Present Value and the Power of Time in Financial Economics (Part 1)
Albert Einstein is remembered for E = mc^2, but it is rumored that he was more enamored of FV(CFn) = CFn *(1+r)^n, supposedly having said that “the most powerful force in the universe is compound interest.” It is an interesting quote on the power of time in economics from a physicist that helped unlock the nuclear age, but it is undermined by an economist who represented the dismal science at its dismal best. As John Maynard Keynes dryly put it, “In the long run, we are all dead.” Leave it to the business valuation profession to combine the best of these sentiments.
A cornerstone of business valuation is the formula for present value, which takes Einstein’s much admired compound interest formula and turns it on its head, using it as a divisor. So PV(CFn) = CFn * 1/(1+r)^n, enabling us to take a cash flow in period n and determine its present value. Business valuation experts use this formula virtually every day, but how often do we think about it like Einstein and Keynes?
A simple example will show what I mean. Below is a chart showing the present value of $1 using two discount rates, 25% and 50%. Let’s assume 25% is the cost of equity for a privately-held company and 50% is the discount rate for a venture capital investment.
Present Value of a $1 per year perpetuity | |||||
% of | % of | ||||
Discount Rate | 25% | Total CF | 50% | Total CF | |
Year | |||||
1 | $0.80 | 20.0% | $0.67 | 33.3% | |
2 | $0.64 | 16.0% | $0.44 | 22.2% | |
3 | $0.51 | 12.8% | $0.30 | 14.8% | |
4 | $0.41 | 10.3% | $0.20 | 9.9% | |
5 | $0.33 | 8.2% | $0.13 | 6.6% | |
6 | $0.26 | 6.6% | $0.09 | 4.4% | |
7 | $0.21 | 5.2% | $0.06 | 2.9% | |
8 | $0.17 | 4.2% | $0.04 | 2.0% | |
9 | $0.13 | 3.4% | $0.03 | 1.3% | |
10 | $0.11 | 2.7% | $0.02 | 0.9% | |
11 | $0.09 | 2.2% | $0.01 | 0.6% | |
12 | $0.07 | 1.7% | $0.01 | 0.4% | |
13 | $0.05 | 1.4% | $0.01 | 0.3% | |
14 | $0.04 | 1.1% | $0.00 | 0.2% | |
15 | $0.04 | 0.9% | $0.00 | 0.1% | |
16 | $0.03 | 0.7% | $0.00 | 0.1% | |
17 | $0.02 | 0.6% | $0.00 | 0.1% | |
18 | $0.02 | 0.5% | $0.00 | 0.0% | |
19 | $0.01 | 0.4% | $0.00 | 0.0% | |
20 | $0.01 | 0.3% | $0.00 | 0.0% | |
21 | $0.01 | 0.2% | $0.00 | 0.0% | |
22 | $0.01 | 0.2% | $0.00 | 0.0% | |
23 | $0.01 | 0.1% | $0.00 | 0.0% | |
24 | $0.00 | 0.1% | $0.00 | 0.0% | |
25 | $0.00 | 0.1% | $0.00 | 0.0% | |
26 | $0.00 | 0.1% | $0.00 | 0.0% | |
27 | $0.00 | 0.1% | $0.00 | 0.0% | |
28 | $0.00 | 0.0% | $0.00 | 0.0% | |
29 | $0.00 | 0.0% | $0.00 | 0.0% | |
30 | $0.00 | 0.0% | $0.00 | 0.0% | |
Sum of cash flows | $4.00 | 100.0% | $2.00 | 100.0% | |
% of total value | |||||
After 10 years | 89.4% | 89.4% | 98.3% | 98.3% | |
After 20 years | 99.0% | 99.0% | 100.0% | 100.0% | |
After 30 years | 100.0% | 100.0% | 100.0% | 100.0% |
From the chart we can see that the present value of $1 using a 25% discount rate is only 1 cent in year 19, and that 99% of the total value of the perpetuity is based on the cash flows in the first 20 years. By year 28, the value of $1 is effectively zero, or to paraphrase Lord Keynes’ pithy terminology, the perpetuity is “dead.” A financial perpetuity is supposed to be immortal, so it may come as a shock that it has zero present value after a relatively short life of 28 years. “Perpetuity, we barely knew ye.”
This is a useful result for business valuation experts if we think of the cash flow to a business as a perpetuity. In developing our cost of equity, we often choose the rate on a 20 year T-bond as a risk free rate. The cash flows of the bond are paid over 20 years, and 99% of our business perpetuity’s value is determined over the same period. That is a nice matching of the interest rate term structure. It might even be a reasonable financial time horizon for an established privately-held company.
For the venture capitalist, the grim reaper comes much sooner. At a 50% discount rate the value of $1 drops to one cent in just 11 years and the perpetuity is essentially “dead” in year 18. No wonder that venture capital funds typically have terms of 10 years, it just isn’t worth sticking around much longer if your discount rate on investments is 50% and 98.3% of your total value is created in the first 10 years.
So far it would appear that Professor Keynes is getting the better of this argument, but in fairness to Professor Einstein, we haven’t unleashed “the most powerful force in the universe.” What would happen if we grew the perpetuity at the long–term growth rate? Or as Einstein would put it, let’s experiment and change PV(CFn) = CFn * 1/(1+r)^n into
PV(CFn) = CFn *(1*(1+g)^(n-1)) / (1+r)^n, so that in each year we grow the cash flow.
Next time in Part II we will find out.
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