Reading: Present Values
3. Discount Factors
We have seen that the present value of a cash flow is a function of time t (how long it takes until the cash flow arrives) and the discount rate R (the return of an alternative investment with identical risk):
\( PV_0 = \frac{C_t}{(1+R)^t} \)
When we slightly rewrite this expression, we get:
\( PV_0 = C_t \times \frac{1}{(1+R)^t} \)
The second element of this latter expression is also known as the discount factor (DFR,t). Given the investment horizon (t) and the discount rate (R), we can find the present value of any future cash flow (Ct) by multiplying it with the corresponding discount factor (DFR,t):
\( \bf{PV_0 = C_t \times DF_{R,t}} \)
with \( \bf{DF_{R,t} = \frac{1}{(1+R)^t}} \)
Since the discount factor solely depends on the investment horizon and the interest rate, we can present it in a simple two-way table (click to enlarge, or use the accompanying Excel file):
Each cell of the table shows the discount factor for a given investment horizon (rows) and a given interest rate (columns) for one unit of currency.
For example, if we are interested in an investment horizon of 8 years and an opportunity cost of capital of 4%, we can read from the table that the corresponding discount factor DF4%,8 is 0.7307. Put differently, if we expect to receive 1 unit of currency in 8 years at an opportunity cost of capital of 4%, the present value of that future payment is 73.07 cents.
Example 3
You expect to receive 25'000 in 12 years. The appropriate discount rate for the risk that is associated with this investment proposal is 8%. What's the present value of the investment proposal?
From the table above, we see that DF8%,12 is 0.3971. Consequently, the present value of the investment proposal is:
\( PV_0 = C_{12} \times DF_{8\%,12} = 25'000 \times 0.3971 = 9'927.50 \)