(1.1)2

3 1,331 1000(1.1)3

4 1,464 1000(1.1)4

5 1,611 1000(1.1)5

9 2,358 1000(1.1)6


6. Finding the Initial Value

Of Exponential Growth/Interest

• Compound Interest Formula with Periodic Compounding

A = P(1 + r/n)nt

• To find the Initial Value, we need to solve for P

• We will be given: A, r, n, t




Ex 1: The future value of an investment at the end of five years is $25,000. What is the initial investment if you earned 10% interest, compounded annually?

A = 25,000P = ?r = 0.10n = 1 (annually)t = 5 (years)

25000 = P(1 + 0.1/1)(1*5)

25000 = P(1.61051)

25000 = 1.61051P1.61051 1.61051

$15,523.03 = PThis initial value was $15,523.03

Check:

15523.03(1.1)5

= 25,000

A = P(1 + r/n)nt

Go to five decimal places




Ex 2: The future value of an investment at the end of seven years is $35,000. What is the initial investment if you earned 5% interest, compounded quarterly?

A = 35,000P = ?r = 0.05n = 4 (quarterly)t = 7 (years)

35000 = P(1 + 0.05/4)(4*7)

35000 = P(1.41599)

35000 = 1.41599P1.41599 1.41599


Check:

24717.69(1.0125)28

= 35,000

A = P(1 + r/n)nt





Ex 3: You decide you need $50,000 to go to graduate school in five years. You find an investment that pays 12% interest, compounded monthly. How much money will you need to invest today, to go to graduate school in five years ?

A = 50,000P = ?r = 0.12n = 12 (monthly)t = 5 (years)

50000 = P(1 + 0.12/12)(12*5)

50000 = P(1.81670)

50000 = 1.81670P1.81670 1.81670


Check:

27522.48(1.01)60

= 50,000

A = P(1 + r/n)nt



7. What is Annual Percentage Yield?

Comparing

APY vs. APR

• The Annual Percentage Rate (APR) is the rate of interest earned per year.

• This is the rate that we’ve used in all of our problems so far

• The Annual Percentage Yield (APY) is the actual annual percent earned when you calculate for compounding

• APY ≥ APR

• Both are rates per one year

1 + APY = (1 + APR/n)(n*t)


Ex 1: Annual Percentage Rate (APR) is 10% and interest is compounded annually. What is APY?

r = APR = 0.10n = 1 (annually)t = 1 (years)

1 + APY = (1 + r/n)(n*t)

1 + APY = (1 + 0.10/1)(1*1)

1 + APY = 1.10000-1 -1 APY = 0.10 = 10%

If annual compoundingAPY = APR



Comparing

APY vs. APR


Ex 2: Annual Percentage Rate (APR) is 10% and interest is compounded quarterly. What is APY?


1 + APY = (1 + r/n)(n*t)

1 + APY = (1 + 0.10/4)(4*1)

1 + APY = 1.10381-1 -1 APY = 0.10381 = 10.381%

10.38% > 10% APY > APR bcs of compounding



Comparing

APY vs. APR


Ex 3: Annual Percentage Rate (APR) is 10% and interest is compounded daily. What is APY?


1 + APY = (1 + r/n)(n*t)

1 + APY = (1 + 0.10/365)(365*1)

1 + APY = 1.10516-1 -1 APY = 0.10516 = 10.516%

10.52% > 10% APY > APR bcs of compounding

Round to five decimal places


Comparing

APY vs. APR



Comparing

APY vs. APR

• Why does it matter?

• Loans will advertise APR (even though you pay higher APY because of compounding)

• Investments will advertise APY (since it is higher than APR)

1. calculating simple interest

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