2.6 Calculating the Interest Rate for an Annuity

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 [latex]PV[/latex], [latex]FV[/latex], and [latex]PMT[/latex] must adhere to the cash flow sign convention.

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

   [latex]3.12\%[/latex]

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

   [latex]9.04\%[/latex]

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

   [latex]7.90\%[/latex]

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

   [latex]9.76\%[/latex]

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

   [latex]3.77\%[/latex]

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

   [latex]8.61\%[/latex]

7. Helen’s husband recently passed away. The life insurance company is offering her a lump-sum payout of [latex]\$250,000[/latex] today, or month-end payments of [latex]\$1,585[/latex] for [latex]20[/latex] 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 [latex]4.75\%[/latex] effectively. How much will her monthly payment increase?
   Solution

   a. [latex]4.53\%[/latex]; b. [latex]\$16.89[/latex]

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

   [latex]8\%[/latex]

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

   [latex]11.56\%[/latex]

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

    [latex]7.25\%[/latex]

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 [latex]\$499[/latex] is better than stating [latex]\$500[/latex]. On a typical [latex]\$5,000[/latex] purchase, the marketing manager wants to offer payments of [latex]\$229[/latex] monthly, [latex]\$699[/latex] quarterly, or [latex]\$1,399[/latex] 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: [latex]9.95\%[/latex], quarterly plan: [latex]10.62\%[/latex], semi-annual plan: [latex]9.54\%[/latex]

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

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

13. Margarite’s goal is to save up [latex]\$100,000[/latex] after [latex]10[/latex] 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 [latex]\$645.19[/latex], under medium-risk conditions her contribution needs to be [latex]\$523.32[/latex], and if she puts her money into high-risk investments she would need [latex]\$401.14[/latex] per month. Based on the adviser’s calculations, what are the effective interest rates on the low-, medium-, and high-risk investments?
    
    Solution

    low: [latex]5\%[/latex], medium: [latex]9\%[/latex], high: [latex]14\%[/latex]