6. Loans, Investments, & Annuities

Lesson

An annuity is a style of investment from which individuals usually withdraw a regular amount of funds. The withdrawal in many cases is repeated until the funds run out, that is the value of the annuity reaches $\$0$$0. A special case of annuities are perpetuities where rather than the funds dwindling the investment value remains constant.

We can solve annuity problems using the finance application of our calculators, or by modelling the problem with a recurrence relation.

We can solve problems involving annuities using a financial application by setting the present value (PV) as the initial value invested and future value (FV) as the remaining funds at the end of the investment (often $\$0$$0). The present value should be entered as a negative to indicate depositing the money for investment and the payment (PMT) will be positive as this is returned to the investor.

Select the brand of calculator you use below to work through an example involving annuities.

Casio Classpad

How to use the CASIO Classpad to complete the following tasks involving annuities using the inbuilt financial solver.

Jordan invests $\$75000$$75000 in an annuity which earns $8%$8% per annum compounded quarterly. At the end of each quarter he withdraws $\$3000$$3000 to spend on a quarterly fishing trip.

If the investment continues in this way, how many fishing trips can Jordan afford to go on from the proceeds of his annuity?

Jordan was hoping to afford $4$4 fishing trips a year for the next $10$10 years. To achieve this he will reduce the amount he withdraws for each trip. How much can he afford to withdraw each quarter?

TI Nspire

How to use the TI Nspire to complete the following tasks involving annuities using the inbuilt financial solver.

Jordan invests $\$75000$$75000 in an annuity which earns $8%$8% per annum compounded quarterly. At the end of each quarter he withdraws $\$3000$$3000 to spend on a quarterly fishing trip.

If the investment continues in this way, how many fishing trips can Jordan afford to go on from the proceeds of his annuity?

Jordan was hoping to afford $4$4 fishing trips a year for the next $10$10 years. To achieve this he will reduce the amount he withdraws for each trip. How much can he afford to withdraw each quarter?

Victoria invests $\$190000$$190000 at a rate of $12%$12% per annum compounded monthly.

We will use the financial solver on our CAS calculator to determine what Victoria's equal monthly withdrawal should be if she wants the investment to last $20$20 years.

Fill in the value for each of the following. Type an $X$

`X`next to the variable we wish to solve for.$N$ `N`$\editable{}$ $I$ `I`$%$%$\left(\editable{}\right)%$()% $PV$ `P``V`$\editable{}$ $PMT$ `P``M``T`$\editable{}$ $FV$ `F``V`$\editable{}$ $P$ `P`$/$/$Y$`Y`$\editable{}$ $C$ `C`$/$/$Y$`Y`$\editable{}$ Hence determine the amount of the monthly withdrawal.

Give your answer to the nearest cent.

This is similar to modelling an investment with regular payment with the payment in this case being withdrawn and hence, negative in the form shown below.

Sequence - Investment with regular withdrawal

For a principal investment, $P$`P`, at the compound interest rate of $r$`r` per period and a payment $d$`d` **withdrawn** per period, the sequence of the value of the investment over time forms a first order linear recurrence.

The sequence which generates the value, $V_n$`V``n`, of the investment at the **end** of each instalment period is:

**Recursive sequence:**

$V_n=V_{n-1}\times(1+r)-d$`V``n`=`V``n`−1×(1+`r`)−`d`, where $V_0=P$`V`0=`P`

**Recursive sequence:**

$V_n=V_{n-1}\times(1+r)-d$`V``n`=`V``n`−1×(1+`r`)−`d`, where $V_1=P$`V`1=`P`

Tahlia receives an inheritance of $\$250000$$250000 and decides to invest the entire amount in an annuity earning $7.2%$7.2% per annum compounded monthly. At the end of each month, once the interest has been paid into her account, Tahlia withdraws $\$2000$$2000 to help pay for living expenses.

**(a) **Calculate the value of her annuity at the end of the first month.

**Think:** To calculate the value at the end of the first month, we will first need to add the interest owed to the investment and then subtract the withdrawal. Remember the annual rate of $7.2%$7.2% must be divided by $12$12 to get a monthly rate of $0.6%$0.6%, or $0.006$0.006.

**Do:**

$\text{Value }$Value | $=$= | $250000\left(1+0.006\right)-2000$250000(1+0.006)−2000 |

$=$= | $\$249500$$249500 |

**(b) **Write a recursive rule that gives the value of the annuity, $V_n$`V``n`, at the end of $n$`n` months.

**Think:** The recursive rule for the value at the end of the month has the form $V_n=V_{n-1}\times(1+r)-d$`V``n`=`V``n`−1×(1+`r`)−`d`, where $V_0$`V`0 is the initial investment. We have an initial investment of $V_0=250000$`V`0=250000, a monthly rate of $r=0.6%$`r`=0.6% and a monthly withdrawal of $d=2000$`d`=2000.

**Do:**

$V_n=1.006V_{n-1}-2000$`V``n`=1.006`V``n`−1−2000; $V_0=250000$`V`0=250000

**(c) **Use the sequence facility of your calculator to determine during which year and month Tahlia's annuity will end.

**Think:** To do this we will need to use our recursive rule and then scroll through our table of values until we find the first time the value of the annuity becomes a negative value.

**Do:** For an example showing how to create a table of values for a sequence inspect the first examples shown in the lesson describing investments with periodic payments. Scrolling through your created table you should see the following:

$n$n |
$V_n$Vn |
---|---|

$229$229 | $\$5422.07$$5422.07 |

$230$230 | $\$3454.60$$3454.60 |

$231$231 | $\$1475.33$$1475.33 |

$232$232 | $-\$515.82$−$515.82 |

We can see the annuity first holds a negative value during month $232$232. Therefore, it will take $19.33$19.33 years. So this occurs after $19$19 years and $4$4 months.

**(d) **How much should Tahlia withdraw each month if she wishes her annuity to last indefinitely (become a perpetuity)?

**Think:** If Tahlia never wants her annuity to run out, then she should only withdraw the amount of interest earned in the first month, so that the value of her investment forever remains at $\$250000$$250000.

**Do: **

$\text{Interest }$Interest | $=$= | $0.006\times250000$0.006×250000 |

$=$= | $\$1500$$1500 |

**(e) **If Tahlia could find a higher interest rate for her investment, but kept the withdrawal amount the same, would the annuity end sooner or later?

**Think:** What does it mean if the interest rate was higher? It means she would earn more from her annuity each month.

**Do:** Therefore, if Tahlia earns more but withdraws the same amount, her annuity will last longer.

**Reflect:** The recursive rule could also be written as $V_{n+1}=1.006V_n-2000$`V``n`+1=1.006`V``n`−2000; $V_0=250000$`V`0=250000.

Lachlan received an inheritance of $\$100000$$100000. He invests the money at $8%$8% per annum with interest compounded annually at the end of the year. After the interest is paid, Lachlan withdraws $\$9000$$9000 and the amount remaining in the account is invested for another year.

How much is in the account at the end of the first year?

Write a recursive rule for $A_n$

`A``n` in terms of $A_{n-1}$`A``n`−1 that gives the value of the account after $n$`n`years and an initial condition $A_0$`A`0.Write both parts on the same line separated by a comma.

What is the value of the investment at the end of year $10$10?

Round your answer to the nearest cent.

By the end of which year will the annuity have run out?

Emma sells her business and with the profit of $\$150000$$150000 sets up an annuity. She will pay herself $\$3500$$3500 monthly from her annuity which earns interest of $7.6%$7.6% per annum compounded quarterly.

Write a recursive rule that gives the value of her annuity, $A_n$

`A``n`, at the end of quarter $n$`n`.Write both parts of the rule (including the initial value $A_0$

`A`0) on the same line, separated by a comma.Use the sequence facility on your calculator to determine the balance of her annuity after $3$3 years, correct to the nearest cent.

Use the sequence facility from the previous part to determine after how many years and quarters the annuity will close.

The annuity will close after $\editable{}$ year(s) and $\editable{}$ quarter(s).

If interest was calculated monthly rather than quarterly, what effect, if any, would this have on the life of her annuity?

No effect

AThe annuity would last longer

BThe annuity would not last as long

CNo effect

AThe annuity would last longer

BThe annuity would not last as long

C

use a recurrence relation to model an annuity, and investigate (numerically or graphically) the effect of the amount invested, the interest rate, and the payment amount on the duration of the annuity

with the aid of a financial calculator or computer-based financial software, solve problems involving annuities (including perpetuities as a special case); for example, determining the amount to be invested in an annuity to provide a regular monthly income of a certain amount