Discounting


It is somewhat easier to understand and valuate an investement's cash flow, if all amounts are in today's euros. The present worth of a future cash flow is its value in today's euros. For a cash flow stream all cash flows must be transformed into present worth using some interest rate, the discount rate.

Discount Rate


Discounting is the process of evaluating future cash flows as an equivalent present worth. The present worth of a future cash flow is usually less than the face value.

The factor by which the future cash flow must be discounted is the discount factor. The one-year discount factor is d1 = 1/(1 + r), where r is the one-year discount rate. If an amount A is to be received in one year, its present worth is the discounted amount d1A (d1 < 1).

For public projects, the UK government recommends a specific real discount rate of 3.5% (Treasury 2003), while the Danish Energy Agency requires a real discount rate of 5% in project analyses.

Obviously, the discount rate is not just a market interest rate, but it must accomodate inflation and risk. The lower bound for the discount rate is the minimum rate of return that will be attractive to the investor, known as the minimum attractive rate of return (MARR). This is for instance the rate that can be achieved by investing in secure bonds. The discount rate is therefore

      r = MARR + risk premium


The Danish discount rate for public projects is based on a 4% real MARR plus a 1% risk premium (Energistyrelsen 2011).

Discounted Cash Flow


For a cash flow stream we define a discount factor dk for each time point k. The future cash flows must be multiplied by these factors to obtain the present worth.

For a given cash flow stream (x0, x1, ..., xK) and discount rate r per period, the discount factor is dk = 1/(1 + r)k. The discounted cash flow stream (DCF) is (Luenberger 1998)

      DCF = (x0, d1x1, d2x2, ..., dKxK)


Each discount factor dk is a weight that transforms its cash flow xk into its present worth. It is a separate concept from inflation, and it is based on the principle that a nearby euro is worth more than a distant euro, time preference.


FigDiscountFactors.png
Fig. 1. Discount factors. Factors using the high rate
(0.10) drop off much faster than those using the low
rate (0.01).



Example (discount factors) Consider a discount rate of 1% corresponding to r = 0.01. The corresponding discount factor for year 1 is

      d1 = 1/(1 + r) = 1/(1 + 0.01) = 0.99


To compare, the ten times larger discount rate of r = 0.1 results in

      d1 = 1/(1 + r) = 1/(1 + 0.1) = 0.91


The difference is noticeable. Continuing for ten years, the results are gathered in the table below and also plotted in Figure 1.

Year 0 1 2 3 4 5 6 7 8 9 10
r = 0.011.000.990.980.970.960.950.940.930.920.910.91
r = 0.101.000.910.830.750.680.620.560.510.470.420.39



The magnitude of the discount factors dk over a period of time (k = 1, 2, ..., K) decreases exponentially. The discount rate controls the decay of the exponential curve. A large discount rate suppresses distant cash flows more than a small discount rate.

We can therefore adjust the curve to suit our personal time preference.

The 72 rule


The 72 rule is a method to calculate, mentally, the halving (doubling) time of an exponential function. In order to quickly estimate the number of periods k0.5 required to halve the initial amount, divide the number 72 by the discounting rate in percent. That is,

      k0.5 = 72/(r * 100)


For r = 0.10, k0.5 = 72/10 = 7.2. Comparing with Figure 1 (r = 0.10), it does seem that at year 7 the discount factor is close to half; it is actually 0.51 according to the table. The event happens after the end of year 7, and due to the discrete nature of the cash flows, it appears at the end of year 8.

We can invert the calculation in order to guess at a discount rate. Let us say that our desired halving time is k0.5 = 5 years, then

      r = 72/(k0.5 * 100) = 72/500 = 0.144


That is, a discount rate of approximately 14.4% will model our time preference.

A discount rate of 3.5%, as in the UK (Treasury 2003), corresponds to a halving time of 21 years. A discount rate of 5%, as in Denmark (ref), corresponds to a halving time of 14 years.

  1. Wikipedia Discounted cash flow
  2. Wikipedia Minimum acceptable rate of return
  3. Wikipedia Rule of 72



The original document is available at http://seacourse.dk/wiki/tiki-index.php?page=Discounting