In the dividend models and discounted cash flow chapters, we discussed the concept of the time value of money. A dollar received today is worth more than a dollar received in the future due to opportunity cost. Money received in the future misses out on potential returns if it were invested today.

In this chapter, we’ll discuss this concept further with a specific emphasis on the following:

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

Present value

This section (Present value) is a repeat from the previous discounted cash flow chapter, but the NPV and IRR sections in this chapter are new. Regardless, you should re-read this section as the following sections build upon the example discussed below.

We already discussed present value in the discounted cash flow chapter, but let’s go ahead and refresh ourselves on the topic. An investor can determine the current value of future money received through this present value calculation:

$PV = \frac{FV}{( 1 + DR ) ^{n}} where: P V F V D R n = present value = future value = discount rate = # of years$

Let’s break down the components of this formula, then work through some numbers to better understand this topic. Future value is the amount of return (money) to be received in the future. The discount rate represents the average rate of return in the market. This is an important factor as it demonstrates the missed opportunity a person experiences if they must wait to receive money in the future. For example, it can be assumed the investor is missing out on a 5% return if the average rate of return in the market is 5%. The ‘n’ in the formula refers to the number of years the investor must wait to receive the future return.

Let’s work through this example:

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?

With the information provided in the question, we can calculate the bond’s present value. This is a two-year bond, so we’ll need to do two present value calculations - one for the return received after one year, and another for the return received after two years.

Present value - year 1
This bond pays a 5% coupon, which is always based on the bond’s par value ($1,000). Therefore, this bond will pay $50 of annual interest to the investor. In the first year of ownership, this is the only return the investor will gain. Let’s do the first year’s present value calculation:

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

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

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

$PV = $47.17$

If the investor must wait a full year to receive $50 when the average rate of return in the market is 6%, then the future return is only worth $47.17 in “today dollars.” Here’s another way to think about it - if the investor had $47.17 today and obtained an average 6% return, they would earn a return of roughly $2.83 ($47.17 x 6%). Earning $2.83 on an original investment of $47.17 results in a total of $50 after one year ($47.17 + $2.83). That’s why $50 received after one full year is considered equivalent to $47.17 today. It’s all about missed opportunities!

Present value - year 2
The bond will pay another $50 in the second year, plus the investor will also receive the $1,000 par value at maturity. With this bond trading at a discount, the investor officially “earns” the discount at maturity. Bottom line - the investor receives $1,050 at the end of year two due to the combination of interest and par value. Let’s do the second year’s present value calculation:

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

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

$PV = \frac{$1,050}{1.0 6 ^{2}}$

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

$PV = $934.50$

If the investor must wait two full years to receive $1,050 when the average rate of return in the market is 6%, then the future return is only worth $934.50 in “today dollars.” Here’s another way to think about it - if the investor had $934.50 today and obtained an average 6% return (compounded over two years), they would earn a return of roughly $115.50 (represents a compounded 6% return on $934.50 over two years). Earning $115.50 on an original investment of $934.50 results in a total of $1,050 after one year ($934.50 + $115.50). That’s why $1,050 received after two full years is considered equivalent to $934.50 today. Again, it’s all about missed opportunities!

Putting it all together
To determine the total present value of the bond, we will now add the two years of present value we just calculated:

$Total PV = Year 1 PV + Year 2 PV$

$Total PV = $47.17 + $934.50$

$Total PV = $981.67$

From a pure “time value of money” perspective, the present value of the bond is $981.67. We discounted the future cash flow from this bond back to its value in today’s dollars. By doing so, we have an indicator of the bond’s value. We can now utilize this information to determine whether the bond is a good or bad deal at its current market price. Here’s a quick clue - it might be underpriced!

Net present value (NPV)

Once a bond’s present value is determined, it should be compared to its market value (a.k.a. cost) to determine if an investment should be made. This can be accomplished through a net present value (NPV) calculation:

$NPV = Present value - investment cost$

In the present value section above, we determined the following:

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

With this information, we can perform an NPV calculation:

$NPV = Present value - investment cost$

$NPV = $981.67 - $970.00$

$NPV = $11.67$

Our present value calculation estimated the value of the bond based on its future cash flows. The present value - $981.67 - is higher than the market price of the bond ($970.00). At the current market price, this bond is underpriced according to our time value of money analysis. Specifically, it’s underpriced by $11.67. A positive NPV indicates an investment is a “good deal” and should be acquired. If an investor is doing a discounted cash flow analysis on multiple investments, the one with the highest positive NPV should be purchased.

A negative NPV indicates the opposite. Let’s reset our numbers and assume the following:

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

Now, let’s perform the NPV calculation:

$NPV = Present value - investment cost$

$NPV = $981.67 - $990.00$

$NPV = -$8.33$

Again, the present value estimates the value of the bond based on its future cash flows. The present value - $981.67 - is lower than the market price of the bond ($990.00). At the current market price, this bond is overpriced according to our time value of money analysis. We think it’s only worth $981.67, but it’s trading for $990.00. Simply put, we believe it’s overpriced by $8.83. A negative NPV indicates an investment is a “bad deal” and should be avoided.

Present value calculations are based on missed opportunities, measured by the average rate of return in the market (the discount rate). While NPV calculations tend to focus on dollar amounts (present value vs. market value), it’s actually a reflection of an investment’s rate of return as compared to average market returns. Even a bond with a negative NPV will likely result in a profit. Assuming the $990 market price above, the investor would still receive $50 in annual interest, plus keep the $10 discount. NPV isn’t as much of a reflection of profitability as it is a reflection of returns above market averages.

With that being said, a positive NPV indicates an investment’s returns are better than the market average. This relates to the investment being considered underpriced; the lower the price, the higher the return. On the other hand, a negative NPV indicates an investment’s returns are worse than the market average. This relates to the investment being considered overpriced; the higher the price, the lower the return.

You might be asking this - what if the NPV is zero? Good question! Quick answer - it means the investment is appropriately* priced. With a present value equivalent to its market value, the investment isn’t a good or bad deal. We can also relate this to market returns; a zero NPV indicates the investment’s returns are equivalent to the average market return. This will be important to note for our next section.

*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) measures its overall rate of return. The term ‘internal’ means this measurement only focuses on the specifics of the investment, not external forces (e.g. inflation, other market risks). There’s a commonly accepted “textbook” definition of IRR:

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

What’s this statement saying? When an investment’s NPV is equal to zero, its overall rate of return is equal to the average market return. To better understand this, let’s refresh ourselves with the numbers referenced above:

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%.

After performing a few calculations, we found the present value of the bond to be $981.67. If the bond were to be trading exactly at $981.67, it would have an NPV of zero. Assuming this were true, we could safely assume the IRR (overall rate of return) of the bond to be equal to the average market return (6%).

Now, let’s go back to the original market price - $970.00. At this price, we calculate a positive NPV of $11.67 (we calculated this above in the NPV section). When an investment demonstrates a positive NPV, we can assume its IRR to be higher than the average market return (6%). Remember, the lower the price (or the more underpriced), the higher the return.

What if the market price was above the present value? For example, let’s assume the market price was $990.00. This would demonstrate an NPV of -$8.33 ($981.67 - $990.00). When an investment demonstrates a negative NPV, we can assume its IRR to be lower than the average market return (6%). Remember, the higher the price (or the more overpriced), the lower the return.

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). As we discussed in the investment vehicles unit, YTM represents a bond’s overall rate of return if held to maturity. This is being mentioned because test questions may use the terms IRR and YTM interchangeably.

As we’ve demonstrated, we can discount the value of future cash flows back to today’s present value. With investments like bonds, it’s very easy to determine future cash flows. Bonds pay fixed semi-annual interest, plus the par value at maturity. There’s no guessing as to what the future cash flow will be. However, these tools become less useful when future cash flow is unpredictable. This is 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 therefore calculating future cash flows is impossible. Even companies that pay dividends on their common stock tend to increase their dividend payments at various rates, and some even suspend or cancel their dividend payments.

*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 calculations are most appropriate when future cash flow is predictable. The less predictable it is, the less relevant and accurate present value, NPV, and IRR calculations are.

Key points

Time value of money

Money received sooner is worth more due to opportunity cost

Opportunity cost

Lost returns from a missed investing opportunity

Discounted cash flow

Tool for determining the present value of future cash flows
Factors in opportunity cost

Present value

Value of future cash flows in today’s dollars

Present value formula

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

Net present value (NPV)

Compares present value to market value
Demonstrates a security’s investment worthiness
Positive NPV = undervalued
Zero NPV = priced appropriately
Negative NPV = overvalued

NPV calculation

$NPV = Present value - investment cost$

Internal rate of return (IRR)

The discount rate that results in the NPV of all future cash flows being equal to zero
Represents a security’s overall rate of return
IRR is equal to a bond’s YTM
Best if analyzing predictable cash flows
Positive NPV = investment’s IRR > average market returns
Zero NPV = investment’s IRR = average market returns
Negative NPV = investment’s IRR < average market returns