Now that we’ve covered the basics of present discounted value analysis, we can apply that framework to evaluate investment choices.

In Section 14.1, all the payments had the same sign. That is, the individual payments have all been considered proceeds for whoever owns the payment stream. But, there are many cases in which some payments are benefits and others are costs. Investment projects, for example, usually have up-front costs and don’t pay benefits until sometime in the future. We can incorporate costs and benefits into a single present discounted value fairly easily: Payments that are benefits enter into the PDV calculation with positive signs, costs with negative signs. You still add up the individual terms of the payment stream to get the PDV. Now, however, some of those terms are negative because costs reduce the PDV.

Let’s suppose you go skiing a few times a year and are considering buying your own skis to save the hassle and cost of renting them. You have found a pair you like that costs $500, and you expect to use them for the next three years. You figure out that you typically spend $200 a year on rental skis. Thus, there are three $200 benefit payments: one a year from now, another in two years, and a third in three years. (For simplicity, assume that the $200 benefit comes in one payment at the end of the season.) The payoff structure of the skis purchase is shown in Table 14.1. Should you buy the skis or continue renting?

The first thing to do is figure out the present discounted value of the investment in your own pair of skis. Applying the PDV formula, we have

The first term is the cost of buying the skis. It happens in the present period, so it is not discounted. Because it is a cost to the decision maker rather than a benefit, it is negative. The other terms account for the future benefits of buying the skis. They have positive signs and are discounted based on how far into the future they occur.

As we have seen in our examples, the PDV of this potential investment also depends on the interest rate. If r = 4%, the PDV of buying the skis is $55.02:

This positive number implies that at a 4% interest rate, the skis’ future benefits outweigh their cost in present value terms, so buying them is a worthwhile idea. If the interest rate is 2% instead, the net PDV is larger: $76.78, an even stronger case for buying the skis. However, if r is higher than 4%, the PDV is less than $55.02. In fact, if r is high enough, the net PDV will be negative. At a 10% interest rate, for example, the PDV is –$2.63; that is, buying the skis creates a loss of $2.63 in present value terms. At an interest rate of 10%, the present value of the skis’ future benefits does not make up for their up-front costs. Figure 14.2 plots the PDV of this example investment versus r. You can see that the skis’ PDV becomes negative at an interest rate just below 10%, at 9.7% to be exact, and continues to drop at higher interest rates.

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Using PDV to evaluate the return of a proposed investment (as we just did in our skis example) is sometimes called net present value (NPV) analysis, a method of evaluating investment projects that uses present discounted value to put all of an investment project’s future cost and benefit flows on a common basis, so they can be compared apples-to-apples. If the PDV of a project’s benefits outweighs the PDV of its costs, the project’s NPV (the sum of these PDVs) will be positive, and the investment is worthwhile. If the PDVs of the costs are greater than the benefits, the project’s NPV is negative, and the investment is not worthwhile.

We can extend the NPV analysis method to more complex costs and benefits payment streams. Costs incurred in the future rather than the present have negative signs and are discounted just like any other payments. Because costs and benefits that occur in the same period are discounted equally, they can first be combined into a net benefit for each period. If costs outweigh benefits, the net benefit is negative; if benefits outweigh costs, the net benefit is positive. Once the PDVs of all net benefits are computed, they are added together to obtain the net present value.

This procedure gives us the generic formula for computing the NPV of any investment decision. Label the periods spanned by the investment project 0, 1, 2, . . . , T, where 0 is the current period and T is the last period in which any costs or benefits associated with the project occur. B₀, B₁, . . . , B_T are the investment’s benefits in the respective periods and C₀, C₁, . . . , C_T are the costs. The NPV formula is

where B_t – C_t is the net benefit in any particular period t. In our ski purchase example, T = 3, B₀ = $0, B₁ = B₂ = B₃ = $200, C₀ = $500, and C₁ = C₂ = C₃ = $0:

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The NPV formula is extremely useful and can be used to evaluate all sorts of investment decisions regardless of the timing of their positive or negative payouts.

For payment streams with both positive and negative terms, the value of the interest rate r can determine whether the net present value NPV is positive or negative. This is because the interest rate, through its discounting role, determines how important future payoffs are relative to those occurring nearer to the present time period. As r rises, more distant payoffs become relatively less important. Mathematically speaking, this is because the term (1 + r)^t in the denominator grows larger with t.

This characteristic means that investment projects tend to become less appealing in NPV terms when the interest rate is higher. Investment projects typically have up-front costs with their benefits realized later on. Higher interest rates reduce the present discounted value of those future benefits relative to the investment’s current and earlier costs, driving down the investment’s NPV.

You can see the economic intuition behind this result if you recognize that the interest rate can be thought of as the opportunity cost of investing. Consider households or firms that are deciding whether to put financial capital toward a potential investment project. They face a choice. They can invest and earn the investment’s returns, or they could instead lend their funds on the financial market to other investors implementing their own projects. A household could do this by holding assets in, say, a mutual fund rather than building an addition to their house. Firms can also use financial markets to offer their uncommitted funds to others. If the firm or household chooses to undertake an investment project, it is giving up the return it could have made by lending to others through the financial market. What is that return? The interest rate r. That’s why r captures the opportunity cost of investing. The higher the market interest rate is, therefore, the more a household or firm gives up by investing. NPV analysis, by putting all payments on a common present-value basis using the market interest rate, implicitly accounts for this opportunity cost. The higher opportunity costs of investment implied by higher interest rates are the reason why investments with a typical “up-front-costs-for-future-returns” payment stream pattern have NPVs that fall as interest rates rise. (These financial markets are very closely related to the market for capital, which we discuss below. Financial markets are the money-based side of capital markets, because capital is bought and sold using funds allocated through financial markets. This connection is the reason why the interest rate is the price that matters for the supply and demand of both productive capital and the financial capital used to buy and sell it.)

You’ve probably heard investment evaluations couched in terms of payback periods, the time it takes for an investment’s initial costs to be recouped in undiscounted future benefits. For our ski purchase example above, the payback period is three years. The total benefits obtained after two years ($200 + $200 = $400) are less than the $500 up-front cost, but the $600 in total benefits recovered after three years are greater.

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Payback periods are extremely easy to compute; that’s the advantage of using them to evaluate investment choices. Their disadvantage is that they don’t take into account the idea of discounting future money by an interest rate. They treat future payments as equivalent to current payments in dollar terms, even though that’s not the case (unless the interest rate is zero). The skis, for example, would be considered worthwhile in payback period terms as long as three years is an acceptable period to the person making a decision about the investment. At high enough interest rates, however, the purchase will have a negative NPV. You would do better by not buying the skis and earning interest on the $500 instead. If you had used the payback period method as an evaluation standard when interest rates are high, you would mistakenly buy the skis. For this reason, while payback periods are a handy “first-pass” way to think about investments, they aren’t as good as NPV methods and can, in fact, lead to some wrong decisions.