Lesson 2.3: Present Value (PV) & Discounting

Lesson Overview

In Lesson 2.2, you learned how to move money forward in time using future value and compounding. In this lesson, you’ll learn the reverse: how to move money backward in time.

Present Value (PV) answers the question: What is a future cash flow worth today? This matters anytime you’re comparing options with different timing such as investments, loans, salaries, business projects, or buying vs. renting.

The key tool is discounting, which uses a discount rate to adjust future dollars into today’s dollars. If you remember one idea from this lesson, make it this: a dollar in the future must be “shrunk” to be comparable to a dollar today.

Learning Objectives

• Define present value (PV) and explain what it represents.
• Use timelines to organize cash flows by date and period.
• Calculate PV of a single future cash flow using a discount rate.
• Explain what a discount rate includes (opportunity cost, inflation, risk).
• Distinguish between discounting annually vs. monthly (period matching).
• Identify common PV mistakes and how to avoid them.

Key Vocabulary

• Present Value (PV): the value today of future money.
• Future Value (FV): the value in the future of money today.
• Discount rate (r): the rate used to convert future money into present value.
• Discount factor: 1 / (1 + r)ⁿ, the multiplier used to shrink future cash.
• Period (n): how many time steps between today and the cash flow date.
• Timeline: a simple map of when cash flows happen.

The Core Present Value Formula

The present value of a single future cash flow is:

PV = FV / (1 + r)ⁿ

• FV = the amount of money in the future
• r = discount rate per period
• n = number of periods until the money is received

Notice how similar this is to the FV formula; you’re simply running the logic in reverse.

Why Discounting Works (Plain-English Intuition)

Discounting is not a trick. It reflects reality:

• Inflation: future dollars typically buy less.
• Opportunity cost: money today could be invested.
• Risk: future cash may not arrive as expected.
• Flexibility: money today gives you options.

The discount rate is how finance “prices” these factors.

Worked Example: PV of a Lump Sum

You will receive $1,000 one year from now. If your discount rate is 8%, what is that worth today?

• FV = 1,000
• r = 0.08
• n = 1

PV = 1,000 / (1.08)¹ = 1,000 / 1.08 = $925.93

Interpretation: Receiving $1,000 in one year is equivalent to having about $925.93 today, if 8% is your required return.

Two-Year Example: The “n” Matters

You will receive $1,000 in 2 years and your discount rate is still 8%.

PV = 1,000 / (1.08)² = 1,000 / 1.1664 = $857.34

Same $1,000, lower PV because you have to wait longer.

The Discount Factor (A Helpful Shortcut)

The discount factor is the multiplier that converts a future amount into PV:

Discount Factor = 1 / (1 + r)ⁿ

Then:

PV = FV × Discount Factor

This is useful when you have many cash flows (you’ll do that in DCF later).

Choosing the Discount Rate: What “r” Should You Use?

The discount rate depends on the situation. A helpful way to think about it is:

• Base return: what you can earn on a very safe alternative (your opportunity cost)
• Inflation expectations: compensation for lost purchasing power
• Risk premium: extra return required for uncertainty

Practical examples:

• Very safe cash flows (e.g., government-backed): lower discount rate.
• Risky business cash flows (e.g., startup profits): higher discount rate.
• Personal decisions: often use your best alternative return (or loan rate) as a starting point.

In Finance 101 your goal is to understand the logic even if the “perfect” discount rate is debated in real life.

Timelines: Your Best Tool for Clarity

Before doing any PV problem, draw a simple timeline:

• t = 0 is today (the present).
• t = 1, 2, 3... are future periods.
• Place each cash flow at the time it occurs.

Example timeline for receiving $1,000 in 2 years:

• t=0: ? (PV)
• t=1: (no cash flow)
• t=2: +$1,000

Period Matching: Annual vs. Monthly Discounting

Just like FV problems, PV problems require that your rate and your periods match.

If cash flows are monthly, you should discount monthly:

• Convert APR to a monthly rate: r_monthly = APR / 12 (for simple compounding assumptions)
• Count months as periods: n = number of months

Example idea (no heavy math needed yet): Getting paid in 6 months means you’re discounting over 6 monthly periods, not 0.5 of a period unless the rate is annual and you handle it consistently.

Common Mistakes (and How to Avoid Them)

• Mistake: Using an annual rate with monthly periods (or vice versa).
  Fix: Always align the rate with the timeline periods.
• Mistake: Forgetting to discount for more than one period.
  Fix: Double-check the exponent n.
• Mistake: Treating the discount rate as a random number.
  Fix: Tie it to opportunity cost and risk: “What return do I need for this?”
• Mistake: Discounting the wrong direction.
  Fix: Future value grows with (1+r)ⁿ; present value divides by it.

Why PV Matters in Real Life

• Loans: your monthly payment is based on the PV of future payments.
• Investing: bonds and stocks are valued by discounting future cash flows.
• Business decisions: projects are evaluated by comparing PV of benefits vs. costs.
• Personal finance: compare a lump-sum offer vs. payments over time.

Practice: Check Your Understanding

1. In one sentence, what does present value (PV) measure?
2. What happens to PV when the discount rate goes up (holding FV and n constant)? Why?
3. Compute PV: What is $500 received in 3 years worth today at 7% annual discounting?
4. Compute PV: What is $1,200 received in 1 year worth today at 10%?
5. Why might you use a higher discount rate for a risky cash flow?

Key Takeaways

• PV converts future cash into today’s dollars so you can compare choices.
• Discounting is the reverse of compounding.
• The discount rate reflects opportunity cost, inflation, risk, and flexibility.
• Longer time and higher rates both make PV smaller.
• Always match rate per period to number of periods.

What’s Next?

In Lesson 2.4: Annuities, you’ll learn how to value steady payment streams like loan payments, savings plans, and retirement contributions using PV and FV as building blocks.