Is a bird in the hand worth two in the bush? Does it sometimes make sense not to do today what can be put off until tomorrow?
Is a bird in the hand worth precisely two in the bush?


Stokey & Zeckhauser Ch. 10, "The Valuation of Future Consequences: Discounting"


When we have our policy hats on, our task is to identify the best choice from among a set of alternatives. "Best" usually depends on the costs and benefits of each alternative. Frequently costs and benefits do not happen all at once; rather, each alternative implies a stream of present and future costs and benefits.

Example. We want to compare option A which will have a net cost of 100 this year but return 200 in benefits next year with option B that has a net benefit of 50 this year and next year. Both "net out" at $100 of benefit, but which should I choose?

Example. Or try this: suppose I offer to buy $1000 from you…for delivery five years from now. What would be a fair price for that deal today?

To resolve questions like these, we need to clarify our assumptions. First, we assume that the money a year from now is guaranteed, period. No risk. Second, we assume you don't need the money right now — that is, you are not going to go hungry or otherwise suffer by waiting. These assumptions allow us to think rather simply about the cash value of the money and allow us to base our analysis on the question of what we would do with the money if we had it today instead of later or, how much would I need to set aside and invest today in order to have that same $1000 in five years?

If you are thinking that the answer to the last question depends on what sort of investment opportunities are available, then you get the gist of discounting.

Suppose we are considering various ways to improve a downtown shopping area with a goal of raising sales tax receipts. Let's think about the increased community policing of the shopping district. How would it have an effect on sales taxes? One chain of logic is:

More Police => Increased Shopper Comfort => More Shoppers => Increased sales

But there is likely to be a time lag. In fact, it might be reasonable to expect that this year's police patrols could only yield an increase in shopper comfort level next year when more people heard about it. So, if we spend 100,000 on police this year the payoff might not be until next year or after. So, how much of an increase in sales tax revenue would make the investment in police worth the price?

Similarly, suppose we put up a new parking garage. The cost is up front for construction and then each year for maintenance. But the payoffs are continuous — we expect the extra shoppers will be shopping each year into the future.

To do our cost benefit analysis correctly, we need to convert these future values into their equivalent present values.

Vocabulary and Concepts

At the completion of this topic the following concepts should be in your tool kit:


Exhortation: This is a fundamental and widely applicable tool. Learn technique first. Caveats and such are easy. Don't get distracted by smokescreens.

Start with interest. We learned previously that we can calculate the future value, FV, one year from now value of an investment with present value, PV, that pays an annual rate, r, as

\begin{align} FV = (1+r) \times PV \end{align}

More generally, what's the value n years from now?

\begin{align} FV = (1+r)^n \times PV \end{align}

Solve this equation for PV:

\begin{align} PV = \frac {FV} {(1+r)^n} \end{align}

The Fundamental Rule: If present value of a future stream of net benefits is greater than 0, it is worth doing. Or, if you are thinking like a business person: a venture with a positive present value is worth taking out a loan on.

Learn the Assumptions: Process unfolds over time. Risk ignored. All returns measurable in same units.

Wait! What about that r?

Q: Why does money cost money? In other words, why does someone who loans you money want both the money and something extra back?

A: Because if she had not given you the money, she could have done something else productive with it. The simplest thing she could have done was put it in a savings account. If the savings account would earn 5% interest, then she will want 5% or better from you before she decides it makes more sense to loan it to you than to put it in the bank.

Our original example said "what's the value next year if the interest rate is r?" Now we are using that r in a slightly different way: the present value, PV, of an amount X at time T in the future is the amount I could invest at rate r and produce X after time T.

Example. What is the present value of $200 3 years from now if the prevailing interest rate is 5%?

\begin{align} PV = \frac {FV} {(1+r)^n} \end{align}
\begin{align} PV = \frac {200} {(1+.05)^3} \end{align}

Example. What is the present value of $121 paid in 2 years if the interest rate is 10%?

Example. What is the present value of $9,261 paid in 3 years if the interest rate is 5%?

Where Does r Come From?

When we calculate the future value of a present investment amount we call R the "interest rate." When we talk about it in the context of calculating the present value of future costs or benefits we call it a "discount rate."

The Discount Rate is a value that represents a reasonable estimate of the return we could earn on a resource if we put it to alternative use.

Review Questions (if you can't write a few coherent sentences, re-study chapter)

What is the "principle of opportunity costs"?

Why does it make sense to distinguish "interest rate" and "discount rate" even though mathematically they are the same thing? (162)

How is this kind of modeling an implementation of our values? Or setting discount rates as a political act?

Explain why this makes sense: "For a yes-no decision on a single project, the choice criterion associated with this concept [internal rate of return] is: 'Undertake a project if its internal rate of return is greater than the appropriate discount rate.'"

Why all the attention to the empirical calculation of appropriate discount rates (pp. 170ff)?

What do we do about risk – the fact that future payoffs are inherently uncertain?

When are negative discount rates appropriate?

What is the present value of a payment X one year from now if discount rate is R?

What is the present value of a payment X two years from now if discount rate is R?

What is the present value of a payment X three years from now if discount rate is R?

What is the present value of a stream of three payments of X, one per year, if the discount rate is R?

Internal Rate of Return

Definitionally, the internal rate of return is "the discount rate at which net present value of project is zero." What does this mean?

Any time you have some resources, there are things you could do with them that might yield some benefit. If nothing else, you could put them in the bank and collect interest. We can think of this collection of opportunities generically as a sort of value creation machine: if we feed our pile of resources in it will generate future value at some rate.

But how do we calculate the IRR? We start with a sequence of costs and benefits

YEAR 1 2 3 4 5 6
COSTS 100 50 10 10 10 10
BENEFITS 0 10 30 30 30 30
NET -100 -40 20 20 20 20

Let's look at the net present value of this stream of costs and benefits, subject to different interest rates. Recall that for any given future benefit, the higher the discount rate, the lower the present value.

This chart tells us that if there are no good investment opportunities out there - the discount rate is low - then the NPV of this project could be 50 or 60. But if there are attractive investment opportunities out there it would be worth less. And the curve crosses the axis at just under 10%. This means that if the opportunities "out there" were paying about 10% then we would be neutral about whether or not to invest in this project. This suggests that we can think of this project as something that has a return on investment of about that rate.

Imagine a project where we have to spend 100 now but we'll get back 200 later. What kind of a bank account is this? Suppose the discount rate is R and that the PV of that later 200 under these conditions is 150. Then the NPV of the project is 50. What if the discount rate were R', a number a little bigger than R. Then the PV(200) would be less, say 130 and the NPV would be 30. And so on. Eventually we'd reach a discount rate R* where the PV(200) is 100 and the NPV of the project is 0.

At this point, we have discovered something fundamental about our project. The discount rate at which the stream of costs and benefits just balances tells us about how "profitable" the project is.

First version of "rule": a project, considered in isolation and without budget constraints, should be undertaken if the internal rate of return is greater than the prevailing discount rate.

Internal Rate of Return Rules and Caveats

Fundamental Rules
  1. For go/no-go decisions on a single project without constraints: go if internal rate of return is greater than appropriate discount rate.
  2. For choosing among alternatives, USUALLY you can choose the project with the highest IRR, BUT
    • can depend on discount rate — different project may be preferred at one discount rate than another and IRR can obscure this
    • may mask the fact that doing nothing may be better option
  3. Variation, "undertake project if IRR > X" can lead to indeterminate results under some conditions

IRR approach only works if

  1. no budget constraints
  2. projects do not preclude one another
  3. net returns start negative and then get positive (more costs up front)

And so, the BOTTOM LINE: choose mix of projects yielding highest present value.

Payback Periods

Popular especially in the private sector. Payback period is usually defined as time it takes for a project to "pay for itself" — that is, when accumulated benefits equal accumulated costs.
Decision criteria can be# Select project with shortest payback period# Go forward on projects with payback period less than X
Results often the same as net present value approach, but not always.

Discounting and Decision Rules

Basic rule. Use same rules as for cost-benefit except base them on the discounted value of the stream of net benefits.