Corporate Finance Fundamentals: Time Value of Money and Discounting Cash Flows

Why Money Has a Time Value

A dollar today is usually worth more than a dollar received later because money can earn a return over time. If you can invest cash (in a bank account, marketable securities, or a business project), waiting has an opportunity cost: you give up the return you could have earned. Time value of money (TVM) is the toolkit for making cash flows at different dates comparable by translating them to a common point in time—most often “today” (present value).

Two core ideas: compounding and discounting

• Compounding moves money forward in time: “If I have money now, what will it grow to?”
• Discounting moves money backward in time: “If I receive money later, what is it worth today?”

Both rely on the same logic: earning a return over time.

Define the Inputs First (Before Any Calculation)

TVM problems become straightforward when you clearly define inputs. Use this checklist:

• Rate (r): the return per period (e.g., 8% per year). Use a decimal in formulas (8% = 0.08).
• Time (t or n): number of periods (years, months, quarters). Rate and time must match (annual rate with years, monthly rate with months).
• Cash flows (CF): amounts and timing (e.g., CF at end of Year 1, Year 2, etc.). Decide whether cash flows occur at the end or beginning of each period.
• Compounding frequency (if applicable): annual, monthly, daily. This changes the effective growth/discount per period.

Common business interpretation of the rate

In corporate finance, the discount rate is often the required return for the risk of the cash flows (sometimes approximated by a hurdle rate). Conceptually: “What return would we demand to tie up capital in this investment instead of the next best alternative of similar risk?”

Single Sum: Future Value (FV)

Future value tells you what a single amount today becomes after earning a return for n periods.

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Formula (end-of-period compounding):

FV = PV × (1 + r)^n

Example: setting aside cash for a planned equipment upgrade

Your company sets aside $50,000 today for an equipment upgrade in 3 years. If the funds can earn 6% annually, what will you have in 3 years?

PV = 50,000  r = 0.06  n = 3

FV = 50,000 × (1.06)^3 = 50,000 × 1.191016 ≈ 59,551

Interpretation: At 6%, $50,000 grows to about $59,551 in 3 years.

Single Sum: Present Value (PV)

Present value tells you what a future amount is worth today given a required return.

Formula:

PV = FV ÷ (1 + r)^n

Example: choosing between “pay now” vs “pay later”

A supplier offers two options for a component purchase:

• Option A: Pay $100,000 today.
• Option B: Pay $108,000 in one year.

If your company’s required return for this type of short-term commitment is 8%, compare the options in today’s dollars.

PV(Option B) = 108,000 ÷ 1.08 = 100,000

Interpretation: At 8%, paying $108,000 in one year is equivalent to paying $100,000 today. If your required return were higher than 8%, the delayed payment would be more attractive in PV terms; if lower, paying today would be better.

Discount factor intuition

The term 1 / (1 + r)^n is the discount factor. It shrinks future dollars because you could have grown today’s dollars at rate r over n periods. The further out the cash flow (larger n) or the higher the rate (larger r), the smaller the present value.

Compounding Frequency (When Periods Aren’t Annual)

If compounding occurs more frequently than once per year, convert the annual rate to a per-period rate and adjust the number of periods.

General approach:

• Periodic rate: r_period = r_annual / m
• Number of periods: n = years × m

where m is the number of compounding periods per year (12 for monthly, 4 for quarterly).

Example: monthly compounding

$20,000 invested for 2 years at 12% APR compounded monthly:

r_period = 0.12 / 12 = 0.01 per month

n = 2 × 12 = 24 months

FV = 20,000 × (1.01)^24 ≈ 20,000 × 1.269735 ≈ 25,395

Practical tip: Always align “rate per period” with “number of periods.” Most mistakes come from mixing annual rates with monthly periods.

From Single Sums to Multiple Cash Flows

Many corporate decisions involve a stream of cash flows rather than one lump sum. The principle stays the same:

• Discount each cash flow to the same point in time (often today).
• Add them up to get total present value.

This is the foundation of discounted cash flow (DCF) analysis.

Annuities: Equal Payments Each Period

An annuity is a series of equal cash flows paid at regular intervals (typically at the end of each period). Common business examples include loan payments, service contracts, and many leases.

Present value of an ordinary annuity (end-of-period payments)

PV = PMT × [1 − 1/(1 + r)^n] ÷ r

Where PMT is the equal payment each period.

Example: comparing a one-time payment vs. a 3-year service contract

You can either:

• Pay $27,000 today for a perpetual software license upgrade, or
• Pay $10,000 at the end of each year for 3 years for a service subscription.

If the appropriate discount rate is 9%, what is the PV of the subscription?

PMT = 10,000  r = 0.09  n = 3

PV = 10,000 × [1 − 1/(1.09)^3] ÷ 0.09

(1.09)^3 = 1.295029

1/(1.09)^3 ≈ 0.772183

PV factor = (1 − 0.772183) / 0.09 ≈ 2.5313

PV ≈ 10,000 × 2.5313 = 25,313

Decision framing: In today’s dollars, the 3-year subscription costs about $25,313. Compared to $27,000 today, the subscription is cheaper on a PV basis (given a 9% rate). If the discount rate were lower, future payments would “hurt more” in PV terms and the subscription PV would rise.

Annuity due (beginning-of-period payments)

If payments occur at the beginning of each period (common for some leases), the PV is higher because each payment is received (or paid) one period earlier.

Shortcut:

PV(annuity due) = PV(ordinary annuity) × (1 + r)

Lease vs. Buy Decision: A Discounting View

Lease vs. buy decisions often boil down to comparing the present value of cash outflows under each option (after considering taxes, maintenance, residual value, and financing structure in more advanced setups). Here we focus on the discounting mechanics.

Example (simplified): buy now vs. lease payments

A company needs a machine for 4 years.

• Buy: Pay $120,000 today. Expected resale value in 4 years: $30,000.
• Lease: Pay $28,000 at the end of each year for 4 years.

Assume a discount rate of 10% and ignore taxes/maintenance for simplicity.

Step 1: PV of buying

Buying has an immediate outflow and a future inflow (resale).

PV(buy) = 120,000 − PV(resale)

PV(resale) = 30,000 ÷ (1.10)^4

(1.10)^4 = 1.4641

PV(resale) ≈ 30,000 / 1.4641 ≈ 20,491

PV(buy) ≈ 120,000 − 20,491 = 99,509

Step 2: PV of leasing (annuity)

PV(lease) = 28,000 × [1 − 1/(1.10)^4] ÷ 0.10

1/(1.10)^4 = 1/1.4641 ≈ 0.6830

PV factor = (1 − 0.6830)/0.10 = 3.170

PV(lease) ≈ 28,000 × 3.170 = 88,760

Interpretation: On a pure PV-of-cash-outflows basis (with these simplified assumptions), leasing has a lower present cost than buying. In real decisions, you would refine the cash flows (tax shields, maintenance, downtime risk, financing constraints), but the discounting structure remains the same.

Uneven Cash Flows: Discount Each One Separately

Projects and investments often have uneven cash flows (different amounts each year). The present value is:

PV = Σ [ CF_t ÷ (1 + r)^t ]

where t is the period number (1, 2, 3, ...).

Example: valuing expected project cash inflows

A project requires an upfront investment of $200,000 today and is expected to generate the following cash inflows:

• Year 1: $70,000
• Year 2: $90,000
• Year 3: $110,000
• Year 4: $60,000

If the required return is 12%, compute the PV of inflows and compare to the upfront cost.

PV of inflows: 62,500 + 71,748 + 78,298 + 38,130 = 250,676

Net present value (NPV) idea (without going deep): Compare PV of inflows to the $200,000 outflow today. Here, PV of inflows exceeds the cost by about $50,676, meaning the project clears a 12% required return under these assumptions.

Guided Practice: Build a Simple Discounting Table

This practice helps you translate any set of cash flows into present value using a repeatable process. You can do it in a spreadsheet or on paper.

Scenario

Your company is considering a marketing initiative that is expected to generate incremental cash inflows (in dollars) at the end of each year:

• Year 1: 40,000
• Year 2: 55,000
• Year 3: 35,000
• Year 4: 25,000

The required return is 10% per year. Build a discounting table and compute the PV of total inflows.

Step-by-step instructions (plain language)

• Step 1: Create columns. Make a table with columns: Year (t), Cash Flow (CF), Discount Factor, Present Value.
• Step 2: Write the timing. List years 1 through 4. Because cash flows occur at the end of each year, Year 1 is discounted by one period, Year 2 by two periods, and so on.
• Step 3: Compute the discount factor for each year. Use 1/(1+r)^t. With r = 10%, Year 1 factor is 1/1.10, Year 2 is 1/(1.10)^2, etc.
• Step 4: Convert each future cash flow into today’s dollars. Multiply each cash flow by its discount factor.
• Step 5: Add the present values. The sum is the PV of all expected inflows.

Work it out

Total PV of inflows: 36,364 + 45,455 + 26,294 + 17,075 = 125,188

Explain each line in plain language (model answer)

• Year 1: We expect $40,000 one year from now. At a 10% required return, $40,000 next year is worth about $36,364 today.
• Year 2: $55,000 arrives two years from now, so we discount it twice. That makes it worth about $45,455 today.
• Year 3: $35,000 arrives three years from now, so it is discounted three times and is worth about $26,294 today.
• Year 4: $25,000 arrives four years from now, so it is discounted four times and is worth about $17,075 today.
• Total: Adding the present values gives the total value today of all expected inflows: about $125,188.

Practical Patterns to Recognize (So You Choose the Right Method)

• One cash flow at one date: use single-sum PV or FV.
• Same payment repeated for a fixed number of periods: use annuity PV/FV formulas (and check whether payments are end or beginning of period).
• Different amounts over time: build a discounting table and sum discounted cash flows.

Quick reference formulas