Tools of Micro-economic Analysis:Discounting

January 2010

2

Time value of money

Discounting is an economic technique for placing a value on financial values in the future.

Many environmental costs will occur far into the future.

What value should we place on the costs of climate change that may be experienced by our grandchildren?

3

A bird in the hand is worth two in the bush

What this means, of course, is that something that is certain is worth more than something that is uncertain.

$100 received next year is worth less than $100 in your hand now.

Why? We may die and not collect

Those who are paying the $100 are not trustworthy

We are impatient and need the money

Therefore, we discount future events.

4

How do we measure value across time?

What is the time value of money?

We start be using the compound formula for interest and saving:

$100 saved for 1 year at 10% will yield $110 in 1 year $100 x (1 ) $100 x (.10) = $100 x (1.10) = $110

$100 saved for 2 years at 10% will yield $121 in 2 years $100 x (1+.10)x(1.+10) = $100 x (1.10)2 = $121

$100 saved for n years at 10% will yield a future value in two years

FV = $100x(1.10)n

5

What is the present value of$110 received in 1 year?

If the interest is 10%, the value of $110 received in 1 year is $100 now.

Since we can save $100 to make $110 in one year, we should not care (be indifferent) between $100 now and $110 in one year.

We can equate any amount received (or spent) in the future to a present value, as long as we know the interest rate.

$110 received in one year has a present value of $110/(1+.10) = $100

$121 received in two years has a present value of $121/(1.10)2 = $100

A future value (FV) to be received in n years has a present value

PV = FV/(1.10)n

6

Compare the two formulas

FV = PV x (1+.10)2 = PV(1+i)2

$121 = $100 x (1+.10)2

PV = FV/(1+.10)2 $100 = $121/(1+.10)2

The interest rate is a number that equates present and future values.General formula

FV = PV x (1+i)n

PV = FV/(1+i)n

7

The higher the interest rate – the more we discount the future and value to the present

$100 received in 1 year when interest is 5% is presently worth

PV = $100/(1.05) = $95.24

$100 received in 1 year when interest is 15% is presently worth

PV = $100/(1.15) = $86. 96

8

Interest rates

Market interest reflects uncertainty and risk.

Low risk investments tend to have low interest rates

Risk averse people want the money now, and need a high interest rate to compensate them for waiting.

9

What is a salary worth

Imagine you expect to get $50,000 in 15 years.

If interest is 15%, what is that worth to you now

PV = $50,000/(1+.15)15

PV = $50,000/(8.14)

= $6143

If i = .05, the present value rises to $24,038

10

The present value of a flow of future income

Imagine that you will receive $10,000 a year (at year end) for the next five years and that i = 5%

What is the present value of this stream of income?

We simply calculate the present value of each year and add.

$10,000/(1.05)1+ $10,000/(1.05)2+ $10,000/(1.05)3+ $10,000/(1.05)4+ $10,000/(1.05)5 =

$9523+$9070+$8638+$8227+$7835 = $43,493

$10,000 received each year for 5 years has a total present value of $43,493 if interest rate = 5%

If i=7% the PV = $41,002

See spreadsheet.