Annuities, Annuities, Loans, and Loans, and MortgagesMortgages

Section 3.6bSection 3.6b

AnnuitiesAnnuitiesThus far, we’ve only looked at investments with one initial lumpsum (the Principal) – but what happens when you keep makingregular deposits into an investment account?

Annuity – a sequence of equal periodic payments. An annuityis ordinary if deposits are made at the end of each period atthe same time the interest is posted in the account.

(in this course, we’ll only consider ordinary annuities…)

AnnuitiesAnnuitiesSuppose you make quarterly $500 payments at the end of eachquarter into an account that pays 8% interest compoundedquarterly. How much will you have after the first year?

End of the year: 2 3

$500 $500 1.02 $500 1.02 $500 1.02 $2060.80

End of Quarter 3: 2$500 $500 1.02 $500 1.02 $1530.20

End of Quarter 2: $500 $500 1.02 $1010

End of Quarter 1:$500 $500

Annuities – Future Annuities – Future ValueValue

Future Value (of an annuity) – the total value of the investment,consisting of all the periodic payments together with all theinterest.

The future value (FV) of an annuity consisting of n equal periodicpayments of R dollars at an interest rate i per compoundingperiod (payment interval) is

1 1ni

FV Ri

Annuities – Future Annuities – Future ValueValue

At the end of each quarter year, you make a $500 payment intoa mutual fund. If your investments earn 7.88% annual interestcompounded quarterly, what will be the value of your annuityin 20 years?

1 1ni

FV Ri

We have: R = 500, i = 0.0788/4, n = (20)(4) = 80

801 0.0788 4 1500

0.0788 4

$95,483.39

Loans and Mortgages Loans and Mortgages – Present Value– Present Value

The net amount returned from an annuity is called its future value.

Present Value – the net amount of money put into an annuity

The periodic and equal payments on a loan or mortgage actuallyconstitute an annuity!!!

To calculate monthly payments on a loan or mortgage, bankscompare the present value to the future value…

Loans and Mortgages Loans and Mortgages – Present Value– Present Value

The present value (PV) of an annuity consisting of n equalpayments of R dollars earning an interest rate i per period(payment interval) is

1 1n

iPV R

i

Annual Percentage Rate (APR) – the annual interest ratecharged on consumer loans (note: The APY for the lender ishigher than the APR).

Loans and Mortgages Loans and Mortgages – Present Value– Present Value

Carlos purchases a new truck for $18,500. What are the monthlypayments for a 4-year loan with a $2000 down payment if theannual interest rate (APR) is 2.9%?

4 121 1 0.029 12

16,5000.029 12

R

The amount borrowed is $16,500 i = 0.029/12, n = (4)(12)

The monthly payment is the solution to

480.029 0.029

1 1 16,50012 12

R

Loans and Mortgages Loans and Mortgages – Present Value– Present Value

48

16,500 0.029 12

1 1 0.029 12R

480.029 0.029

1 1 16,50012 12

R

$364.49R Carlos will have to pay $364.49Carlos will have to pay $364.49per month for 47 months, andper month for 47 months, andslightly less the last month.slightly less the last month.

364.487

Guided PracticeGuided PracticeWhich investment is more attractive, 5.125% compoundedannually or 5% compounded continuously? We need APYs!!!We need APYs!!!

APY for 5.125% account:1

0.05125APY 1 1

1 5.125%

APY for 5% account:

0.05APY 1e

5.1271%The second investment is a better deal!!!The second investment is a better deal!!!

Guided PracticeGuided PracticeAndrew contributes $50 per month into a mutual fund that earns15.5% annual interest. What is the value of Andrew’s investmentafter 20 years?

We have: R = 50, i = 0.155/12, n = (12)(20)

1 1ni

FV Ri

12 201 0.155 12 1

500.155 12

$80,367.73

Guided PracticeGuided PracticeJuana contributes to a money market account that earns 4.5%annual interest. What should her monthly payments be if shewants to accumulate $120,000 in 30 years?

We have: i = 0.045/12, n = (12)(30), FV = 120,000

1 1ni

FV Ri

12 301 0.045 12 1

120,0000.045 12

R

$158.03R

Guided PracticeGuided PracticeWhat is Ericka’s monthly payment for a 3-year $4500 car loanwith an APR of 10.25%?

We have: i = 0.1025/12, n = (12)(3), PV = 4500

12 31 1 0.1025 12

45000.1025 12

R

$145.74R

1 1n

iPV R

i