# Why We Discount

Cash now is worth more than cash in the future. There are several reasons for this:

- People are impatient — You'd probably rather have a beer now than two beers a year from now.

- The institution of interest — If you have cash now but don't expect to use it for awhile, you can lend it to someone else and charge them interest on it.

- Uncertainty about the future — Maybe you're an exceptionally patient person and would rather have two beers next year than one beer now, but what if the folks at CERN accidentally destroy the universe before then?

# Exponential vs Hyperbolic Discounting

Exponential and hyperbolic discounting are two ways to discount the value of a future event.

Exponential discounting is what the financial markets use and what economists use as a simplifying assumption. The exponential model is time consistent. That means that the relative benefits of events in the future will not change because time passes, only if there is new information or something else changes.

Humans intuitively use hyperbolic discounting or some other form of nonexponential discounting. People have a preference for events in the near future.

# Assumptions

Since we don't want to get too deep into the math here, we're going to make some assumptions.

- Costs and benefits of equal size have equal value in all periods

This isn't true. The value of a cost or benefit usually depends on what else is going on. $100 on the day you win the lottery is worth less than $100 on the day you lost your home.

- The values of costs and benefits are independent of costs and benefits in other periods.

This also is not true. Anyone that has had a hangover or tried walking up stairs the day after leg day knows this.

- Benefits can offset costs — they are equal but opposite

You guessed it, also not true. People tend put more effort into avoiding losses than acquiring gains of the same size. It is commonly known as loss aversion or gain/loss asymmetry.

- Costs and benefits are known in advance with certainty

This is obviously false.

# Discounting a Single Value

## Rules about Discounting

# A Graphical and Symbolic Notation

## Graphical

Watch the above video on Time Value Notation for graphical notation.

## Symbolic

$P$ - Present worth in period zero

$P_n$ - Present worth in period n

$F$ - Period N present worth. Same as P_N

$i$ - Interest rate

$N$ - Last period

$n$- Some time period

$A_n$ - Cost or benefit in period n

$A$ - The cost or benefit in a constant series

$G$- Change in cost or benefit every period in a linear gradient series

$g$- Growth rate in a geometric gradient series

# Questions

The following represent good sample questions that would help you prepare for an exam:

- Midterm 1 Exponential Discounting
- Exponential Discount Sample Question
- Time Value of Money Assumptions