Transcript Present Value and the Interest Rate

APPENDIX
Present Value and
Discounting
17
After studying this appendix, you will be able to
Explain how to calculate the present value of a future
amount of money
Explain how a firm uses a present value calculation to
make an investment decision
Explain the relationship between present value and the
interest rate
Comparing Current and Future Dollars
Discounting is converting a future amount of money into
a present value.
The present value of a future amount of money is the
amount that, if invested today, will grow to be as large as
that future amount when the interest that it will earn is
taken into account.
The easiest way to understand discounting is to consider
how a present value grows to a future amount of money
because of compound interest.
Comparing Current and Future Dollars
Compound Interest
Compound interest is the interest on an initial investment
plus the interest on the interest that the investment has
previously earned.
Because of compound interest, a present amount of
money (a present value) grows into a larger future
amount.
Future amount = Present value + Interest income
Comparing Current and Future Dollars
If the interest rate is r per year, then the amount of money
a person has one year in the future is
Amount after I year = Present value + (r  Present value)
Or
Amount after I year = Present value  (1 + r)
For example, if the present value is $100 and r is 0.1 (10
percent a year), then the amount after 1 year is $110.
Comparing Current and Future Dollars
After 2 years:
Amount after 2 years = Present value  (1 + r)2
For example, if the present value is $100 and r is 0.1 (10 percent a
year), then
Amount after 2 years = $100  (1 + 0.1)2
Amount after 2 years = $100  (1.1)2
Amount after 2 years = $121
Similarly, the amount of money after n years is
Amount after n years = Present value  (1 + r)n
Comparing Current and Future Dollars
Discounting a Future Amount
To find the present value of an amount one year in the
future, we divide the future amount by (1 + r).
So an amount after 1 year has a
Present value = Amount 1 year in future/(1 + r)
Similarly, an amount n years in the future has a
Present value = Amount n years in future/(1 + r)n
Comparing Current and Future Dollars
Present Value of a Sequence of Future Amounts
The return a firm earns from capital accrues over a
number of future years.
So to calculate the present value of these returns, the firm
must calculate the present value of each year’s return and
then sum them.
Let’s now look at the present value of an investment
decision.
Present Value and Investment Decision
Tina runs Taxfile, Inc., a firm that sells advice to taxpayers.
Tina is considering buying a new computer that costs
$2,000 and has a life of two years.
If Tina buys the computer, she will pay $2,000 now.
She expects new business to bring in an additional $1,150
at the end of each of the next two years.
To calculate the present value, PV, of the marginal
revenue product (MRP) of a new computer, Tina
calculates
PV = MRP1/(1 + r) + MRP2/(1 + r)2
Present Value and Investment Decision
PV = MRP1/(1 + r) + MRP2/(1 + r)2
If the interest rate is 4 percent a year:
PV = $1,150/(1 + 0.04) + $1,150 /(1 + 0.04)2
PV = $1,150/(1.04) + $1,150 /(1.04)2
PV = $1,106 + $1,063
PV = $2,169
Present Value and Investment Decision
The Decision To Buy
Tina decides whether to buy the computer by comparing
the present value of its future flow of marginal revenue
product with its purchase price.
She makes this comparison by calculating the net present
value of the computer.
The net present value (NVP) is the present value of the
future flow of marginal revenue product minus the price of
the capital good.
If NVP is positive, the firm buys the capital; if NVP is
negative, the firm does not buy the capital.
Present Value and Investment Decision
Tina decides whether to buy the computer by comparing
the present value of $2,169 with its purchase price
($2,000).
The net present value is positive, so Tina buys the
computer.
Table A17.2 provide examples of a net present value
calculation.
Present Value and the Interest Rate
The higher the interest rate, the smaller is the present
value of a given future amount of money.
As the interest rate rises, fewer projects have positive
NVP, other things remaining the same.
Table A17.2 shows the calculations using three interest
rates—4%, 8%, and 12% a year.
The higher the interest rate, the smaller is the present
value and the smaller is NVP.
Present Value and the Interest Rate
The higher the interest rate, the smaller is the quantity of
physical capital demanded.
But to finance the purchase of physical capital, firms
demand financial capital.
So the higher the interest rate, the smaller is the quantity
of financial capital demanded.
Figure A17.1 on the next slide shows Taxfile’s demand
for capital.
Present Value and the Interest Rate
Figure A17.1 shows a
firm’s demand curve for
capital.
This demand curve is
based on the calculations
in Table A17.2.
THE END