Mathematics of Mathematics of FinanceFinance

Mathematics of Mathematics of FinanceFinance

We can use our knowledge of exponential functions and logarithms to see how interest works. When customers put money into a savings account, the money is a loan to the bank and the bank pays interest to the customer for the use of their money. If money is borrowed the customer will pay interest to the bank for the use of the banks money.

Suppose that a principal (beginning) amount P dollars in invested in an account earning 3% annual interest. How much money would be there at the end of n years if no money is added or taken away? (Let t = time and A(t) = amount in account after time t)

Suppose that a principal (beginning) amount P dollars in invested in an account earning 3% annual interest. How much money would be there at the end of n years if no money is added or taken away?

)03.01(03.0)1( PPPA

)03.01(03.0)03.01()2( PPA

))03.01(out (Factor )03.01)(03.01( PP2)03.01( P

By extending this pattern we find that, where P is the principal and r is the constant interest rate expressed as a decimal. This is the compound interest formula where interest is compounded annually.

( ) (1 )tA t P r

Example. Joe invests $500 in a savings account earning 2% annual interest compounded annually. How much will be in his account after 5 years?

trPtA )1()( 5)02.01(500)5( A

04.552$)5( A

What happens when interest is compounded more than one time a year?

Let P =principal, r=annual interest rate, k=number of times the account is compounded per year, and t=time in years. Thus, r/k=interest rate per compounding period, and kt=the number of compounding periods. The amount A in the account after t years is:

Let P =principal, r=annual interest rate, k=number of times the account is compounded per year, and t=time in years. Thus, r/k=interest rate per compounding period, and kt=the number of compounding periods. The amount A in the account after t years is:

A(t) P 1r

k

kt

Example. Suppose P=$1500, r=7%, t=5, k=4

54

4

07.011500)5(

A

17.2122$)5( A

Example. Finding time. If John invests $2300 in a savings account with 9% interest rate compounded quarterly, how long will it take until John’s account has a balance of $4150?

Example. Finding time. If John invests $2300 in a savings account with 9% interest rate compounded quarterly, how long will it take until John’s account has a balance of $4150? t4

4

09.0123004150

t4)0225.1(8043.1 0225.1ln48043.1ln tt4525.26 tyears 63.6

Compounding Continuously

A(t) Pert

Example. Suppose P=$3350, r=6.2%, t=8 yrs

8062.03350)8( eA

17.5501$)8( A

It can be very difficult to compare investment with all the different compounding options. A common basis for comparing investments is the annual percentage yield (APY) – the percentage that, compounded annually, would yield the same return as the given interest rate with the given compounding period.

Example. What is the APY for a $8000 investment at 5.75% compounded daily?

Let x=APY. Therefore, the value of the investment using the APY after 1 year is A=8000(1+x). So,

365

365

0575.018000)1(8000

x

365

365

0575.011

x

1365

0575.01

365

x

%92.50592.0 x

Example. Which investment is more attractive, 6% compounded monthly, or 6.1% compounded semiannually?

Let

ratetheforAPYther

ratetheforAPYther

%1.6

%6

2

1

0619.0 0617.0 21 rr

2

2

12

1 2

061.011

12

06.011

rr

12

061.011

12

06.01

2

2

12

1

rr

The 6.1% compounded semiannually is more attractive because the APY=6.19% compared with APY=6.17% for the 6% compounded monthly.

Guided Practice

1. Jean deposits $3000 into a savings account earning 3% annual interest compounded semiannually. How much will be in the account in 5 years?

2. Becky Jo deposits $10,000 into an account earning 2% annual interest compounded continuously. How much will be in the account in 7 years?

3. Which investment is more attractive, 4% compounded daily, or 4.1% compounded semiannually?

Annuities – Future Value

So far we have only discussed when the investor has made a single lump-sum deposit. But what if the investor makes regular deposits monthly, quarterly, yearly – the same amount each time. This is an annuity

Annuity: a sequence of equal periodic payments

We will be studying ordinary annuities – deposits are made at the end of each period at the same time the interest is posted in the account.

Suppose Jill makes quarterly $200 payments at the end of each quarter into a retirement account that pays 6% interest compounded quarterly. How much will be in Jill’s account after 1 year?

Since interest is compounded quarterly, Jill will not earn the full 6% each quarter.

She will earn 6%/4=1.5% each quarter. Following is the growth pattern of Jill’s account:

End of quarter 2: $200 + 200(1+0.015)=$403

End of quarter 1: $200

End of quarter 3: $200 + 200(1.015) + 200(1.015)2=$609.05

2

End of the year: 200+200(1.015)+200(1.015)2+200(1.015)3= $818.19

This is called future value. It includes all periodic payments and the interest earned. It is called future value because it is projecting the value of the annuity into the future.

Future Value (FV) of an Annuity:

krkr

RFV

kt

11

where R=payments, k=number of times compounded per year, r=annual interest rate, and t=years of investment.

Example. Matthew contributes $50 per month into the Hoffbrau Fund that earns 15.5% annual interest. What is the value of Matthew’s investment after 20 years?

Example. Matthew contributes $50 per month into the Hoffbrau Fund that earns 15.5% annual interest. What is the value of Matthew’s investment after 20 years?

12

155.0

112

155.01

50

2012

FV

73.367,80$FV

Loans and Mortgages – Present Value

Present Value

the net amount of money put into an annuity

This is how a bank determines the amount of the periodic payments of a loan/mortgage.

Present value (PV) of an annuity:

PV R1 1 r

k

kt

r

k

**Note that the annual interest rate charged on consumer loans is the annual percentage rate (APR).

Example. Calculating a Car Loan Payment:

What is Kim’s monthly payment for a 4-year $9000 car loan with an APR of 7.95% from Century

Bank?

12

0795.012

0795.011

9000

)4(12

R

R

48

12

0795.011

12

0795.09000

R51.219$

Kim’s monthly payment will pay $219.51 for 48 months.

Review:

When would you use the compound interest or continuous interest formulas?

What are the similarities and differences Future Value and Present Value?

Can you determine what you would use

to answer the following?

1. Sally purchases a $1000 certificate of deposit (CD) earning 5.6% annual interest compounded quarterly. How much will it be worth in 5 years?

2. Luke contributes $200 a month into a retirement account that earns 10% annual interest. How long will it take the account to grow to $1,000,000?

3. Gina is planning on purchasing a home. She will need to apply for a mortgage and can only afford to make $1000 monthly payments. The current 30-yr mortgage rate is 5.8% (APR). How much can she afford to spend on a home?