Transcript Annuities, Loans, and Mortgages
Annuities, Loans, and Mortgages
Section 3.6b
Annuities
Thus far, we’ve only looked at investments with one initial lump sum (the Principal) – but what happens when you keep making regular deposits into an investment account?
Annuity
– a sequence of equal periodic payments. An annuity is
ordinary
if deposits are made at the end of each period at the same time the interest is posted in the account.
(in this course, we’ll only consider
ordinary
annuities…)
Annuities
Suppose you make quarterly $500 payments at the end of each quarter into an account that pays 8% interest compounded quarterly. How much will you have after the first year?
End of Quarter 1: $500 $500 End of Quarter 2: End of Quarter 3: End of the year: $1010 2 $1530.20
2 3 $2060.80
Annuities – Future Value
Future Value
(of an annuity) – the total value of the investment, consisting of all the periodic payments together with all the interest.
The future value (FV) of an annuity consisting of
n
equal periodic payments of
R
dollars at an interest rate
i
per compounding period (payment interval) is
FV
R
1
i i
n
1
Annuities – Future Value
At the end of each quarter year, you make a $500 payment into a mutual fund. If your investments earn 7.88% annual interest compounded quarterly, what will be the value of your annuity in 20 years?
FV
We have:
R R
1
i
n i
= 500,
i
1 = 0.0788/4,
n
500 = (20)(4) = 80 0.0788 4 80 1
$95, 483.39
Loans and Mortgages – Present Value
The periodic and equal payments on a loan or mortgage actually constitute an annuity!!!
The net amount returned from an annuity is called its future value.
Present Value
– the net amount of money put into an annuity To calculate monthly payments on a loan or mortgage, banks compare the present value to the future value…
Loans and Mortgages – Present Value
The present value (PV) of an annuity consisting of
n
equal payments of
R
dollars earning an interest rate
i
(payment interval) is per period
PV
R
1
i i
n
Annual Percentage Rate (APR)
– the annual interest rate charged on consumer loans (note: The APY for the lender is higher than the APR).
Loans and Mortgages – Present Value
Carlos purchases a new truck for $18,500. What are the monthly payments for a 4-year loan with a $2000 down payment if the annual interest rate (APR) is 2.9%?
The amount borrowed is $16,500
i
= 0.029/12, n = (4)(12) The monthly payment is the solution to 1 16,500
R
0.029 12
R
1 0.029
12 48 16, 500 0.029
12
Loans and Mortgages – Present Value
R
1
R
1 0.029
12
R
$364.49
48 16, 500 0.029
12 48 364.487
Carlos will have to pay $364.49
per month for 47 months, and slightly less the last month.
Guided Practice
Which investment is more attractive, 5.125% compounded annually or 5% compounded continuously?
We need APYs!!!
APY for 5.125% account: APY for 5% account: APY 0.05125
1 5.125% 1 1 APY
e
0.05
1 5.1271%
The second investment is a better deal!!!
Guided Practice
Andrew contributes $50 per month into a mutual fund that earns 15.5% annual interest. What is the value of Andrew’s investment after 20 years?
We have:
R
= 50,
i
= 0.155/12,
n
= (12)(20)
FV
R
1
i
n i
1 50 0.155 12 1
$80, 367.73
Guided Practice
Juana contributes to a money market account that earns 4.5% annual interest. What should her monthly payments be if she wants to accumulate $120,000 in 30 years?
FV
We have:
i
1
R i
= 0.045/12,
n i
n
1 120, 000
R
= (12)(30),
FV
0.045 12 1 = 120,000
$158.03
Guided Practice
What is Ericka’s monthly payment for a 3-year $4500 car loan with an APR of 10.25%?
PV
We have:
i R
1 = 0.1025/12,
n i
n i
4500
R
1 0.1025 12 = (12)(3),
PV
= 4500