5.2 Present value

Present value is a core concept in financial accounting that lets you compare cash flows happening at different points in time. By discounting future cash flows back to what they're worth today, you can make better decisions about investments, valuations, and financial planning.

This matters because a dollar today and a dollar five years from now are not the same thing. Present value shows up constantly in accounting: bond pricing, capital budgeting, lease analysis, pension obligations. If you don't understand how to discount cash flows, you'll struggle with nearly every valuation topic from here on out.

Definition of Present Value

Present value is the current worth of a future sum of money (or a stream of cash flows) given a specified rate of return. Think of it as answering the question: How much would you need to invest today to end up with that future amount?

This is the foundational tool for comparing cash flows that happen at different times. You can't just add up dollars from Year 1 and Year 5 as if they're equivalent. Present value puts them on the same footing by expressing everything in today's dollars.

Importance of Present Value in Financial Accounting

• Enables comparison of cash flows occurring at different times by converting them all to present-day terms
• Drives investment decisions by letting you determine whether a project or asset is worth its cost
• Required for proper financial reporting whenever companies need to reflect the time value of money on their statements (bonds payable, lease liabilities, pension obligations, etc.)

Time Value of Money

Relationship Between Time and Money

The time value of money principle states that a dollar received today is worth more than a dollar received in the future. Two reasons drive this:

• Investment potential: Money received today can be invested and earn interest, so it grows over time. A dollar today becomes more than a dollar tomorrow.
• Inflation: Rising prices mean that a fixed amount of money buys less in the future than it does now.

Because of these factors, receiving money sooner is always preferable (all else being equal), and future cash flows must be discounted to reflect this reality.

Impact of Inflation on Money's Value

Inflation erodes purchasing power over time. As prices increase, a fixed amount of money buys fewer goods and services. To account for this, you incorporate expected inflation into the discount rate when calculating present value.

For example, if inflation runs at 3% per year, $100 today has the same purchasing power as roughly $103 one year from now. The real value of any future cash flow must reflect this erosion.

Present Value of a Single Amount

Present Value Formula for a Single Amount

To find the present value of one future lump sum, use:

$PV = \frac{FV}{(1+r)^n}$

Where:

• PV = present value (what the future amount is worth today)
• FV = future value (the cash flow you'll receive later)
• r = discount rate per period
• n = number of compounding periods

This formula works by "undoing" compound interest. Instead of growing a present amount forward, you're shrinking a future amount backward.

Components of the Present Value Formula

• Future value (FV): The specific dollar amount to be received at a later date.
• Discount rate (r): The rate of return used to discount the future cash flow. It reflects the opportunity cost of capital (what you could earn elsewhere) and the riskiness of the cash flow. A higher discount rate produces a lower present value.
• Number of periods (n): How many compounding periods sit between now and when you receive the cash flow. More periods means more discounting, which means a lower present value.

Examples of Calculating Present Value of a Single Amount

Example 1: What is the present value of $10,000 to be received in 5 years at a 6% annual discount rate?

$PV = \frac{10{,}000}{(1+0.06)^5} = \frac{10{,}000}{1.3382} = 7{,}473$

You'd need to invest $7,473 today at 6% to have $10,000 in five years.

Example 2: What is the present value of $50,000 to be received in 3 years at an 8% annual discount rate?

$PV = \frac{50{,}000}{(1+0.08)^3} = \frac{50{,}000}{1.2597} = 39{,}692$

Notice how a higher discount rate (8% vs. 6%) and shorter time frame still produces significant discounting.

Present Value of an Annuity

Definition of an Annuity

An annuity is a series of equal payments occurring at regular intervals over a set period. There are two types you need to know:

• Ordinary annuity: Payments occur at the end of each period (more common in practice).
• Annuity due: Payments occur at the beginning of each period (think rent payments or insurance premiums paid in advance).

Present Value Formula for an Annuity

For an ordinary annuity:

$PV = PMT \times \frac{1 - (1+r)^{-n}}{r}$

Where:

• PMT = the equal payment amount each period
• r = discount rate per period
• n = total number of payments

This formula is really just a shortcut for discounting each individual payment back to the present and adding them all up.

Ordinary Annuity vs. Annuity Due

With an ordinary annuity, the first payment happens one period from now, so every payment gets discounted at least once.

With an annuity due, the first payment happens immediately (at time zero), so it isn't discounted at all. To convert an ordinary annuity PV to an annuity due PV, multiply by $(1 + r)$:

$PV_{\text{annuity due}} = PV_{\text{ordinary annuity}} \times (1+r)$

Because each payment is received one period earlier, an annuity due always has a higher present value than an otherwise identical ordinary annuity.

Examples of Calculating Present Value of an Annuity

Example 1 (Ordinary Annuity): Annual payments of $5,000 for 4 years, discounted at 5%.

$PV = 5{,}000 \times \frac{1 - (1.05)^{-4}}{0.05} = 5{,}000 \times 3.5460 = 17{,}730$

Step by step:

1. Calculate $(1.05)^{-4} = 0.8227$

2. Subtract from 1: $1 - 0.8227 = 0.1773$

3. Divide by r: $0.1773 \div 0.05 = 3.5460$

4. Multiply by PMT: $5{,}000 \times 3.5460 = 17{,}730$

Example 2 (Annuity Due): Quarterly payments of $2,000 for 3 years (12 payments), quarterly discount rate of 2%.

1. First, calculate as an ordinary annuity: $PV = 2{,}000 \times \frac{1 - (1.02)^{-12}}{0.02} = 2{,}000 \times 10.5753 = 21{,}151$

2. Then adjust for annuity due: $PV = 21{,}151 \times 1.02 = 21{,}574$

Present Value of an Uneven Cash Flow Stream

Definition of an Uneven Cash Flow Stream

Not all cash flows come in neat, equal payments. An uneven cash flow stream involves amounts that vary from period to period. Since the annuity formula only works for equal payments, you have to discount each cash flow individually and then add them up.

Approach to Calculating Present Value of Uneven Cash Flows

1. List each cash flow and the period in which it occurs.
2. Determine the appropriate discount rate.
3. Discount each cash flow individually using the single-amount formula: $PV = \frac{FV}{(1+r)^n}$
4. Sum all the individual present values to get the total.

Examples of Present Value for Uneven Cash Flows

Example 1: Cash flows discounted at 8% per year:

Example 2: An investment with an initial outflow, discounted at 10%: