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

Present Value: How Much Do I Need Today?

C3.3About 65 minutes
Learning goals
  • I can explain what present value means.
  • I can solve A = P(1 + i)ⁿ for P, and write the result as P = A(1 + i)⁻ⁿ.
  • I can use present value to decide how much to invest now to reach a goal.
  • I can compare money at different times by moving amounts to the same date.

Words to know

present value (PV or P)
The amount that must be invested now, at a given rate, to grow to a given future amount. It is also the value today of money to be received or paid later.
future value (FV or A)
What an amount invested now will be worth at a later date.
discounting
Moving an amount back in time by dividing by (1 + i)ⁿ, which is the same as multiplying by (1 + i)⁻ⁿ.
time value of money
The idea that a dollar today is worth more than a dollar later, because today's dollar can earn interest.

Working backward

Amira wants $10 000 in 5 years to help pay for college or university. Her bank offers 4.2% compounded monthly. She does not want to know what an amount will grow to. She wants to know how much to put in today. That amount is the present value.

Present value answers questions like "How much do I need now?" and "What is a future payment worth today?" It is the key to comparing offers, valuing debts and, later in this unit, understanding loans.

The present value formula

Start with A = P(1 + i)ⁿ and divide both sides by (1 + i)ⁿ:

P = A ÷ (1 + i)ⁿ, which is the same as P = A(1 + i)⁻ⁿ

The negative exponent comes from the exponent law x⁻ⁿ = 1/xⁿ that you used in the unit Exponential Functions. Both forms give the same answer; use whichever is easier on your calculator.

As long as the interest rate is positive, the present value is always less than the future value: the difference is the interest the money will earn.

Moving money back in time: present valuemultiply by (1.0035)^(−60): 60 months at 0.35% per month012345time (years)P = ?$10 000
Future value moves money forward in time (multiply by (1 + i)ⁿ). Present value moves it back (multiply by (1 + i)⁻ⁿ).
Worked example

Example 1: Saving for a goal

How much must Amira invest today at 4.2% compounded monthly to have $10 000 in 5 years?

  1. i = 0.042 ÷ 12 = 0.0035 and n = 5 × 12 = 60.
  2. P = 10000(1.0035)⁻⁶⁰.
  3. Calculator: 10000 × 1.0035 ^ (−60) = 8108.8150…, or 10000 ÷ 1.0035 ^ 60.
  4. P ≈ $8 108.81.
  5. Check: $8 108.81 × 1.0035⁶⁰ ≈ $10 000 ✓. The interest she will earn is about 10000 − 8 108.81 = $1 891.19.

She must invest about $8 108.81 today.

Worked example

Example 2: Which offer is worth more?

You win a contest. You can take $5 000 now or $5 600 in 3 years. You could invest money at 3.5% compounded annually. Which choice is worth more?

  1. You cannot compare amounts at different times directly. Move them to the same date. Use today.
  2. Present value of $5 600: P = 5600(1.035)⁻³ ≈ $5 050.88.
  3. $5 050.88 is more than $5 000, so the $5 600 later is worth more at this rate.
  4. Another way: $5 000 invested now grows to 5000(1.035)³ ≈ $5 543.59 in 3 years, which is less than $5 600. Same conclusion.

Take the $5 600 in 3 years (if you can earn 3.5% compounded annually and do not need the money sooner).

Worked example

Example 3: Value of a debt today

Jordan owes his aunt $2 500, due in 18 months. She agrees that he can pay it off today instead, at a rate of 6% compounded monthly. How much should he pay today?

  1. i = 0.06 ÷ 12 = 0.005 and n = 18.
  2. P = 2500(1.005)⁻¹⁸ ≈ $2 285.34.

About $2 285.34 today settles the debt.

Present value of $10 000 due in 10 years (compounded annually)02 0004 0006 0008 00010 0002%$8 203.484%$6 755.646%$5 583.958%$4 631.93interest rate
The higher the interest rate, the less you need today to reach the same $10 000 in 10 years, because the money grows faster.
Worked example

Example 4: Two future amounts

What single amount invested today at 6% compounded annually would cover $1 000 needed in 2 years and $2 000 needed in 5 years?

  1. Present value of $1 000: 1000(1.06)⁻² ≈ $890.00.
  2. Present value of $2 000: 2000(1.06)⁻⁵ ≈ $1 494.52.
  3. Add the present values (they are now at the same date): ≈ $2 384.51.

About $2 384.51 today.

Present value and future value are a pair

FV = PV(1 + i)ⁿ and PV = FV(1 + i)⁻ⁿ. If you are not sure which one you need, ask: "Do I know the amount now or the amount later?" Then check your answer: present value should be smaller than future value (when the rate is positive).

Just for fun

Present value is the only part of math class where you are officially told to live in the present... and then immediately asked to plan for the future.

Hands-on activity

A letter to your future self

You need: a calculator, paper and a pen.

  1. Pick a real goal for when you finish high school or soon after: a laptop, a first-year residence deposit, a used car, a trip. With a grown-up, look up a realistic price and write it down.
  2. Count the months from now until you need the money.
  3. Calculate how much you would need to invest today at 3% compounded monthly, at 4% compounded monthly and at 5% compounded monthly.
  4. Write a short note to your future self: "To have $___ in ___ months at ___%, I needed $___ today."

What to look for: for a $1 500 laptop in 24 months, the present values are about $1 412.75 at 3%, $1 384.86 at 4% and $1 357.54 at 5%. The higher the rate, the less you need today. (Your numbers will differ; check by growing your answer forward with A = P(1 + i)ⁿ.)

Your turn

1

How much must be invested today at 5% compounded annually to have $5 000 in 4 years?

2

Find the present value of $8 000 due in 6 years at 4% compounded quarterly.

3

How much should be deposited now at 3.6% compounded monthly to have $3 000 in 30 months?

4

In the first question, how much interest will the investment earn?

5

A grandparent wants a child who just turned 6 to have $15 000 on their 18th birthday. How much must be invested today at 4.5% compounded semi-annually?

6

Write 1 ÷ (1.02)⁸ using a negative exponent.

7

Which is worth more today at 3.5% compounded annually: $5 000 now or $5 600 in 3 years?

More practice: warm-up

1

In your own words, what is present value?

2

When the interest rate is positive, which is larger: the present value or the future value?

3

Find the present value of $110 due in 1 year at 10% compounded annually.

4

Solve A = P(1 + i)ⁿ for P.

5

What are i and n for 6% compounded monthly for 18 months?

More practice: on your own

1

A debt of $2 500 is due in 18 months. Find its value today at 6% compounded monthly.

2

Find the single amount needed today, at 6% compounded annually, to cover $1 000 in 2 years and $2 000 in 5 years.

3

A bond will pay $6 000 in 5 years. What is it worth today at 3.25% compounded semi-annually?

4

Show that the present value of $10 000 due in 10 years is smaller at 8% than at 4% (both compounded annually), and explain why.

5

You invested some money 7 years ago at 5% compounded annually. It is worth $750 today. How much did you invest?

Challenge

1

At 5% compounded annually, which is worth more today: $10 000 in 8 years or $12 000 in 12 years?

2

Find the present value at 4% compounded annually of three payments of $1 000, at the end of years 1, 2 and 3.

3

Riley owes $4 000 due in 2 years and $3 000 due in 5 years. The lender agrees to replace both with one payment today, at 6% compounded monthly. How much is that payment?

Check your understanding

Quiz

1.

Present value is:

2.

Which expression gives the present value of $4 000 due in 3 years at 5% compounded annually?

3.

Find the present value of $1 000 due in 2 years at 6% compounded semi-annually.

4.

If the interest rate increases, the present value of a fixed future amount:

5.

(1.04)⁻⁵ is equal to:

6.

To compare $3 000 now with $3 300 in 2 years, you should:

For parents and teachers

Big idea: Expectation C3.3 includes solving the compound interest formula for the present value. Present value is the foundation for loans and for the present value of annuities in Lessons 9 and 10.

How to help: Talk about the "time value of money": would your teen rather have $100 today or $100 next year? Why? (Today's $100 can earn interest.) Ask them to check each answer by growing it forward.

Common difficulties:

  • Multiplying by (1 + i)ⁿ instead of dividing.
  • Entering the negative exponent incorrectly. On many calculators the negative sign key (−) is different from the subtraction key; brackets also help: 1.0035 ^ (−60).
  • Comparing amounts at different dates without moving them to the same date.