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 three related tools:
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:
Here’s what each piece means:
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:
In year 1, the investor receives only this $50 interest payment. Discount it back one year at the 6% market rate:
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:
So the total cash flow at the end of year 2 is $1,050. Discount that back two years at 6%:
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:
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.
Once you’ve calculated present value, you compare it to the investment’s market price (its cost). That comparison is net present value (NPV):
From the example above:
Now compute NPV:
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 negative NPV suggests the opposite. Reset the market price and assume:
Now compute NPV:
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.
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.
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.
Let’s go ahead and summarize what we’ve learned:
| NPV | IRR |
|---|---|
| Positive | Greater than average market return |
| Zero | Equal to average market return |
| Negative | Lower than average market return |
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.
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