# Present Value and the Power of Time in Financial Economics (Part 2)

In Part 1, we looked at two views of the power of time in financial economics. While Albert Einstein is remembered for E = mc^2, we were more interested in his rumored fascination with FV(CFn) =CF* (1+r)^n, as he supposedly said that “the most powerful force in the universe is compound interest.” As counterpoint, economist John Maynard Keynes dismissed this notion by saying, “In the long run, we are all dead.”

Testing the two professors’ theories produces some dramatic results. Our experiment looked at the present value of a cash flow perpetuity, assuming it was a reasonable proxy for the cash flow from a business. We used the inverse of the future value formula, where PV(CFn) =CFn * 1/(1+r)^n, enabling us to take a $1 cash flow in period n and determine its present value. Using a hypothetical, privately-held company’s cost of equity of 25% as our discount rate, we found that 99% of the value of the perpetuity is in the first 20 years. In fact the perpetuity’s annual cash flow is worth only a penny on the dollar after 19 years. This result was even more pronounced for a venture capital investor using a 50% discount rate. 98.3% of the value is created in the first ten years, and the $1 cash flow was only worth a penny on the dollar after 11 years. As Thomas Hobbes might have put it, the life of the perpetuity is “nasty, brutish and short.”

So far Professor Keynes’s theory is looking rather strong, just like the NY Giants looked against the Eagles when they were up 31-10 with seven minutes to go last Sunday. Can Professor Einstein rally like the Eagles by unleashing “the most powerful force in the universe?” (I’m referring to compound interest, not Michael Vick).

We can incorporate a growth rate by using the following formula for present value:

PV(CFn) = CFn* (1*(1+g)^(n-1))/(1+r)^n

So let’s add a 3% growth rate, which could be seen as representing the long-term inflation rate. Using our example of a $1 perpetuity, we get the present value for each year as shown below:

Present Value of a $1 per year perpetuity growing at 3% | |||||

Growth Rate | 3% | 3% | |||

Discount Rate | 25% | % of | 50% | % of | |

Capitalization Rate | 22% | Total CF | | 47% | Total CF |

Year | |||||

1 | $0.80 | 17.6% | $0.67 | 31.3% | |

2 | $0.66 | 14.5% | $0.46 | 21.5% | |

3 | $0.54 | 12.0% | $0.31 | 14.8% | |

4 | $0.45 | 9.9% | $0.22 | 10.1% | |

5 | $0.37 | 8.1% | $0.15 | 7.0% | |

6 | $0.30 | 6.7% | $0.10 | 4.8% | |

7 | $0.25 | 5.5% | $0.07 | 3.3% | |

8 | $0.21 | 4.5% | $0.05 | 2.3% | |

9 | $0.17 | 3.7% | $0.03 | 1.5% | |

10 | $0.14 | 3.1% | $0.02 | 1.1% | |

11 | $0.12 | 2.5% | $0.02 | 0.7% | |

12 | $0.10 | 2.1% | $0.01 | 0.5% | |

13 | $0.08 | 1.7% | $0.01 | 0.3% | |

14 | $0.06 | 1.4% | $0.01 | 0.2% | |

15 | $0.05 | 1.2% | $0.00 | 0.2% | |

16 | $0.04 | 1.0% | $0.00 | 0.1% | |

17 | $0.04 | 0.8% | $0.00 | 0.1% | |

18 | $0.03 | 0.7% | $0.00 | 0.1% | |

19 | $0.02 | 0.5% | $0.00 | 0.0% | |

20 | $0.02 | 0.4% | $0.00 | 0.0% | |

21 | $0.02 | 0.4% | $0.00 | 0.0% | |

22 | $0.01 | 0.3% | $0.00 | 0.0% | |

23 | $0.01 | 0.2% | $0.00 | 0.0% | |

24 | $0.01 | 0.2% | $0.00 | 0.0% | |

25 | $0.01 | 0.2% | $0.00 | 0.0% | |

26 | $0.01 | 0.1% | $0.00 | 0.0% | |

27 | $0.01 | 0.1% | $0.00 | 0.0% | |

28 | $0.00 | 0.1% | $0.00 | 0.0% | |

29 | $0.00 | 0.1% | $0.00 | 0.0% | |

30 | $0.00 | 0.1% | $0.00 | 0.0% | |

31 | $0.00 | 0.1% | $0.00 | 0.0% | |

32 | $0.00 | 0.0% | $0.00 | 0.0% | |

33 | $0.00 | 0.0% | $0.00 | 0.0% | |

34 | $0.00 | 0.0% | $0.00 | 0.0% | |

35 | $0.00 | 0.0% | $0.00 | 0.0% | |

Sum of cash flows | $4.54 | 100.0% | $2.13 | 100.0% | |

% of total value | |||||

After 10 years | 85.7% | 85.7% | 97.7% | 97.7% | |

After 20 years | 98.0% | 98.0% | 99.9% | 99.9% | |

After 30 years | 99.8% | 99.8% | 100.0% | 100.0% |

Adding the 3% growth rate appears to have done little to increase the effective life of our perpetuity’s cash stream. Instead of 99% of the value being realized in the first 20 years with no growth, 98% is realized in the first 20 years with 3% growth. The point at which a dollar perpetuity is worth a penny occurs in year 22, an increase of only three years over the no growth scenario. Our venture capitalist still gets 97.7% of his value in the first ten years, little changed from the 98.3% he received before we added the growth rate. Keynes is smiling.

What if we used a 6% growth rate, which could represent 3% inflation plus 3% long-term GDP growth? Look at the chart below:

Present Value of a $1 per year perpetuity growing at 6% | |||||

Growth Rate | 6% | 6% | |||

Discount Rate | 25% | % of | 50% | % of | |

Capitalization Rate | 19% | Total CF | | 44% | Total CF |

Year | |||||

1 | $0.80 | 15.2% | $0.67 | 29.3% | |

2 | $0.68 | 12.9% | $0.47 | 20.7% | |

3 | $0.58 | 10.9% | $0.33 | 14.6% | |

4 | $0.49 | 9.3% | $0.24 | 10.4% | |

5 | $0.41 | 7.9% | $0.17 | 7.3% | |

6 | $0.35 | 6.7% | $0.12 | 5.2% | |

7 | $0.30 | 5.7% | $0.08 | 3.7% | |

8 | $0.25 | 4.8% | $0.06 | 2.6% | |

9 | $0.21 | 4.1% | $0.04 | 1.8% | |

10 | $0.18 | 3.4% | $0.03 | 1.3% | |

11 | $0.15 | 2.9% | $0.02 | 0.9% | |

12 | $0.13 | 2.5% | $0.01 | 0.6% | |

13 | $0.11 | 2.1% | $0.01 | 0.5% | |

14 | $0.09 | 1.8% | $0.01 | 0.3% | |

15 | $0.08 | 1.5% | $0.01 | 0.2% | |

16 | $0.07 | 1.3% | $0.00 | 0.2% | |

17 | $0.06 | 1.1% | $0.00 | 0.1% | |

18 | $0.05 | 0.9% | $0.00 | 0.1% | |

19 | $0.04 | 0.8% | $0.00 | 0.1% | |

20 | $0.03 | 0.7% | $0.00 | 0.0% | |

21 | $0.03 | 0.6% | $0.00 | 0.0% | |

22 | $0.03 | 0.5% | $0.00 | 0.0% | |

23 | $0.02 | 0.4% | $0.00 | 0.0% | |

24 | $0.02 | 0.3% | $0.00 | 0.0% | |

25 | $0.02 | 0.3% | $0.00 | 0.0% | |

26 | $0.01 | 0.2% | $0.00 | 0.0% | |

27 | $0.01 | 0.2% | $0.00 | 0.0% | |

28 | $0.01 | 0.2% | $0.00 | 0.0% | |

29 | $0.01 | 0.2% | $0.00 | 0.0% | |

30 | $0.01 | 0.1% | $0.00 | 0.0% | |

31 | $0.01 | 0.1% | $0.00 | 0.0% | |

32 | $0.00 | 0.1% | $0.00 | 0.0% | |

33 | $0.00 | 0.1% | $0.00 | 0.0% | |

34 | $0.00 | 0.1% | $0.00 | 0.0% | |

35 | $0.00 | 0.1% | $0.00 | 0.0% | |

36 | $0.00 | 0.0% | $0.00 | 0.0% | |

Sum of cash flows | $5.26 | 100.0% | $2.27 | 100.0% | |

% of total value | |||||

After 10 years | 80.8% | 80.8% | 96.9% | 96.9% | |

After 20 years | 96.3% | 96.3% | 99.9% | 99.9% | |

After 30 years | 99.3% | 99.3% | 100.0% | 100.0% |

Instead of 99% of the value being realized in the first 20 years with no growth, 96.3% is realized in the first 20 years with 6% growth. The point at which a dollar perpetuity is worth a penny occurs in year 26, a gain of 7 years. The venture capitalist gets 96.9% of his value in the first ten years, little changed from 98.3% with no growth. So far, increasing the growth rate seems to have little effect, as most of the value is still produced in the first 20 years.

This is an interesting result for business valuation experts, as it appears that using a long-term growth rate of 6% does not push the financial time horizon out to a level that would appear unrealistic for an established privately-held company. Going from a 3% growth rate with 98% of the value determined in 20 years to a 6% growth rate with 96.3% lowers the portion of the value determined in the first 20 years by only 1.7%.

Note, however, that while the time horizon changed little, going from a 3% growth rate to a 6% growth rate (with a 25% discount rate) changed the total present value of the perpetuity a lot, increasing it from $4.54 to $5.26, a 16% increase. So while the growth rate increase has little impact on the time horizon, it can have a significant impact on value.

Getting beaten badly by Professor Keynes, Professor Einstein decides to unleash the bomb (No, not THAT bomb!) He pumps the long-term growth rate up to 10%, arguing that he is valuing a high tech company where the industry has been growing at double-digit rates for decades. Let’s look at the results.

Present Value of a $1 per year perpetuity growing at 10% | |||||

Growth Rate | 10% | 10% | |||

Discount Rate | 25% | % of | 50% | % of | |

Capitalization Rate | 15% | Total CF | | 40% | Total CF |

Year | |||||

1 | $0.80 | 12.0% | $0.67 | 26.7% | |

2 | $0.70 | 10.6% | $0.49 | 19.6% | |

3 | $0.62 | 9.3% | $0.36 | 14.3% | |

4 | $0.55 | 8.2% | $0.26 | 10.5% | |

5 | $0.48 | 7.2% | $0.19 | 7.7% | |

6 | $0.42 | 6.3% | $0.14 | 5.7% | |

7 | $0.37 | 5.6% | $0.10 | 4.1% | |

8 | $0.33 | 4.9% | $0.08 | 3.0% | |

9 | $0.29 | 4.3% | $0.06 | 2.2% | |

10 | $0.25 | 3.8% | $0.04 | 1.6% | |

11 | $0.22 | 3.3% | $0.03 | 1.2% | |

12 | $0.20 | 2.9% | $0.02 | 0.9% | |

13 | $0.17 | 2.6% | $0.02 | 0.6% | |

14 | $0.15 | 2.3% | $0.01 | 0.5% | |

15 | $0.13 | 2.0% | $0.01 | 0.3% | |

16 | $0.12 | 1.8% | $0.01 | 0.3% | |

17 | $0.10 | 1.6% | $0.00 | 0.2% | |

18 | $0.09 | 1.4% | $0.00 | 0.1% | |

19 | $0.08 | 1.2% | $0.00 | 0.1% | |

20 | $0.07 | 1.1% | $0.00 | 0.1% | |

21 | $0.06 | 0.9% | $0.00 | 0.1% | |

22 | $0.05 | 0.8% | $0.00 | 0.0% | |

23 | $0.05 | 0.7% | $0.00 | 0.0% | |

24 | $0.04 | 0.6% | $0.00 | 0.0% | |

25 | $0.04 | 0.6% | $0.00 | 0.0% | |

26 | $0.03 | 0.5% | $0.00 | 0.0% | |

27 | $0.03 | 0.4% | $0.00 | 0.0% | |

28 | $0.03 | 0.4% | $0.00 | 0.0% | |

29 | $0.02 | 0.3% | $0.00 | 0.0% | |

30 | $0.02 | 0.3% | $0.00 | 0.0% | |

31 | $0.02 | 0.3% | $0.00 | 0.0% | |

32 | $0.02 | 0.2% | $0.00 | 0.0% | |

33 | $0.01 | 0.2% | $0.00 | 0.0% | |

34 | $0.01 | 0.2% | $0.00 | 0.0% | |

35 | $0.01 | 0.2% | $0.00 | 0.0% | |

36 | $0.01 | 0.1% | $0.00 | 0.0% | |

37 | $0.01 | 0.1% | $0.00 | 0.0% | |

38 | $0.01 | 0.1% | $0.00 | 0.0% | |

39 | $0.01 | 0.1% | $0.00 | 0.0% | |

40 | $0.01 | 0.1% | $0.00 | 0.0% | |

41 | $0.00 | 0.1% | $0.00 | 0.0% | |

42 | $0.00 | 0.1% | $0.00 | 0.0% | |

43 | $0.00 | 0.1% | $0.00 | 0.0% | |

44 | $0.00 | 0.0% | $0.00 | 0.0% | |

45 | $0.00 | 0.0% | $0.00 | 0.0% | |

Sum of cash flows | $6.67 | 100.0% | $2.50 | 100.0% | |

% of total value | |||||

After 10 years | 72.1% | 72.1% | 95.5% | 95.5% | |

After 20 years | 92.2% | 92.2% | 99.8% | 99.8% | |

After 30 years | 97.8% | 97.8% | 100.0% | 100.0% |

Now Professor Einstein is feeling “the Force!” Instead of 99% of the value being realized in the first 20 years with no growth, only 92.2% is realized in the first 20 years with 10% growth. We have to go out 30 years to get the percentage up to 97.8%. The point at which a dollar is worth a penny occurs in year 33, a gain of 14 years (74%) over the original 19 years. The venture capitalist still gets 95.5% of his value in the first ten years, not that different from 98.3% with no growth.

The effects on value are more dramatic. The value of the $1 perpetuity with 10% growth is $6.67, a large 67% increase over the value of a $1 perpetuity with no growth discounted at 25%, which is only $4. Professor Einstein finally scores big.

What can we conclude from this? The use of a 3% or 6% long term growth rate in a small, privately-held company valuation model, if appropriate to the individual circumstances of the company, does not seem to present business valuation experts with an extraordinarily long time horizon for the business to exist to realize most of the value. It can, however, have a significant impact on value. On the other hand, the use of a 10% long-term growth rate, selected because the industry is a “high tech growth industry”, may present issues of whether it is realistic to assume a small, privately-held business will exist the 30 or more years needed to realize most of its value. However, it may be appropriate for valuing certain types of venture capital investments using appropriately high venture capital discount rates. Finally, it appears Professor Einstein was somewhat better at describing the relationship between space and time than he was at describing the relationship between time and value.