5.4: Payout Annuities and Loans

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In the last section you learned about savings annuities. In a savings annuity, you accumulate money by making many deposits on a regular schedule and the total of these deposits grows by accumulating interest.

In this section, we will learn about a second type of annuity called a payout annuity. A payout annuity begins as a large amount of money that is then "paid out" through many withdrawals of the same amount of money on a regular schedule. As the withdrawals are paid out, the remaining money in the account continues to earn interest. After a certain amount of time paying out the withdrawals, the account will end up empty.

While a savings annuity can be used to save for retirement, a payout annuity is used as income once reaching retirement. The same payout annuity formulas can be also used to describe paying off conventional loans in which you borrow a certain amount of money and then pay it off periodically by making equal payments. Examples include car loans and home mortgages.

Payout Annuities

Mathematical formulas for a payout annuity are given below. As in the previous sections, we will emphasize using the TVM Solver application on the TI calculator but will also demonstrate several examples with a formula.

Payout Annuity Formulas

$A = \dfrac{d\left[1-\left(1+\dfrac{r}{n}\right)^{-n t}\right]}{\left(\dfrac{r}{n}\right)}$

$d = \dfrac{A \cdot \left(\dfrac{r}{n}\right)}{1-\left(1+\dfrac{r}{n}\right)^{-n t}}$

where

$A$ is the balance in the account at the beginning (the starting amount or principal)
$d$ is the regular withdrawal (the amount you take out each year, each month, etc.)
$r$ is the annual interest rate (APR) in decimal form
$n$ is the number of compounding periods in one year
$t$ is the number of years the withdrawals will continue

In these formulas we assume that withdrawals happen on the same schedule as interest is compounded.

One common scenario involving a payout annuity is to determine the amount you will need to save by the time you retire so that you can withdraw a certain monthly or yearly payment from your savings for several years.

Example 1

After retiring, you want to be able to take $1,000 every month for a total of 20 years from your retirement account. The account earns 6% interest. How much will you need in your account when you retire?

Solution

In this example,

$\begin{array} {ll} d = \$1,000 & \text{the monthly withdrawal} \\ r = 0.06 & 6\% \text{ annual rate} \\ n = 12 & \text{since we’re doing monthly withdrawals, we’ll compound monthly} \\ t = 20 & \text{ we are taking withdrawals for 20 years} \end{array}$

We’re looking for how much money needs to be in the account at the beginning, $A$.

Putting these values into the first formula:

$A = \dfrac{d\left[1-\left(1+\dfrac{r}{n}\right)^{-n t}\right]}{\left(\dfrac{r}{n}\right)} \; \; = \; \; \dfrac{1,000\left[1-\left(1+\dfrac{0.06}{12}\right)^{-12 \cdot 20}\right]}{\left(\dfrac{0.06}{12}\right)} \; \; = \; \; \dfrac{1,000\left[1-\left(1.005\right)^{-240}\right]}{0.005} \; \; \approx \; \; 139,580.77 $

You will need to have $139,580.77 in the account when you retire.

Notice that you withdrew a total of $240,000 ($1,000 a month for 240 months). The difference between what you pulled out and what you started with is the interest earned. In this case, the interest earned is $\$ 240,000-\$ 139,580.77=\$100,419.29. $

We can also use the TVM Solver to solve payout annuities. Recall to access the TVM Solver press the APPS key and then choose the Finance... option from the menu. We need to solve for $PV$ where

$N = 240 $: The number of withdrawals and times compounding is done is $20 \text{ years } \times 12 \text { months } = 240$.
$I\%=6$: Remember to leave this as a percentage. Do not change to a decimal.
$FV =0$: There is no money in the account after the last withdrawal is made.
$PMT = 1000$: The withdrawal amount is positive because you will receive money from the annuity. The direction of the cash flow is "coming to you."
$P\text{/}Y = C\text{/}Y = 12$ because deposits are made and the interest is compounded 12 times per year.

Solve for present value $PV$ by moving the cursor back to the value for $PV$ and pressing ALPHA and then ENTER.

$TI graphing calculator screenshot showing TVM Solver with N equals 240, I percent equals 6, PV equals negative 139580.7717, PMT equals 1000, FV equals 0, P over Y equals 12, C over Y equals 12, PMT end.$

Here, the value of $PV$ is negative and indicates that this is the amount that you have deposited into the annuity account. The cash flow of this amount is "away from you."

Try it Now 1

Brandon believes he will need $60,000 for living expenses after he retires. How much money should Brandon have at retirement so he can withdraw this amount each year for 20 years from an annuity earning 8% interest compounded annually?

Answer

$TI graphing calculator screenshot showing TVM Solver with N equals 20, I percent equals 8, PV equals 589088.8444, PMT equals negative 6000, FV equals 0, P over Y equals 1, C over Y equals 1, PMT end.$

$589,088.84

Once reaching retirement with a certain amount in your account, you might want to know how much you can withdraw each month. The next example will determine the amount of the regular withdrawal when the beginning amount in the account is known.

Example 2

Denise has $500,000 in her account when she retires. She wants to take monthly withdrawals from the account during the next 30 years. Her retirement account earns 8% interest. How much will Denise be able to withdraw each month?

Solution

In this example, we’re solving for $d$ so use the second formula for a payout annuity with the following values:

$\begin{array} {ll} r = 0.08 & 8\% \text{ annual rate} \\ n = 12 & \text{since we’re doing monthly withdrawals} \\ t = 30 & \text{since we are taking withdrawals for 30 years} \\ A = \$500,000 & \text{we are beginning with }\$500,000 \end{array}$

$d = \dfrac{A \cdot \left(\dfrac{r}{n}\right)}{1-\left(1+\dfrac{r}{n}\right)^{-n t}} \; \; = \; \; \dfrac{500,000 \cdot \left(\dfrac{0.08}{12}\right)}{1-\left(1+\dfrac{0.08}{12}\right)^{-12 \cdot 30}} \; \; \approx \; \; 3668.82$

Denise can withdraw $3,668.82 each month for 30 years.

Solving for $PMT$ with the TVM Solver requires that

$N = 360 $: The number of monthly payments made is $30 \text{ years } \times 12 \text { months } = 360$.
$I\%=8$: Remember to leave this as a percentage. Do not change to a decimal.
$PV = -500,000$: This is the amount Denise has invested into the annuity. The direction of the cash flow is "going away from Denise."
$FV =0$: Nothing will be owed at the end of the loan.
$P\text{/}Y = C\text{/}Y = 12$ : Payments are made and the interest is compounded monthly (12 times per year.)

$TI graphing calculator screenshot showing TVM Solver with N equals 360, I percent equals 8, PV equals negative 500000, PMT equals 3668.822869, FV equals 0, P over Y equals 12, C over Y equals 12, PMT end.$

Solving for $PMT$, Denise can expect to receive $3,668.82 monthly for 30 years.

Try it Now 2

A university has received $100,000 from a donor who specifies that it is to be used for annual scholarships for the next 20 years. If the university invests the donation into an annuity that pays 4% interest, how much can the university give in scholarships each year?

Answer

$TI graphing calculator screenshot showing TVM Solver with N equals 20, I percent equals 4, PV equals negative 100000, PMT equals 7358.17033, FV equals 0, P over Y equals 1, C over Y equals 1, PMT end.$

$7,358.18

Paying Off Loans

Now you will learn about conventional loans (also called amortized loans or installment loans). Examples of these types of loans include auto loans and home mortgages. One great thing about solving problems about loans is that they use exactly the same formulas as the payout annuity. This is because we begin with an amount we owe and make regular payments that reduce the balance until it is $0.

Example 3

You can afford $200 per month as a car payment. If you get an auto loan at 3% interest for 60 months (5 years), how much can you afford to pay for the car? In other words, what loan amount can you pay off with $200 per month?

Solution

In this example, re’re looking for the starting value of the loan, $A$.

$\begin{array}{ll} d = \$200 & \text{the monthly loan payment} \\ r = 0.03 & 3\% \text{ annual rate} \\ n = 12 & \text{since we’re doing monthly payments, we’ll compound monthly} \\ t = 5 & \text{we’re making monthly payments for 5 years} \end{array}$

Putting these values into the first formula:

$A = \dfrac{d\left[1-\left(1+\dfrac{r}{n}\right)^{-n t}\right]}{\left(\dfrac{r}{n}\right)} \; \; = \; \; \dfrac{200\left[1-\left(1+\dfrac{0.03}{12}\right)^{-12 \cdot 5}\right]}{\left(\dfrac{0.03}{12}\right)} \; \; = \; \; \dfrac{200\left[1-\left(1.0025\right)^{-60}\right]}{0.0025} \; \; \approx \; \; 11,130.47 $

You can afford a loan in the amount of $11,130.47.

You will pay a total of $12,000 ($200 per month for 60 months) to the loan company. The difference between the amount you pay and the amount of the loan is the interest paid. In this case, the amount of interest you’re paying to finance the loan is $\$ 12,000-\$ 11,130.47=\$ 869.53.$

To confirm this answer, solve for $PV$ with the TVM Solver:

$N = 60 $: The number of monthly payments made is $5 \text{ years } \times 12 \text { months } = 60$.
$I\%=3$: Remember to leave this as a percentage. Do not change to a decimal.
$PMT = -200$: The amount of monthly payment is negative. You are making this payment to the loan company and the direction of the cash flow is "going away from you."
$FV =0$: Nothing will be owed at the end of the loan.
$P\text{/}Y = C\text{/}Y = 12$ : Payments are made and the interest is compounded monthly (12 times per year.)

$TI graphing calculator screenshot showing TVM Solver with N equals 60, I percent equals 3, PV equals 11130.47154, PMT equals negative 200, FV equals 0, P over Y equals 12, C over Y equals 12, PMT end.$

One of the most common loans people have is a home mortgage. A home mortgage is a long-term loan in which the property is pledged as security for payment of the difference between the sales price and the down payment.

Example 4

You want to take out a $140,000 mortgage for a new home. The interest rate on the loan is 6%, and the loan is for 30 years. How much will your monthly payments be?

Solution

In this example, we’re looking for $d$ because we want to know the payment amount. Use the second formula for a payout annuity with the following values:

$\begin{array}{ll} r = 0.06 & 6\% \text{ annual rate} \\ n = 12 & \text{since we’re doing monthly payments, we’ll compound monthly} \\ t = 30 & \text{we’re making monthly payments for 30 years} \\ A = \$140,000 & \text{the starting loan amount} \end{array}$

$d = \dfrac{A \cdot \left(\dfrac{r}{n}\right)}{1-\left(1+\dfrac{r}{n}\right)^{-n t}} \; \; = \; \; \dfrac{140,000 \cdot \left(\dfrac{0.06}{12}\right)}{1-\left(1+\dfrac{0.06}{12}\right)^{-12 \cdot 30}} \; \; = \; \; \dfrac{140,000 \cdot \left(0.005\right)}{1-\left(1.005\right)^{-240}}\; \; \approx \; \; 839.37 $

You will make payments of $839.37 per month for 30 years.

You're paying a total of $302,173.20 to the loan company ($839.37 per month for 360 months.) Over the life of the loan, you will pay interest in the amount of $\$ 302,173.20 - \$ 140,000=\$ 162,173.20.$

Solving for $PMT$ with the TVM Solver:

$N = 360 $: The number of monthly payments made is $30 \text{ years } \times 12 \text { months } = 360$.
$I\%=6$: Remember to leave this as a percentage. Do not change to a decimal.
$PV = 140,000$: The amount of mortgage is positive because the loan company has given you this money. The direction of the cash flow is "coming to you."
$FV =0$: Nothing will be owed at the end of the loan.
$P\text{/}Y = C\text{/}Y = 12$ : Payments are made and the interest is compounded monthly (12 times per year.)

$TI graphing calculator screenshot showing TVM Solver with N equals 360, I percent equals 6, PV equals 140000, PMT equals negative 839.37073, FV equals 0, P over Y equals 12, C over Y equals 12, PMT end.$

The monthly payment is displayed as a negative amount on the TVM Solver because the direction of cash flow is "going away from you.” That is, you are paying this amount to the loan company.

Try it Now 4

In order to buy a house Janine is taking a mortgage of $312,000. Her lender will finance it for 25 years at an annual interest rate of 5.5%. How much will Janine's monthly payment be and how much interest will she pay over the life of the loan?

Answer

$TI graphing calculator screenshot showing TVM Solver with N equals 300, I percent equals 5, PV equals 312000, PMT equals negative 1915.9529, FV equals 0, P over Y equals 12, C over Y equals 12, PMT end.$

Janine's monthly payment will be $1,915.95, and she will pay $262,785 in interest.

Example 5

Liz is buying a house that costs $180,000. Her bank requires a 15% down payment. The mortgage rate is 6% for a 30 year loan. To obtain this interest rate, her bank requires an upfront payment of "2 points" at the time of closing. What is Liz's monthly mortgage payment? What is the total amount that Liz actually pays for her house?

Solution

First we need to find Liz's required down payment: $\$180,000 \times 0.15 = \$27,000.$

Liz must pay the down payment amount on her own, so Liz's mortgage will be in the amount of $$180,000 - \$27,000 = \$153,000.$

The bank also requires Liz to pay "2 points" upfront. Points refer to "percentage points" of the mortgage: 2 points amount to 2% of the mortgage or $\$153,000 \times 0.02 = \$3,060.00.$

Now, we find the monthly mortgage payment using TVM Solver settings similar to those in Example 4.

$TI graphing calculator screenshot showing TVM Solver with N equals 360, I percent equals 6, PV equals 153000, PMT equals negative 917.3123035, FV equals 0, P over Y equals 12, C over Y equals 12, PMT end.$

Therefore, Liz's monthly mortgage payment will be $917.31.

The total amount Liz pays for the house includes the down payment + points + monthly payments for 30 years: $\$27,000 + \$3,060 + (\$917.31 \times 12 \times 30) = \$360,291.60.$

Try it Now 5

Charles buys a house that costs $280,000. His bank requires a 20% down payment. The mortgage rate is 5% when 1 point is paid upfront at the time of closing. The term is 25 years.

Find Charles' down payment, mortgage amount, monthly mortgage payment, and total house cost over the 25 years.

Answer

$TI graphing calculator screenshot showing TVM Solver with N equals 300, I percent equals 5, PV equals 224000, PMT equals negative 1309.4816, FV equals 0, P over Y equals 12, C over Y equals 12, PMT end.$

The down payment is $56,000. The mortgage amount is $224,000. The monthly payment is $1,309.48. Charles will pay a total of $451,084 to buy the house.

Example 6

You want to buy a house, but how much can you afford to pay for it? The highest monthly payment you can afford is $300. You are sure that you can secure a 30-year mortgage from your credit union at 4% interest, but a condition of the mortgage requires at least a 10% down payment. What is the price of the house you can afford to buy?

Solution

First, find the amount of a mortgage you can pay off with a monthly payment of $300 every month for 30 years. Using the TVM Solver, we solve for $PV$:

$TI graphing calculator screenshot showing TVM Solver with N equals 360, I percent equals 4, PV equals 62838.37214, PMT equals negative 300, FV equals 0, P over Y equals 12, C over Y equals 12, PMT end.$

Your mortgage amount can be as high as $62,838.37.

Your down payment is 10% of the sales price which means the mortgage amount is 90% of the sales price. To find the sales price you can afford, we can set up a proportion and solve for the sale price:

$\dfrac{\text{mortgage amount}}{\text{sales price}}=\dfrac{$62,838.37}{\text{sales price}}=\dfrac{90\%}{100\%}$

Solve this equation by cross multiplying to obtain: $90\%\text{(sales price)} = 100\%($62,838.37)$.

Finally, divide by 90% to get $\frac{$62,838.37}{90\%}= $69,820.41$.

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