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finite mathematics & applied calculus topic summary:
Mathematics of finance: Part 2 of 2
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Annuities

An annuity account or just annuity is an account earning interest into which you make periodic deposits or from which you make periodic withdrawals. The term annuity by itself sometimes also refers to the sequence of deposits or withdrawals.

The time during which you are making deposits into an annuity account is the accumulation phase, during which the value of the account increases.

The time during which you are making withdrawals from an annuity account is the annuitization phase or payout phase, during which the value of the account decreases.
Examples: Annuities

A pension fund that earns interest is an example of an annuity account. Its accumulation phase is the period during which you make deposits into the fund, and the payout phase is the period during which you make withdrawals from the fund, (for instance, once you retire).


When a business or government accumulates money in an annuity for some future goal or obligation, the account is referred to as a sinking fund.


A home or auto loan is an example of an annuitizing (or payout) annuity: The lender "deposits" the amount of the loan with the borrower and makes periodic "withdrawals" (the loan payments) until the balance is zero.

 

Accumulation: Future value

If you make a payment of $PMT$ at the end of each period into an account with an interest rate of $i$ per period, then, after $n$ periods, the future value will be

    $FV = PMT\frac{(1 + i)^{n} - 1}{i}.$     
    Payments of PMT for n periods at an interest rate of i per period.
Examples: Accumulation: Future value

To find the future value at the end of 10 years of a fund earning 3% per year compounded monthly into which you make payments of \$500 per month, use
    $PMT = 500, \ \ i = \frac{0.03}{12} = 0.0025,\ $ $n = 10 \times 12 = 120,$
so that
$FV$$=PMT\frac{(1 + i)^{n} - 1}{i}$
$= 500\frac{(1 + 0.0025)^{120} - 1}{0.0025} \approx \$69,870.71$


Practice:

 

Annuitization: Present value

If you receive a payment of $PMT$ at the end of each period from an account with an interest rate of $i$ per period, so that the account is drawn down to zero $n$ periods from now, the present value must be

    $PV = PMT\frac{1 - (1 + i)^{-n}}{i}.$
Examples: Annuitization: Present value

You wish to establish a trust fund from which your niece can withdraw \$10,000 every six months for 15 years. The trust will be invested at 5% per year compounded every six months. How large should the trust be?

Answer:
    $PMT = 10\,000, \ \ i = \frac{0.05}{2} = 0.025,\ $ $n = 15 \times 2 = 30,$
so that the trust should be worth at least
$PV$$=PMT\frac{1 - (1 + i)^{-n}}{i}$
$=10\,000\frac{1 - (1 + 0.025)^{-30}}{0.025} \approx \$209,302.93$


Practice:

 

Amortization: installment loans and mortgages

In a typical installment loan or home mortgage, we borrow an amount of money and then pay it back with interest by making fixed payments over some number of years. The process of paying off a loan is called amortizing the loan, meaning killing the debt owed.

From the point of view of the lender, this situation is an annuitizing annuity: the lender invests the money with you and "withdraws" monthly payments until the balance is zero. Thus, loan calculations are identical to annuity calculations, and so we can use the formula

    $PV = PMT\frac{1 - (1 + i)^{-n}}{i},$
in which $PV$ is the amount of the loan.
Examples: Amortization: installment loans and mortgages

Marco and Joaquín are buying a house and have taken out a 15-year, \$180,000 mortgage at 8.4% interest per year. What will their monthly payments be?

Answer:
    $PV = 180\,000, \ \ i = \frac{0.084}{12} = 0.007,\ $ $n = 15 \times 12 = 360,$
    $PV = PMT\frac{1 - (1 + i)^{-n}}{i}$
$180\,000$$=PMT\frac{1 - (1 + 0.007)^{-360}}{0.007}$
$=PMT\frac{1 - 1.007^{-360}}{0.007}$
Thus, the monthly payments would be
    $PMT = \frac{180\,000 \times 0.007}{1 - 1.007^{-360}} \approx \$1\,371.31.$


Practice:

 

Time value of money calculator