Grade by Grade
B. Exponential Functions · Financial Mathematics: Compound Interest and Annuities

Present Value: How Much Do I Need to Invest Now?

B3.2 B3.3About 70 minutes
Learning goals
  • I can explain what present value means and why money today is worth more than the same amount in the future.
  • I can find the principal (present value) with P = A(1 + i)⁻ⁿ, using a scientific calculator.
  • I can use present value to compare offers that pay at different times.
  • I can recognize that PV = A(1 + i)⁻ⁿ is an exponential function of n that decreases as n grows.

Words to know

present value (PV)
The amount that must be invested now, at a given rate, to grow to a certain amount in the future. It is the principal, P.
future value (FV)
What an amount invested now will be worth at a later date. It is the amount, A.
discounting
Working backwards from a future value to a present value by dividing by (1 + i) once for each period.
negative exponent
x⁻ⁿ = 1 ÷ xⁿ (for x ≠ 0). So A(1 + i)⁻ⁿ means A ÷ (1 + i)ⁿ.
time value of money
The idea that a dollar now is worth more than a dollar later, because a dollar now can be invested and earn interest.

A goal with a deadline

Leila wants $5 000 for her first year of college in 4 years. A GIC pays 3.6% compounded monthly. How much should she put in today so the GIC grows to exactly $5 000?

Until now you started with P and found A. This time you know A, the future value, and you need P, the present value. You work the compound interest formula backwards.

The present value formula

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

P = A ÷ (1 + i)ⁿ

Using a negative exponent (x⁻ⁿ = 1 ÷ xⁿ, from the unit Exponential Functions), this is written:

P = A(1 + i)⁻ⁿ or PV = FV(1 + i)⁻ⁿ

Going forward in time you multiply by (1 + i) each period. Going backward you divide by (1 + i) each period. This is called discounting.

Working backwards: $1 000 needed in 3 years, 5% compounded annuallynow$863.841 year$907.032 years$952.383 years$1 000.00÷ 1.05÷ 1.05÷ 1.05present value (PV)future value (FV)PV = 1000 ÷ (1.05)³ = 1000(1.05)⁻³ ≈ $863.84
Present value is compound interest in reverse. Each step back in time divides by 1.05. Investing $863.84 now at 5% compounded annually grows to exactly $1 000 in 3 years.
Worked example

Example 1: Leila's college fund

How much must Leila invest now at 3.6% compounded monthly to have $5 000 in 4 years? How much interest will she earn?

  1. A = 5000, i = 0.036 ÷ 12 = 0.003, n = 12 × 4 = 48.
  2. P = 5000(1.003)⁻⁴⁸.
  3. Calculator: 5000 × 1.003 ^ (−) 48 ≈ 5000 × 0.8660744 ≈ 4330.37. (Use the (−) negation key for the negative exponent, or type 5000 ÷ 1.003 ^ 48.)
  4. Check: 4330.37(1.003)⁴⁸ ≈ 5000.00. ✓
  5. Interest earned: 5000 − 4330.37 = 669.63.

Leila needs to invest $4 330.37 now. The GIC will earn $669.63 of the $5 000.

Worked example

Example 2: Comparing offers at different times

Daniel wins a contest. He can take $10 000 now or $12 000 in 5 years. He could invest money now at 4% compounded semi-annually. Which prize is worth more?

  1. You cannot compare dollars at different times directly. Bring the future prize back to today.
  2. PV of $12 000: i = 0.02, n = 10. PV = 12 000(1.02)⁻¹⁰ ≈ 9844.18.
  3. So the $12 000 prize is worth about $9 844.18 in today's dollars, less than $10 000.
  4. Another way to check: $10 000 now invested for 5 years grows to 10 000(1.02)¹⁰ ≈ $12 189.94, which is more than $12 000.

At 4% compounded semi-annually, $10 000 now is the better prize (by about $155.82 in today's dollars). If Daniel could only earn a much lower rate, the later prize might win.

Worked example

Example 3: How much was borrowed?

A loan with no payments is settled with a single payment of $3 000 after 2 years. The interest rate was 9% compounded quarterly. How much was borrowed, and how much interest was charged?

  1. A = 3000, i = 0.09 ÷ 4 = 0.0225, n = 8.
  2. P = 3000(1.0225)⁻⁸ ≈ 2510.82.
  3. Interest = 3000 − 2510.82 = 489.18.

About $2 510.82 was borrowed and $489.18 was interest.

Present value as a function

Fix the future value at $10 000 and the rate at 5% compounded annually. The present value depends on how many years away the goal is:

PV(n) = 10 000(1.05)⁻ⁿ = 10 000(1 ÷ 1.05)ⁿ ≈ 10 000(0.9524)ⁿ

This is an exponential function of n with a base between 0 and 1, so it decreases: the further away the goal, the less you need to put in today. Each extra year multiplies the present value by about 0.9524. You studied this kind of exponential decay in the unit Exponential Functions.

Present value of $10 000 at 5% compounded annually02 0004 0006 0008 00010 000051015202530Years until the money is needed (n)Present value ($)$6 139.13$3 768.89$2 313.77$10 000 (goal is now)
The present value of a $10 000 goal shrinks as the goal moves further into the future: $6 139.13 if it is 10 years away, $3 768.89 if it is 20 years away. The curve never reaches zero.
Just for fun

Riddle: What is always on its way but never arrives?

Tomorrow. That is also why a dollar tomorrow is worth a little less than a dollar today.

Hands-on activity

Would you rather? Survey and calculate

You need: paper and a calculator. Optional: 3 or 4 family members or friends.

  1. Ask each person: "Would you rather have $100 today or $110 in one year?" Record their answers and their reasons.
  2. Calculate the present value of $110 one year from now at three rates, compounded annually: 2%, 5% and 12%.
  3. Decide which choice is better at each rate.
  4. Write two sentences: how does the interest rate affect the answer? What other reasons (not math) did people give?

What to look for: PV at 2% ≈ $107.84, at 5% ≈ $104.76, at 12% ≈ $98.21. At 2% and 5%, waiting for $110 is worth more. At 12%, $100 today is better (it would grow to $112). People may also mention needing money now, or trusting that the $110 will really be paid: real reasons that the math does not capture.

Your turn

1

Write 2000(1.04)⁻⁵ as a division, and evaluate it.

2

How much must be invested now at 4% compounded quarterly to have $8 000 in 3 years?

3

Find the present value of $2 500 due in 5 years at 3% compounded annually.

4

Find the present value of $15 000 due in 3 years at 5.4% compounded monthly.

5

Grandparents want a child to have $20 000 at age 18. How much should they invest at the child's birth at 5% compounded annually?

6

Without calculating, which has the larger present value: $1 000 due in 2 years or $1 000 due in 10 years (same rate)? Why?

More practice: warm-up

1

Evaluate 5⁻² as a fraction.

2

Rewrite P = A ÷ (1 + i)ⁿ using a negative exponent.

3

Find the present value of $1 000 due in 2 years at 10% compounded annually.

4

Is present value bigger or smaller than future value when the interest rate is positive?

5

For 6% compounded semi-annually for 4 years, find i and n.

More practice: on your own

1

A loan is repaid with one payment of $6 000 after 4 years at 5% compounded semi-annually. How much was borrowed? How much interest?

2

A $1 000 savings bond matures in 10 years. At 6% compounded annually, what is it worth today?

3

Which needs a bigger deposit today to reach $2 000 in 5 years: 4% compounded annually or 8% compounded annually? Show both.

4

Option A: $4 000 now. Option B: $4 500 in 3 years. Use 2.75% compounded semi-annually to decide which is worth more today.

5

Explain why the graph of PV against n for a fixed future value is a decreasing exponential curve.

Challenge

1

Malik will need $25 000 in 10 years. He can invest at 4% compounded annually. He has $10 000 now. How much more must he add today?

2

A debt of $50 000 is due in 15 years. At 4.5% compounded semi-annually, what single payment today would settle it? Explain in one sentence why the answer is so much less than $50 000.

3

Show algebraically that PV(n + 1) = PV(n) ÷ (1 + i). What does this mean in words?

Check your understanding

Quiz

1.

Which formula gives the present value?

2.

How much must be invested now at 5% compounded annually to have $1 000 in 2 years?

3.

As the time until a goal increases (same rate), the present value:

4.

As the interest rate increases (same goal and time), the present value:

5.

$12 000 in 5 years at 4% compounded semi-annually is worth about how much today?

6.

Which describes PV = 5000(1.03)⁻ⁿ as a function of n?

For parents and teachers

Big ideas: B3.2 includes finding the principal, P (present value, PV), with the compound interest formula. B3.3 asks students to see that the compound interest and present value formulas are exponential functions of the number of periods n: A grows and PV decays.

How to help: Use real goals: "If you want $2 000 for a trip in 3 years, what would you need to put away today at a given rate?" Talk about the time value of money: a dollar now can earn interest, so it is worth more than a dollar later.

Common difficulties:

  • Entering the negative exponent. On most scientific calculators use the (−) key, not the subtraction key, or simply divide: A ÷ (1 + i)ⁿ.
  • Mixing up which value is known. Ask: "Is the money now or later?"
  • Comparing amounts at different dates without discounting them to the same date.

All rates are practice examples, not current offers.