We will first look at discounting
a single cash flow or amount. The cash flow can be discounted
back to a present value by using a discount rate that
accounts for the factors mentioned above (present consumption
preference, risk, and inflation). Conversely, cash flows
in the present can be compounded to arrive at an expected
future cash flow.
The present value of a single cash flow can be written
as follows:
|
PV = FVn
/ (1 + i)n |
PV |
the present value (or initial principal) |
FVn |
future value at the end of n periods |
i |
the interest rate paid each period |
n |
the number of periods |
This means that if you know what a future payment will
be, when it will be made and what interest rate that
we would be paid to achieve comparable future payments
-- you can compute that payment's present value! Armed
with this basic formula, you can compute a present value
quite easily if you know what the future payment will
be (or is expected to be), when it will be made, and
the discount rate applied.
To illustrate: What would you be willing to give up
to have $1,200 a year from now? Stated in our valuation
lexicon we ask: What is the present value of a $1,200
cash flow to be received one year from now? Assuming
an appropriate discount rate of 20%, we can apply our
present value formula:
|
PV = FV / (1 + discount
rate)
= $1,200 / (1 + .20)....
= $1,200 / 1.2 ..............
= $1,000.......................
|
Notice that the higher the discount rate, the
smaller the present value. This inverse relationship
reflects the reality that an amount in the future
is worth less today if present investment opportunities
promise high returns (discount rate). |
Simplifying present value calculations
You can simplify present value calculations
by using a table that shows present value of $1
discounted at i percent for n
periods. Remember the attached tables:
Using the present value table, what is the present
value of $2,500 in 8 years, assuming a discount
rate (the opportunity cost of other investments)
of 12%? of 6%? Answers: $2,500*.4039 = $1,009.70
(12%); $2,500*.6274 = $1,568.50 (6%).
Notice a couple truths:
- The lower the discount rate, the more valuable
are future amounts -- in a low-inflation economy,
the promise of being a millionaire in ten years
means something.
- The higher the discount rate, the less valuable
are future amounts -- in a high-inflation economy,
the promise of becoming a millionaire in ten
years means little.
|
Example:
Suppose the government offered to pay you $150,000
in five years. You determine that you can invest
today in a five-year government note that yields
8.5%. What is the present value of this government
offer?
Answer:
Spreadsheet
solution
We can solve our problem using a calculator,
spreadsheet or table :
...PV
= FVn / (1 + r)n
= $150,000 / (1 + .085)5
= $150,000 / 1.503657
= $99,756.81
That is, having $99,756.81 now (and
investing it at 8.5% for five years) is the same
as having $150,000 in five years. Take your pick.
Notice that the problem assumed
that the $150,000 payment was a near certainty—just
as certain as the government paying on a five-year
note. Things change if the payment had been uncertain.
We’ll begin to explore how uncertainty (risk)
affects our choice of discount rate and ultimate
valuation decisions in the next chapter. - see
Chapter 2 (Risk
and Return).
|
|
1.3.1 Present Value - First Principles