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C. Discrete Functions · Financial Applications: Interest, Present Value and Annuities

Financial Applications: Interest, Present Value and Annuities: Unit Review

C3.1 C3.2 C3.3 C3.4 C3.5 C3.6 C3.7About 40 minutes
Learning goals
  • I can connect simple interest to arithmetic sequences and linear growth, and compound interest to geometric sequences and exponential growth.
  • I can solve A = P(1 + i)ⁿ for the amount, the present value, the rate or the number of periods.
  • I can find the future value, present value and regular payment of an ordinary simple annuity.
  • I can use technology to investigate savings plans and loans and to compare total interest with the principal.

Words to know

simple interest
I = Prt; interest on the principal only (linear growth).
compound interest
A = P(1 + i)ⁿ; interest on principal and earlier interest (exponential growth).
present value
The value today of a future amount: P = A(1 + i)⁻ⁿ.
ordinary simple annuity
Equal payments at the end of each period, with payment period = compounding period.
annuity formulas
FV = R[(1 + i)ⁿ − 1] ÷ i and PV = R[1 − (1 + i)⁻ⁿ] ÷ i.

Let's review!

In this unit you connected money to the mathematics of sequences, series and exponential functions. Simple interest grows linearly, like an arithmetic sequence. Compound interest grows exponentially, like a geometric sequence, and the formula A = P(1 + i)ⁿ let you find the future value, the present value, the rate and the number of periods. Then you added up many equal payments, an annuity, using geometric series, and found future values, present values and regular payments for savings plans and loans. Finally you used technology to ask "what if?" and to build amortization tables.

This review has a few questions from every lesson. Try each one, then press Show answer to check.

How to use this review

Do one or two sections at a time. For every question, first write down what kind of problem it is (simple interest, one compound amount, or an annuity) and i and n. Sketch a timeline for annuity questions. Round money to the cent only at the end. For every question you get wrong, write down why (wrong formula? i or n? calculator brackets? sign in a TVM Solver?) and go back to that lesson's worked examples. Finish with the 10-question quiz.

Which formula? Start with the question: one amount, or regular payments?Simple interestI = PrtA = P(1 + rt)linear, arithmeticOne amount, compoundA = P(1 + i)ⁿP = A(1 + i)⁻ⁿi = (A ÷ P)^(1/n) − 1Finding nguess and checkgraph or TVM Solverround up to whole periodsAnnuity: amount laterFV = R[(1 + i)ⁿ − 1] ÷ isavings goalsR = FV·i ÷ [(1 + i)ⁿ − 1]Annuity: amount nowPV = R[1 − (1 + i)⁻ⁿ] ÷ iloans, withdrawalsR = PV·i ÷ [1 − (1 + i)⁻ⁿ]Every problemi = annual rate ÷ kn = years × kinterest = total − principal
A one-page map of the unit. The top row is about a single amount of money; the bottom row is about regular payments (annuities).
Just for fun

You now know more about loans than many people who have one. Use this power wisely: the next time someone says "only $19.99 a month!", you are allowed to ask, "For how many months, and at what rate?"

Simple Interest, Arithmetic Sequences and Linear Growth

1

Find the simple interest on $2 400 at 3.5% for 3 years.

2

A simple-interest account shows $1 060, $1 120, $1 180 at the ends of years 1 to 3. Find P and r.

3

Why are the year-end amounts of simple interest an arithmetic sequence?

Compound Interest, Geometric Sequences and Exponential Growth

1

Find the first three year-end amounts for $500 at 6% compounded annually.

2

Is $800, $840, $882, $926.10 simple or compound interest?

3

Write the general term for $500 at 6% compounded annually.

The Compound Interest Formula: Compounding Periods and Future Value

1

Find i and n for 5.4% compounded monthly for 3 years.

2

Find the amount when $3 000 is invested at 4% compounded quarterly for 5 years.

3

Which gives more on $1 000 for 3 years: 4.5% compounded annually or 4.4% compounded monthly?

Present Value: How Much Do I Need Today?

1

How much must be invested today at 5% compounded annually to have $6 000 in 7 years?

2

Find the present value of $2 500 due in 18 months at 6% compounded monthly.

3

At 4% compounded annually, which is worth more today: $3 000 now or $3 300 in 2 years?

Finding the Interest Rate

1

$2 000 grew to $2 600 in 5 years, compounded annually. Find the rate.

2

$1 000 grew to $1 200 in 3 years, compounded monthly. Find the annual rate.

3

What is wrong with typing 1.2 ^ 1 ÷ 36 for 1.2^(1/36)?

Finding the Number of Compounding Periods

1

How many years for $1 000 to reach at least $1 500 at 5% compounded annually?

2

Use the rule of 72 to estimate the doubling time at 9% compounded annually.

3

Money doubles in about 12 years. About how long to grow 8 times?

What Is an Annuity? Annuities and Geometric Series

1

What makes an annuity an ordinary simple annuity?

2

Write the future value of $400 at the end of each year for 3 years at 5% compounded annually as a series, and find it.

3

In that series, what are a and r?

The Future Value of an Ordinary Simple Annuity

1

Find the future value of $250 at the end of every month for 4 years at 4.8% compounded monthly.

2

How much of that is interest?

3

Find the future value of $1 000 at the end of every half-year for 6 years at 6% compounded semi-annually.

The Present Value of an Ordinary Simple Annuity

1

Find the present value of $400 at the end of every month for 3 years at 6% compounded monthly.

2

For that loan, find the total interest.

3

Which is worth more today at 6% compounded annually: $20 000 now or $2 500 at the end of each year for 10 years?

An annuity seen from both ends01R2R3R4R5R6RPVFVstart: loansend: savings goalsFV = PV(1 + i)ⁿ (payments at the end of each period)
The same payments can be valued at the start (present value: loans, withdrawals) or at the end (future value: savings goals). The two values are linked by FV = PV(1 + i)ⁿ.

Finding the Regular Payment: Savings Goals and Loans

1

Find the monthly deposit needed to have $6 000 in 3 years at 3.6% compounded monthly.

2

Find the monthly payment on a $12 000 loan at 6% compounded monthly over 4 years.

3

Find the total interest on that loan.

What If? Changing the Payment, Frequency, Rate or Term

1

If the monthly deposit of a savings plan is doubled, what happens to the future value?

2

A loan's term is changed from 5 years to 3 years. What happens to the payment and the total interest?

3

Which grows more: $1 200 at the end of each year (5% compounded annually) or $100 at the end of each month (5% compounded monthly), over 20 years?

Using Technology: TVM Solver, Spreadsheets and Amortization Tables

1

Write TVM Solver entries for the monthly payment on a $12 000 loan at 6% compounded monthly over 4 years.

2

A loan balance is $6 000 at 7.2% compounded monthly and the payment is $250. Find the interest and principal parts of the next payment.

3

Why do the interest parts of loan payments decrease over time?

Check your understanding

Quiz

1.

The year-end amounts of $1 000 at 5% simple interest form:

2.

Find the amount after 2 years for $2 000 at 6% compounded semi-annually.

3.

Which gives the present value of $5 000 due in 4 years at 4% compounded quarterly?

4.

$1 000 grew to $1 210 in 2 years, compounded annually. The rate was:

5.

1.04¹⁷ ≈ 1.948 and 1.04¹⁸ ≈ 2.026. How many years does it take money to double at 4% compounded annually?

6.

Which is an ordinary simple annuity?

7.

Find the future value of $100 at the end of each year for 2 years at 10% compounded annually.

8.

A loan of $10 000 is repaid with 24 payments of $450. The total interest is:

9.

For a loan with fixed principal and rate, choosing a longer term:

10.

On a TVM Solver, for a savings plan with $50 deposited at the end of each month, enter:

For parents and teachers

Use the review to find what to revisit:

  • Mixing up simple and compound interest, or arithmetic and geometric sequences → Lessons 1 and 2.
  • Wrong i or n (forgetting to divide the rate or multiply the years) → Lesson 3.
  • Multiplying when you should divide for present value, or negative-exponent calculator errors → Lesson 4.
  • Forgetting to subtract 1, missing brackets around 1/n, or giving a monthly rate as an annual rate → Lesson 5.
  • Rounding n down instead of up, or trouble with guess and check → Lesson 6.
  • Not recognizing an ordinary simple annuity, or giving the first payment too many periods of interest → Lesson 7.
  • Bracket errors in the future value formula, or forgetting the lump-sum step in two-stage problems → Lesson 8.
  • Choosing FV when PV is needed (loans, withdrawals) → Lesson 9.
  • Using the wrong formula to find a payment, or not finding total interest → Lesson 10.
  • Weak explanations of how payment, rate, frequency or term affect the result → Lesson 11.
  • TVM Solver signs and P/Y, C/Y settings, or amortization table steps → Lesson 12.

This unit finishes the Discrete Functions strand of MCR3U. In Grade 12, Advanced Functions (MHF4U) adds logarithms, which solve for n directly.