# 2.6 Calculating the Interest Rate for an Annuity

## Learning Objectives

• Calculate the interest rate for an annuity

## Formula & Symbol Hub

#### Symbols Used

• $FV$ = Future value or maturity value
• $PV$ = Present value of principal
• $PMT$ = Annuity payment amount
• $I/Y$ = Nominal interest rate
• $P/Y$ = Number of payments per year or payment frequency
• $C/Y$ = Number of compounds per year or compounding frequency
• $n$ or $N$ = Total number of annuity payments

#### Formulas Used

• ##### Formula 2.1 – Total Number of Payments (Annuity)

$n=P/Y \times t$

• ##### Formula 2.2 – Future Value of Ordinary Annuity

$FV=PMT \times \left[\frac{(1+i_2)^n-1}{i_2}\right]$

• ##### Formula 2.3 – Future Value of Annuity Due

$FV=PMT \times (1+i_2) \times \left[\frac{(1+i_2)^n-1}{i_2}\right]$

• ##### Formula 2.4 – Present Value of Ordinary Annuity

$PV=PMT \times \left[\frac{1-(1+i_2)^{-n}}{i_2}\right]$

• ##### Formula 2.5 – Present Value of Annuity Due

$PV=PMT \times (1+i_2) \times \left[\frac{1-(1+i_2)^{-n}}{i_2}\right]$

## Introduction

You must calculate the interest rate on an annuity in a variety of situations:

• To determine the interest rate being charged on any debt.
• To determine the interest rate that an investment is earning.
• To calculate the required interest rate for savings to reach a goal within a certain time period.
• To calculate the required interest rate needed for a series of payments to be sustained over a certain time period.

## Using a Financial Calculator

In the future value and present value formulas for annuities, there is no way to algebraic isolate the periodic interest rate. Consequently, calculating the interest rate for an annuity cannot be found by using the formulas. However, a financial calculator can quickly find the required interest rate. You use the financial calculator in the same way as described previously, but the only difference is that the unknown quantity is I/Y (the nominal interest rate). You must still load the other six variables into the calculator, set the calculator to the correct payment setting, and apply the cash flow sign conventions carefully.

## Using the TI BAII Plus Calculator to Find the Interest Rate for Annuities

• Set the calculator to the correct payment setting (END or BGN).
• Enter values for the known variables ($PV$, $FV$, $PMT$, $I/Y$, $P/Y$ and$C/Y$), paying close attention to the cash flow sign convention for $PV$, $PMT$, and $FV$.
• After all of the known quantities are loaded into the calculator, press $CPT$ and then $I/Y$ to solve for the interest rate.

### Notes

1. Ensure that the calculator is set to the required payment setting. The interest rate will be different for END and BGN.
2. Because the calculator is using a trial-and-error technique to calculate the interest rate, it may take your calculator a few seconds to compute the answer. If your screen goes blank and your calculator hesitates, this is normal.
3. The values you enter for $PV$, $FV$, and $PMT$ must adhere to the cash flow sign convention.

### Example 2.6.1

Smartchoice, a rent-to-own store, offers a Dell 10″ Mini Inspiron Netbook for a cash n’ carry price of $\399$. Alternatively, under its rent-to-own plan you could make $\59.88$ monthly payments in advance and own the laptop after one year. What effective interest rate is being charged on the rent-to-own plan?

Solution

The timeline for the annuity appears below.

 PMT Setting BGN $N$ $12 \times 1=12$ $PV$ $399$ $FV$ $0$ $PMT$ $-59.88$ $I/Y$ ? $P/Y$ $12$ $C/Y$ $1$

$I/Y=337.98\%$

The effective interest rate being charged under the rent-to-own payment plan is $337.9759\%$.

### Example 2.6.2

Cubonic Industries deposits $\30,000$ at the end of every quarter to save up $\550,000$ for a capital project in four years. To achieve its goal, what nominal interest rate compounded quarterly does Cubonic Industries require on its investment?

Solution
 PMT Setting END $N$ $4 \times 4=16$ $PV$ $0$ $FV$ $550,000$ $PMT$ $-30,000$ $I/Y$ ? $P/Y$ $4$ $C/Y$ $4$

$I/Y=7.15\%$

For Cubonic Industries to achieve its savings goal, the savings annuity must earn $7.15\%$ compounded quarterly.

### Try It

1) Francisco just changed occupations. Unfortunately, he is not able to transfer his company pension with him to his new company. The administrators of the pension plan offer him the choice of a lump-sum payout of $\103,075$ today or beginning-of-month payments of $\535$ for the next $25$ years. What semi-annually compounded rate of return are the pension administrators using in their calculations?

Solution
 PMT Setting BGN $N$ $12 \times 25=300$ $PV$ $103,075$ $FV$ $0$ $PMT$ $-535$ $I/Y$ ? $P/Y$ $12$ $C/Y$ $2$

$I/Y=3.90\%$

### Example 2.6.3

Amadeus has already saved $\5,000$ in his RRSP today. Suppose he continues to make $\250$ contributions at the beginning of each month for the next $14$ years. For him to achieve his goal of having $\100,000$, what effective rate of return must his investment earn?

Solution

The timeline for RRSP contributions appears below.

 PMT Setting BGN $N$ $12 \times 14=168$ $PV$ $-5,000$ $FV$ $100,000$ $PMT$ $-250$ $I/Y$ ? $P/Y$ $12$ $C/Y$ $1$

$I/Y=9.17\%$

The nominal rate of interest that Amadeus must earn on his investment is $9.17\%$ effective.

### Example 2.6.4

Gemma is looking to purchase a new Nissan Pathfinder for $\54,904.64$ including all fees and sales taxes. She can afford to pay no more than $\1,500$ at the end of every month, and she wants to have the balance owing reduced to $\30,000$ after two years, when she can pay off the vehicle with her trust fund. What is the maximum effective rate of interest she could be charged on the car loan to meet her goals?

Solution

The timeline for the car loan appears below.

 PMT Setting END $N$ $12 \times 2=24$ $PV$ $54,904.64$ $FV$ $-30,000$ $PMT$ $-1,500$ $I/Y$ ? $P/Y$ $12$ $C/Y$ $1$

$I/Y=13.53\%$

Gemma will be able to purchase the car if she can obtain a car loan that has an effective interest rate lower than $13.53\%$.

### Try It

2) An investment today requires $\1,125.51$ to purchase. In return, the investment pays out $\30$ after every six months for the next $20$ years, along with an additional final lump-sum payout of $\1,000$. What semi-annually compounded interest rate is being earned on the investment?

Solution
 PMT Setting END $N$ $2 \times 20=40$ $PV$ $-1,125.51$ $FV$ $1,000$ $PMT$ $30$ $I/Y$ ? $P/Y$ $2$ $C/Y$ $2$

$I/Y=5\%$

### Try It

3) When you buy a car, a cash rebate is usually available if you finance the vehicle through your bank instead of the dealership. If you finance the vehicle through the dealership, you are not eligible for the cash rebate. Assume you can purchase a vehicle for $\24,960$ and finance it for four years with month-end payments at $0\%$ through the dealership. Alternatively, you could get a loan from a bank and pay cash for your vehicle, which would entitle you to receive a $\3,500$ cash rebate. What monthly compounded interest rate would the bank have to charge to arrive at the same monthly payment as the dealership alternative? What decision rule can you create from this calculation?

Solution

Payments for financing through the dealership at $0\%$ interest:

 PMT Setting END $N$ $12 \times 4=48$ $PV$ $-24,960$ $FV$ $0$ $PMT$ ? $I/Y$ $0$ $P/Y$ $12$ $C/Y$ $12$

$PMT=\520$

Interest rate for the bank with monthly payments of $\520$ and applying the $\3,500$ rebate to the present value:

 PMT Setting END $N$ $12 \times 4=48$ $PV$ $-21,460$ $FV$ $0$ $PMT$ $-520$ $I/Y$ ? $P/Y$ $12$ $C/Y$ $12$

$I/Y=7.61\%$

## Section 2.6 Exercises

1. You purchase an annuity for $\500,000$. The annuity will pay you $\8,000$ every quarter for $21.5$ years. What rate compounded quarterly does the annuity earn?
Solution

$3.12\%$

2. You invest $\300$ at the beginning of every month for $33.5$ years. At the end of the $33.5$ years, you have saved $\750,000$. What rate compounded semi-annually did your investment earn?
Solution

$9.04\%$

3. You have the opportunity to make an investment of $\100,000$ today and receive beginning of the month payments of $\1,200$ for $10$ years. Calculate the rate compounded monthly on the investment.
Solution

$7.90\%$

4. Today, your RRSP has $\25,000$. Over the next $8$ years you plan to deposit $\6,250$ every six months into the RRSP with the goal of having $\200,000$ in the account. Calculate the rate compounded semi-annually the RRSP must earn for you to achieve your goal.
Solution

$9.76\%$

5. You have $\300,000$ in your RIF. Over the next $13$ years and $3$ months, you receive beginning-of-quarter payments of $\4,975$ from the RIF. At the end of the $13$ years and $3$ months, the balance in your RIF is $\150,000$. What rate compounded quarterly does the RIF earn?
Solution

$3.77\%$

6. Following his financial adviser’s recommendations, Sanchez starts monthly contributions of $\375$ today to his RRSP. The plan is to have $\240,000$ in his RRSP after $20$ years of monthly compounding. What nominal interest rate does the financial adviser think Sanchez’s RRSP will be able to realize?
Solution

$8.61\%$

7. Helen’s husband recently passed away. The life insurance company is offering her a lump-sum payout of $\250,000$ today, or month-end payments of $\1,585$ for $20$ years.
1. What monthly compounded rate is the life insurance company using in its calculations?
2. Helen thinks she can take the lump sum and invest it herself at $4.75\%$ effectively. How much will her monthly payment increase?
Solution

a. $4.53\%$; b. $\16.89$

8. Under a wrongful dismissal suit, a court awarded a former employee $\100,835.25$ for end-of-month wages of $\4,500$ for the past $21$ months. What effective interest rate is the court using in the judgment?
Solution

$8\%$

9. Jake’s Electronics wants to match a competitor’s advertised payment plan on an identical stereo system. If the system retails for $\1,011.35$ including all sales taxes and Jake wants to advertise six equal end-of-month payments of $\174$, what effective rate of interest does he need to charge?
Solution

$11.56\%$

10. The marketing manager for Gold’s Gym offers members a two-year membership for $\650$ in advance or beginning-of-month payments of $\29$. What monthly compounded interest is the marketing manager using in his pricing?
Solution

$7.25\%$

11. A retail store wants to offer its clients different two-year ordinary payment plans on their product purchases. The marketing manager understands the importance of odd-number pricing in these plans, where $\499$ is better than stating $\500$. On a typical $\5,000$ purchase, the marketing manager wants to offer payments of $\229$ monthly, $\699$ quarterly, or $\1,399$ semi-annually. The Competition Act of Canada requires full disclosure of the annual interest rate being charged on any plan. What interest rate must be published for each plan?
Solution

monthly plan: $9.95\%$, quarterly plan: $10.62\%$, semi-annual plan: $9.54\%$

12. On a $\3,500$ purchase, a company is thinking of offering a year-long month-end payment plan that requires payments of $\299, \319, \334$, or $\349$.
1. If the goal of the plan is to offer a competitive interest rate comparable to a bank credit card that averages $18\%$ effectively, which payment plan should be chosen?
2. If the goal of the plan is to offer a competitive interest rate comparable to a department store credit card that averages $28\%$ effectively, which payment plan should be chosen?
Solution

$\299$ payments: $4.71\%, \319$ payments: $18.24\%, \334$ payments: $29.07\%, \349$ payments: $40.51\%$; a. choose $\319$ plan with $18.24\%$ effective; b. choose $\334$ plan with $29.07\%$ effective

13. Margarite’s goal is to save up $\100,000$ after $10$ years of monthly contributions into an investment annuity starting today. Depending on the level of risk she chooses, her adviser tells her that under low-risk conditions she would need to contribute $\645.19$, under medium-risk conditions her contribution needs to be $\523.32$, and if she puts her money into high-risk investments she would need $\401.14$ per month. Based on the adviser’s calculations, what are the effective interest rates on the low-, medium-, and high-risk investments?
Solution

low: $5\%$, medium: $9\%$, high: $14\%$