Donella Meadows Thinking In Systems Pdf To Excel

In finance, the rule of 72, the rule of 70^[1] and the rule of 69.3 are methods for estimating an investment's doubling time. The rule number (e.g., 72) is divided by the interest percentage per period (usually years) to obtain the approximate number of periods required for doubling. Although scientific calculators and spreadsheet programs have functions to find the accurate doubling time, the rules are useful for mental calculations and when only a basic calculator is available.^[2]

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These rules apply to exponential growth and are therefore used for compound interest as opposed to simple interest calculations. They can also be used for decay to obtain a halving time. The choice of number is mostly a matter of preference: 69 is more accurate for continuous compounding, while 72 works well in common interest situations and is more easily divisible.There are a number of variations to the rules that improve accuracy. For periodic compounding, the exact doubling time for an interest rate of r percent per period is

{displaystyle t={frac {ln(2)}{ln(1+r/100)}}approx {frac {72}{r}}}

where t is the number of periods required. The formula above can be used for more than calculating the doubling time. If one wants to know the tripling time, for example, replace the constant 2 in the numerator with 3. As another example, if one wants to know the number of periods it takes for the initial value to rise by 50%, replace the constant 2 with 1.5.

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4Adjustments for higher accuracy
5Derivation

Using the rule to estimate compounding periods[edit]

To estimate the number of periods required to double an original investment, divide the most convenient 'rule-quantity' by the expected growth rate, expressed as a percentage.

For instance, if you were to invest $100 with compounding interest at a rate of 9% per annum, the rule of 72 gives 72/9 = 8 years required for the investment to be worth $200; an exact calculation gives ln(2)/ln(1+0.09) = 8.0432 years.

Similarly, to determine the time it takes for the value of money to halve at a given rate, divide the rule quantity by that rate.

To determine the time for money's buying power to halve, financiers divide the rule-quantity by the inflation rate. Thus at 3.5% inflation using the rule of 70, it should take approximately 70/3.5 = 20 years for the value of a unit of currency to halve^[1].
To estimate the impact of additional fees on financial policies (e.g., mutual fund fees and expenses, loading and expense charges on variable universal life insurance investment portfolios), divide 72 by the fee. For example, if the Universal Life policy charges an annual 3% fee over and above the cost of the underlying investment fund, then the total account value will be cut to 1/2 in 72 / 3 = 24 years, and then to just 1/4 the value in 48 years, compared to holding exactly the same investment outside the policy.

Choice of rule[edit]

The value 72 is a convenient choice of numerator, since it has many small divisors: 1, 2, 3, 4, 6, 8, 9, and 12. It provides a good approximation for annual compounding, and for compounding at typical rates (from 6% to 10%). The approximations are less accurate at higher interest rates.

For continuous compounding, 69 gives accurate results for any rate. This is because ln(2) is about 69.3%; see derivation below. Since daily compounding is close enough to continuous compounding, for most purposes 69, 69.3 or 70 are better than 72 for daily compounding. For lower annual rates than those above, 69.3 would also be more accurate than 72.^[3]

Graphs comparing doubling times and half lives of exponential growths (bold lines) and decay (faint lines), and their 70/t and 72/t approximations. In the SVG version, hover over a graph to highlight it and its complement.

Rate	Actual Years	Rate * Actual Years	Rule of 72	Rule of 70	Rule of 69.3	72 adjusted	E-M rule
0.25%	277.605	69.401	288.000	280.000	277.200	277.667	277.547
0.5%	138.976	69.488	144.000	140.000	138.600	139.000	138.947
1%	69.661	69.661	72.000	70.000	69.300	69.667	69.648
2%	35.003	70.006	36.000	35.000	34.650	35.000	35.000
3%	23.450	70.349	24.000	23.333	23.100	23.444	23.452
4%	17.673	70.692	18.000	17.500	17.325	17.667	17.679
5%	14.207	71.033	14.400	14.000	13.860	14.200	14.215
6%	11.896	71.374	12.000	11.667	11.550	11.889	11.907
7%	10.245	71.713	10.286	10.000	9.900	10.238	10.259
8%	9.006	72.052	9.000	8.750	8.663	9.000	9.023
9%	8.043	72.389	8.000	7.778	7.700	8.037	8.062
10%	7.273	72.725	7.200	7.000	6.930	7.267	7.295
11%	6.642	73.061	6.545	6.364	6.300	6.636	6.667
12%	6.116	73.395	6.000	5.833	5.775	6.111	6.144
15%	4.959	74.392	4.800	4.667	4.620	4.956	4.995
18%	4.188	75.381	4.000	3.889	3.850	4.185	4.231
20%	3.802	76.036	3.600	3.500	3.465	3.800	3.850
25%	3.106	77.657	2.880	2.800	2.772	3.107	3.168
30%	2.642	79.258	2.400	2.333	2.310	2.644	2.718
40%	2.060	82.402	1.800	1.750	1.733	2.067	2.166
50%	1.710	85.476	1.440	1.400	1.386	1.720	1.848
60%	1.475	88.486	1.200	1.167	1.155	1.489	1.650
70%	1.306	91.439	1.029	1.000	0.990	1.324	1.523

History[edit]

An early reference to the rule is in the Summa de arithmetica (Venice, 1494. Fol. 181, n. 44) of Luca Pacioli (1445–1514). He presents the rule in a discussion regarding the estimation of the doubling time of an investment, but does not derive or explain the rule, and it is thus assumed that the rule predates Pacioli by some time.

“

A voler sapere ogni quantità a tanto per 100 l'anno, in quanti anni sarà tornata doppia tra utile e capitale, tieni per regola 72, a mente, il quale sempre partirai per l'interesse, e quello che ne viene, in tanti anni sarà raddoppiato. Esempio: Quando l'interesse è a 6 per 100 l'anno, dico che si parta 72 per 6; ne vien 12, e in 12 anni sarà raddoppiato il capitale. (emphasis added).

”

Roughly translated:

“

In wanting to know of any capital, at a given yearly percentage, in how many years it will double adding the interest to the capital, keep as a rule [the number] 72 in mind, which you will always divide by the interest, and what results, in that many years it will be doubled. Example: When the interest is 6 percent per year, I say that one divides 72 by 6; 12 results, and in 12 years the capital will be doubled.

”

Adjustments for higher accuracy[edit]

For higher rates, a bigger numerator would be better (e.g., for 20%, using 76 to get 3.8 years would be only about 0.002 off, where using 72 to get 3.6 would be about 0.2 off). This is because, as above, the rule of 72 is only an approximation that is accurate for interest rates from 6% to 10%. For every three percentage points away from 8% the value 72 could be adjusted by 1.

{displaystyle tapprox {frac {72+(r-8)/3}{r}}}

or for the same result, but simpler:

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{displaystyle tapprox {frac {70+(r-2)/3}{r}}}

{displaystyle tapprox {frac {69.3}{r}}+0.33}

E-M rule[edit]

The Eckart–McHale second-order rule (the E-M rule) provides a multiplicative correction for the rule of 69.3 that is very accurate for rates from 0% to 20%. The rule of 69.3 is normally only accurate at the lowest end of interest rates, from 0% to about 5%. To compute the E-M approximation, multiply the rule of 69.3 result by 200/(200−r) as follows:

{displaystyle tapprox {frac {69.3}{r}}times {frac {200}{200-r}}}

For example, if the interest rate is 18%, the rule of 69.3 says t = 3.85 years. The E-M rule multiplies this by 200/(200−18), giving a doubling time of 4.23 years, where the actual doubling time at this rate is 4.19 years. (The E-M rule thus gives a closer approximation than the rule of 72.)

To obtain a similar correction for the rule of 70 or 72, one of the numerators can be set and the other adjusted to keep their product approximately the same. The E-M rule could thus be written also as

{displaystyle tapprox {frac {70}{r}}times {frac {198}{200-r}}}

{displaystyle tapprox {frac {72}{r}}times {frac {192}{200-r}}}

In these variants, the multiplicative correction becomes 1 respectively for r=2 and r=8, the values for which the rules of 70 and 72 are most precise.

Similarly, the third-order Padé approximant gives a more accurate answer over an even larger range of r, but it has a slightly more complicated formula:

{displaystyle tapprox {frac {69.3}{r}}times {frac {600+4r}{600+r}}}

Derivation[edit]

Periodic compounding[edit]

For periodic compounding, future value is given by:

{displaystyle FV=PVcdot (1+r)^{t}}

where ${displaystyle PV}$ is the present value, ${displaystyle t}$ is the number of time periods, and ${displaystyle r}$ stands for the interest rate per time period.

The future value is double the present value when the following condition is met:

{displaystyle (1+r)^{t}=2,}

This equation is easily solved for ${displaystyle t}$ :

{displaystyle {begin{array}{ccc}ln((1+r)^{t})&=&ln 2tln(1+r)&=&ln 2t&=&{frac {ln 2}{ln(1+r)}}end{array}}}

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A simple rearrangement shows:

{displaystyle {frac {ln {2}}{ln {(1+r)}}}={bigg (}{frac {ln 2}{r}}{bigg )}{bigg (}{frac {r}{ln(1+r)}}{bigg )}}

If r is small, then ln(1 + r) approximately equals r (this is the first term in the Taylor series). That is, the latter term grows slowly when ${displaystyle r}$ is close to zero.

Calling this latter term ${displaystyle f(r)}$ , the function ${displaystyle f(r)}$ is shown to be accurate in the approximation of ${displaystyle t}$ for a small, positive interest rate when ${displaystyle r=.08}$ (see derivation below). ${displaystyle f(.08)approx 1.03949}$ , and we therefore approximate time ${displaystyle t}$ as:

{displaystyle t={bigg (}{frac {ln 2}{r}}{bigg )}f(.08)approx {frac {.72}{r}}}

Written as a percentage:

{displaystyle {frac {.72}{r}}={frac {72}{100r}}}

This approximation increases in accuracy as the compounding of interest becomes continuous (see derivation below). ${displaystyle 100r}$ is ${displaystyle r}$ written as a percentage.

In order to derive the more precise adjustments presented above, it is noted that ${displaystyle ln(1+r),}$ is more closely approximated by ${displaystyle r-{frac {r^{2}}{2}}}$ (using the second term in the Taylor series). ${displaystyle {frac {0.693}{r-r^{2}/2}}}$ can then be further simplified by Taylor approximations:

{displaystyle {begin{array}{ccc}{frac {0.693}{r-r^{2}/2}}&=&{frac {69.3}{R-R^{2}/200}}&&&=&{frac {69.3}{R}}{frac {1}{1-R/200}}&&&approx &{frac {69.3(1+R/200)}{R}}&&&=&{frac {69.3}{R}}+{frac {69.3}{200}}&&&=&{frac {69.3}{R}}+0.34end{array}}}

Replacing the 'R' in R/200 on the third line with 7.79 gives 72 on the numerator. This shows that the rule of 72 is most precise for periodically composed interests around 8%.

Alternatively, the E-M rule is obtained if the second-order Taylor approximation is used directly.

Continuous compounding[edit]

For continuous compounding, the derivation is simpler and yields a more accurate rule:

{displaystyle {begin{array}{ccc}(e^{r})^{p}&=&2e^{rp}&=&2ln e^{rp}&=&ln 2rp&=&ln 2p&=&{frac {ln 2}{r}}&&p&approx &{frac {0.693147}{r}}end{array}}}

References[edit]

^ ^a^bDonella Meadows, Thinking in Systems: A Primer, Chelsea Green Publishing, 2008, page 33 (box 'Hint on reinforcing feedback loops and doubling time').
^Slavin, Steve (1989). All the Math You'll Ever Need. John Wiley & Sons. pp. 153–154. ISBN0-471-50636-2.
^Kalid Azad Demystifying the Natural Logarithm (ln) from BetterExplained

External links[edit]

The Scales Of 70 – extends the rule of 72 beyond fixed-rate growth to variable rate compound growth including positive and negative rates.

Retrieved from 'https://en.wikipedia.org/w/index.php?title=Rule_of_72&oldid=902893569'

At Quest to Learn, a New York City public school which uses a systems thinking approach to secondary education (February 2013)
Born	1947
Alma mater	MIT Ph.D,1978; M.S.,1972 Stanford University B.S.
Known for	The Fifth Discipline, Learning organization
Scientific career
Fields	Systems science
Institutions	MIT, New England Complex Systems Institute
Influences	David Bohm

Peter Michael Senge (born 1947) is an American systems scientist who is a senior lecturer at the MIT Sloan School of Management, co-faculty at the New England Complex Systems Institute, and the founder of the Society for Organizational Learning. He is known as the author of the book The Fifth Discipline: The Art and Practice of the Learning Organization (1990, rev. 2006).

2Work

Life and career[edit]

Peter Senge was born in Stanford, California. He received a B.S. in Aerospace engineering from Stanford University. While at Stanford, Senge also studied philosophy. He later earned an M.S. in social systems modeling from MIT in 1972, as well as a PhD in Management from the MIT Sloan School of Management in 1978.^[1]^[2]

He is the founding chair of the Society for Organizational Learning (SoL). This organization helps with the communication of ideas between large corporations. It replaced the previous organization known as the Center for Organizational Learning at MIT.

He is co-Founder, and sits on the Board of Directors, of the Academy for Systems Change. This non-profit organization works with leaders to grow their ability to lead in complex social systems that foster biological, social and economic well-being. The focus is on awareness-based systems thinking tools, methods and approaches.

He has had a regular meditation practice since 1996 and began meditating with a trip to Tassajara, a Zen Buddhist monastery, before attending Stanford.^[3] He recommends meditation or similar forms of contemplative practice.^[3]^[4]^[5]

Work[edit]

An engineer by training, Peter was a protégé of John H. Hopkins and has followed closely the works of Michael Peters and Robert Fritz and based his books on pioneering work with the five disciplines at Ford, Chrysler, Shell, AT&T Corporation, Hanover Insurance, and Harley-Davidson, since the 1970s.

Organization development[edit]

Senge emerged in the 1990s as a major figure in organizational development with the book The Fifth Discipline, in which he developed the notion of a learning organization. This conceptualizes organizations as dynamic systems (as defined in Systemics), in states of continuous adaptation and improvement.

In 1997, Harvard Business Review identified The Fifth Discipline as one of the seminal management books of the previous 75 years.^[6] For this work, he was named 'Strategist of the Century' by Journal of Business Strategy, which said that he was one of a very few people who 'had the greatest impact on the way we conduct business today.'^[6]

The book's premise is that too many businesses are engaged in endless search for a heroic leader who can inspire people to change. This effort creates grand strategies that are never fully developed. The effort to change creates resistance that finally overcomes the effort.^[7]

Senge believes that real firms in real markets face both opportunities and natural limits to their development. Most efforts to change are hampered by resistance created by the cultural habits of the prevailing system. No amount of expert advice is useful. It's essential to develop reflection and inquiry skills so that the real problems can be discussed. ^[7]

According to Senge, there are four challenges in initiating changes.

There must be a compelling case for change.
There must be time to change.
There must be help during the change process.
As the perceived barriers to change are removed, it is important that some new problem, not before considered important or perhaps not even recognized, doesn't become a critical barrier. ^[7]

Learning organization and systems thinking[edit]

According to Senge 'learning organizations' are those organizations where people continually expand their capacity to create the results they truly desire, where new and expansive patterns of thinking are nurtured, where collective aspiration is set free, and where people are continually learning to see the whole together.'^[6] He argues that only those organizations that are able to adapt quickly and effectively will be able to excel in their field or market. In order to be a learning organization, there must be two conditions present at all times. The first is the ability to design the organization to match the intended or desired outcomes, and second, the ability to recognize when the initial direction of the organization is different from the desired outcome and follow the necessary steps to correct this mismatch. Organizations that are able to do this are exemplary.

Senge also believed in the theory of systems thinking which has sometimes been referred to as the 'Cornerstone' of the learning organization. Systems thinking focuses on how the individual that is being studied interacts with the other constituents of the system.^[8] Rather than focusing on the individuals within an organization, it prefers to look at a larger number of interactions within the organization and in between organizations as a whole.

Meadows Thinking In Systems

Publications[edit]

Peter Senge has written several books and articles throughout his career. A selection of his works:

1990, The Fifth Discipline: The art and practice of the learning organization, Doubleday, New York.
1994, The Fifth Discipline Fieldbook
1999, The Dance of Change
2000, Schools that Learn: A Fifth Discipline Fieldbook for Educators, Parents, and Everyone Who Cares about Education
2004, Presence: Human Purpose and the Field of the Future
2005, Presence: An Exploration of Profound Change in People, 'Organizations, and Society'
2008, The Necessary Revolution: How Individuals and Organizations Are Working Together to Create a Sustainable World

References[edit]

Notes