Time value of money

In the dividend models and discounted cash flow chapters, we introduced the time value of money. A dollar received today is worth more than a dollar received in the future because of opportunity cost. If you don’t have the money today, you can’t invest it today - and you miss out on potential returns.

In this chapter, the focus is on four related tools:

• Present value (review)
• Future value
• Net present value (NPV)
• Internal rate of return (IRR)

Present value

This section (Present value) repeats material from the previous discounted cash flow chapter. The NPV and IRR sections are new. This review matters because the later sections build directly on the example below.

Present value (PV) tells you what a future cash flow is worth in today’s dollars, given a required rate of return (the discount rate). The basic present value formula is:

$$\begin{gathered} \text{PV}=\frac{\text{FV}}{(1+\text{DR}{)}^n} \\ \begin{array}{rl}\text{where:}& \\ PV& =\text{present value}\\ FV& =\text{future value}\\ DR& =\text{discount rate}\\ n& =\text{\# of years}\end{array} \end{gathered}$$

Here’s what each piece means:

• Future value (FV): the cash you expect to receive in the future.
• Discount rate (DR): the market’s required/expected rate of return. It represents the return you give up by waiting.
• n: how long you must wait (in years) to receive the cash flow.

Let’s apply this to a bond.

An investor is considering the purchase of a $1,000 par, 2-year, 5% corporate debenture currently trading at 97. The rate of return in the market is 6%. What is the present value of the debenture?

Because this bond pays cash flows in two different years, we discount each year’s cash flow separately and then add them together.

Present value - year 1
This bond pays a 5% coupon, and coupon payments are based on the bond’s par value ($1,000). So the annual interest payment is:

• $1,000 × 5% = $50

In year 1, the investor receives only this $50 interest payment. Discount it back one year at the 6% market rate:

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

$$\text{PV}=\frac{\text{\$50}}{(1+\text{0.06}{)}^1}$$

$$\text{PV}=\frac{\text{\$50}}{1.06}$$

$$\text{PV}=\text{\$47.17}$$

Interpretation: receiving $50 one year from now is equivalent to having $47.17 today if the market return is 6%. If you invested $47.17 today at 6%, you’d earn about $2.83 in one year ($47.17 × 6%), ending with about $50.

Present value - year 2
In year 2, the investor receives:

• another $50 interest payment, and
• the $1,000 par value at maturity

So the total cash flow at the end of year 2 is $1,050. Discount that back two years at 6%:

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

$$\text{PV}=\frac{\text{\$1,050}}{(1+\text{0.06}{)}^2}$$

$$\text{PV}=\frac{\text{\$1,050}}{1.06^2}$$

$$\text{PV}=\frac{\text{\$1,050}}{1.1236}$$

$$\text{PV}=\text{\$934.50}$$

Interpretation: receiving $1,050 two years from now is equivalent to having $934.50 today if the market return is 6%. If you invested $934.50 today at 6% compounded for two years, it would grow to about $1,050.

Putting it all together
Add the present values of each year’s cash flow:

$$\text{Total PV}=\text{Year 1 PV + Year 2 PV}$$

$$\text{Total PV}=\text{\$47.17 + \$934.50}$$

$$\text{Total PV}=\text{\$981.67}$$

So, based purely on time value of money (discounting the bond’s future cash flows at 6%), the bond’s estimated value is $981.67. Next, we compare that value to the bond’s actual market price.

Future value

Future value (FV) tells you what a present amount grows into after compounding at a given rate over time. It’s the same relationship as present value, just solved for the other variable:

$$\text{FV}=\text{PV}\times (1+\text{DR}{)}^n$$

Instead of discounting a future amount back to today, you’re compounding today’s amount forward. When deposits happen at different times, find the future value of each deposit on its own, then add them together - the same approach used above to value the bond’s two cash flows.

Example: Future value of multiple deposits

An investor deposits money into an account paying 5% compounded annually: nothing in year 1, $3,000 at the start of year 2, and $2,000 at the start of year 3. What is the account’s value at the end of year 3?

Count the compounding periods between each deposit and the valuation date - not the year label. The year-2 deposit compounds for 2 years (through years 2 and 3) before reaching the end of year 3, and the year-3 deposit compounds for only 1 year:

$${\text{FV}}_{\text{year 2 deposit}}=\$3,000\times (1.05{)}^2=\$3,307.50$$

$${\text{FV}}_{\text{year 3 deposit}}=\$2,000\times (1.05{)}^1=\$2,100.00$$

$$\text{Total FV}=\$3,307.50+\$2,100.00=\$5,407.50$$

Answer: $5,407.50

If these same deposits were made at the end of each year instead of the start, each would compound one year less. The exponent always comes from counting periods on the timeline, not from the year label itself.

Net present value (NPV)

Once you’ve calculated present value, you compare it to the investment’s market price (its cost). That comparison is net present value (NPV):

$$\text{NPV}=\text{Present value - investment cost}$$

From the example above:

• Bond’s market price = $970.00
• Bond’s present value = $981.67

Now compute NPV:

$$\text{NPV}=\text{Present value - investment cost}$$

$$\text{NPV}=\text{\$981.67 - \$970.00}$$

$$\text{NPV}=\text{\$11.67}$$

Because the present value is higher than the market price, the bond appears underpriced by $11.67 (based on a 6% discount rate). In general:

• A positive NPV suggests the investment offers returns above the market’s required return (given the discount rate used).
• When comparing multiple opportunities using the same discount rate, the investment with the highest positive NPV is preferred.

A negative NPV suggests the opposite. Reset the market price and assume:

• Bond’s market price = $990.00
• Bond’s present value = $981.67

Now compute NPV:

$$\text{NPV}=\text{Present value - investment cost}$$

$$\text{NPV}=\text{\$981.67 - \$990.00}$$

$$\text{NPV}=\text{-\$8.33}$$

Here, the bond appears overpriced relative to its discounted cash flows. We estimate it’s worth $981.67, but it costs $990.00.

One important nuance: NPV is not simply “profit vs. loss.” Even with a negative NPV, the investor may still earn a dollar profit (for example, by receiving interest and principal). NPV is mainly telling you whether the investment’s return is better or worse than the market return used as the discount rate.

• Positive NPV → returns are better than the market average (underpriced → lower price → higher return)
• Negative NPV → returns are worse than the market average (overpriced → higher price → lower return)

What if NPV is zero? That means the investment is appropriately priced relative to the discount rate used. In return terms, a zero NPV implies the investment’s return is equal to the average market return.

*When an investment is appropriately priced, the market it trades in is efficient. The more efficient a market, the more its prices reflect true value. On the other hand, an inefficient market has over and/or underpriced investments, which would reflect positive and/or negative NPVs.

Internal rate of return (IRR)

An investment’s internal rate of return (IRR) is its overall rate of return based only on the investment’s own cash flows and price. “Internal” means the calculation focuses on the investment itself, not outside factors like inflation.

A common textbook definition is:

The IRR is the discount rate that results in the NPV of all future cash flows being equal to zero

Here’s the key idea: when NPV equals zero, the investment’s return equals the discount rate used. So IRR is the rate that makes the present value of the cash flows exactly match the investment’s price.

Return to the earlier bond:

An investor is considering the purchase of a $1,000 par, 2-year, 5% corporate debenture currently trading at 97. The rate of return in the market is 6%.

We calculated the bond’s present value (discounted at 6%) as $981.67.

• If the bond traded at $981.67, then NPV = 0, and the bond’s IRR would be equal to the market return used in the discounting (6%).
• At the original market price of $970.00, NPV is positive ($11.67), so the bond’s IRR is higher than 6%.
• If the market price were $990.00, NPV is negative (-$8.33), so the bond’s IRR is lower than 6%.

Let’s go ahead and summarize what we’ve learned:

A bond’s IRR is equal to its yield to maturity (YTM). YTM represents a bond’s overall rate of return if held to maturity. Test questions may use IRR and YTM interchangeably.

Present value, NPV, and IRR work best when future cash flows are predictable. With bonds, cash flows are relatively easy to estimate because the bond pays fixed semi-annual interest and returns par value at maturity.

These tools are less useful when future cash flows are uncertain. That’s why present value, NPV, and IRR calculations are not typically associated with securities like common stock*. Some common stocks don’t pay cash dividends at all, and even dividend-paying companies may raise, suspend, or cancel dividends.

*While present value, NPV, and IRR calculations are not typically utilized for common stock due to its unpredictable future cash flow, it can be used for preferred stock. As a reminder, preferred stock pays a fixed, predictable dividend rate.

Bottom line: time value of money tools are most appropriate when future cash flows are predictable. As predictability decreases, present value, NPV, and IRR become less relevant and less accurate.