1 The Simugram: A Holistic Paradigm for Risk Analysis, James R. Thompson, Noah Harding Professor of Statistics Rice University Copyright 2003, Simugram

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The Simugram: A Holistic Paradigm for Risk Analysis,

James R. Thompson, Noah Harding Professor of Statistics

Rice University

Copyright 2003, Simugram Systems, Inc.Patent Pending

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Horatio: You will lose this wager, my lord.

Hamlet: I do not think so. Since he went intoFrance, I have been in continual practice.I shall win at the odds....

*************************************************

Hamlet: O, I die, Horatio;

The potent poison quite o'er-crows my spirit.


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Will Rodgers

It’s not the things we don’t know that hurt us; it’s the things we know that

just ain’t so.


4

John Maynard Keynes

The market can remain irrationallonger than you can remain solvent


5

Fifty years ago, Harry Markowitz posed and solved thefollowing problem:

Given a set of n stocks and a capital to be invested of C,what is the allocation of capital which maximizes theexpected return, at a future time t, of the portfolio P(t)for an acceptable volatility of the total portfolio (t)?


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For this contribution, Markowitz receivedthe Nobel Prize. His result is the foundationof portfolio analysis. However, it is flawed.“Volatility” is the square root of thevariance of the value of the portfolio. It is apoor surrogate for risk. The concept of riskis a hard one to grasp. Laurence Siegel,treasurer of the Ford Foundation, definesrisk rather forcefully, if imprecisely:

… risk is the possibility that, in the longrun, stock returns will be terrible.


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–1.0 –0.5 0.0 0.50.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

µ

Volatility versus Growth (including Dividends) 1933

1938

1932

1931

1937

1930

19291987

1974

19391940

Cover of Models for Investors in Real World Markets

=–.317


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Far better an approximate answerto the right question, which is often vague,than an exact answer to the wrong question,which can always be made precise.

John W. Tukey


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There are dangers with maximizing the expectation of growth.

0 5 10 150.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

log2(X)

Prob(winningsX)

St. Petersburg Paradox


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The modern computer enables us to use historicaldata to simulate many possible values of a stock at a desired time horizon.

S(t) S(0)exp[( 2

2) t t]


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Look at a simugram for an Ibbotson Index Portfolio of initial value $100,00 over a 3 year period, using data using data from 1926-2000.

$100,000 $200,000 $300,000 $400,000

5

10

15

20

Percent

Histogram of Portfolio Values after 3 Years

Portfolio Value

The Simugram Using Ibbotson Index Data


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From this diagram we can note

•that the value of the portfolio is less than $142,000 50% of the time

•and that it is less than $86,000 10 % of the time.

50000 100000 150000 200000 250000 3000000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Value in Dollars after 3 Years

Cumulative

Proportion

Cumulative Simugram

for 3 Years Forecast of

the Ibbotson Annual

Growth Index

Easier to use is the cumulative simugram shown below


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The mean value of a $100,000 portfolio after five years is $192,676.

The median value is $175,530 (growth rate of .1125).

However, the lower ten percentile is $92,747 (growth rate of -.015).

0 100000 200000 300000 400000 5000000

20

40

60

80

100

Value

Cumulative

PercentileCumulative Simugram of

100,000 Portfolio

After 5 Years


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Next, we consider the same scenario except looking 20 years into the future.

The median value is $873,100, an annual increase of 10.8%.

Even the lower ten percentile value of $285,590 represents a growth rate of 5.2%. 0

20

40

60

80

100

Cumulative

Percentile

1E6 2E6 3E6 4E6 5E6 6E6

100,000 invested in

Large Cap Stocks Index Simugram

Results after 20 Years


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Mean = $4,850,000Median = $2,725,000Lower Ten Percentile =

$695,000

0

20

40

60

80

100

Cumulative Percent

5.0 E6 1.0 E7 1.5 E7 2.0 E7 2.5 E7 3.0 E7 3.5 E7

Portfolio Value after 45 Years

Privatized Social Security Simugram Investing $2,000 per Year for 45 Years Large Cap Market Performance


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0 5000000 10000000 150000000

20

40

60

80

100

Value

Cumulative Percentile

$2,000/yearValues and Inputs Assume 3% Inflation

Lower 10 Percentile = $269,000Median = $993,000Mean = $1,745,000

45 Year


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The Bear Bites AllBear Market Occurred Duration of Bear Market Amount of Decline Recovery Period

08/25/56 - 10/22/56 14.7 months 21.60% 11.1 months

12/12/61 - 06/26/62 6.4 months 28.00% 14.3 months

02/09/66 - 10/07/66 7.9 months 22.20% 6.9 months

11/29/68 - 05/26/70 17.9 months 36.10% 21.3 months

01/11/73 - 10/03/74 20.7 months 48.20% 69.4 months

09/21/76 - 03/06/78 17.5 months 19.40% 17.3 months

01/06/81 - 08/12/82 19.2 months 25.80% 2.3 months

08/25/87 - 12/04/87 3.3 months 33.50% 19.7 months

07/16/90 - 10/11/90 2.9 months 19.90% 4.3 months

Bear Markets 1950-2000


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Resampling vs ModeledThree Years Out

0 20 40 60 80 1000

$500,000

$1,000,000

$1,500,000

Percentile

Value after 10 Years Distribution of Value

of $100,000 Investment in Ibbotson Index after Ten Years


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Resampling Versus Model 1 Year Out

0 20 40 60 80 100$50,000

$100,000

$150,000

$200,000

Value after One Year

Cumulative Distribution Function of $100,000 Investment in Ibbotson Index After One Year

Resampling

Resampling

Percentile


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Ways of Dealing with Bears

• Go to Levy Stable Distributions

• Stay with Resampling

• Patch the Geometric Brownian Model

• Every Year the Market Has a 10% Bear

• Every Five Years the Market Has a 20% Bear


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Black-Scholes-Merton and Their Amazing Money Machine

Definition: A Call Option is the right (but not the obligation)to buy a security of current price S(0) for strike price X at a future time T.

What is the “fair price” C of such a call option?

Answer: There ain’t no such thing.


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Wrong Answer!

How about

C = exp(–µt) E{Max(0,S(T) –X)}

e t etS(0) log(S(0) / X) ( 2 /2)T

T

Xlog(S(0) / X) ( 2 /2)T

T


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No. What is wanted is:

CBS e rt ertS(0) log(S(0)/ X ) (r 2 /2)T

T

Xlog(S(0) / X) (r 2 /2)T

T

Transforms a noisy game into asure thing.


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Some Problems With Black-Scholes

Transaction costs are not really free. The closer the hedge gets to being riskless, the more frequently one must rebalance (and this results in material transaction costs).

The realistic value of r will be significantly higher than that of a Treasury bill.


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Historical records show that the Black-Scholes formula, generally does not give the actual market price of a call option. To correct thisimperfection in nature, it is customary for some traders to plug in whatever valueis necessary for to give the market price for the option.


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In the case of horse betting, there is an arbitrary mechanism which sets payoffs at the instant the race starts. The bookmaker is allowed to set payoffs with his profit margin locked into the payoff. Suppose that the betis placed a week before the instant the race starts. What costs must the bookmakerincur in the intervening time period to rebalance the payoffs for the bets he hascovered? The answer is ``zero.'' And this is a reason the analogy between bookmakingand selling options is flawed.


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Looking at expectations and variances doesnot tell the story.We need to look at the distributionfunction of the payoffs.


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0. 1

0. 2

0. 3

0. 4

0. 5

0. 6

0 16 32 48

PR

OPO

RT

ION

PE

R B

AR

Value of Option

µ=.15; T=0.5; X=$108; S(0) = $100

C=$3.54 55%of the time V=0. E= $7.23


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Cvendor e t etS(0) log(S(0) / X) ( 2 /2)T

T

Xlog(S(0) / X) ( 2 /2)T

T

Cbuyer e t etS(0) log(S(0) / X) ( 2 /2)T

T

Xlog(S(0) / X) ( 2 /2)T

T

r


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A Portfolio Case Study

Next we take the 90 stocks in the S&P 100 that were in business prior to 1991.

Optimal investment allocation maximizing the one year lower 20 percentile.

Constraining each stock to no more than 5% of the portfolio share


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simugramsystems 17

T able A.3. P ortfolio Allo cation from S&P 100.Maximizing One-Y ear 20 P ercen tile

with Max 5% in An y Sto ck (con tin ued).id permno tic ker š æ par allo c npar allo c36 22111 JNJ 0.20 0.26 0.00 0.0237 22592 MMM 0.14 0.25 0.00 0.0038 22752 MRK 0.17 0.31 0.00 0.0039 22840 SLE 0.11 0.31 0.00 0.0040 23077 HNZ 0.06 0.25 0.00 0.0041 23819 HAL ° 0.03 0.47 0.00 0.0042 24010 ETR 0.11 0.29 0.00 0.0143 24046 CCU 0.27 0.38 0.00 0.0044 24109 AEP 0.04 0.22 0.00 0.0045 24643 AA 0.21 0.37 0.00 0.0046 24942 RTN 0.02 0.44 0.00 0.0047 25320 CPB 0.04 0.29 0.00 0.0048 26112 DAL 0.00 0.33 0.00 0.0049 26403 DIS 0.05 0.33 0.00 0.0050 27828 HWP 0.11 0.48 0.00 0.0051 27887 BAX 0.21 0.24 0.05 0.0552 27983 XRX 0.04 0.61 0.00 0.0053 38156 WMB 0.14 0.33 0.00 0.0054 38703 WF C 0.20 0.31 0.00 0.0055 39917 WY 0.07 0.32 0.00 0.0056 40125 CSC 0.17 0.48 0.00 0.0057 40416 AVP 0.24 0.47 0.00 0.0058 42024 BCC 0.00 0.33 0.00 0.0059 43123 ATI ° 0.05 0.39 0.00 0.0060 43449 MCD 0.05 0.26 0.00 0.0061 45356 TYC 0.37 0.33 0.05 0.0562 47896 JPM 0.15 0.38 0.00 0.0063 50227 BNI 0.03 0.27 0.00 0.0064 51377 NSM 0.35 0.69 0.05 0.0565 52919 MER 0.31 0.44 0.00 0.0166 55976 WMT 0.32 0.31 0.05 0.0567 58640 NT 0.23 0.64 0.00 0.0068 59176 AXP 0.19 0.31 0.00 0.0069 59184 BUD 0.20 0.21 0.05 0.0570 59328 INTC 0.37 0.52 0.00 0.0071 59408 BA C 0.14 0.34 0.00 0.0072 60097 MDT 0.28 0.28 0.05 0.0573 60628 FD X 0.23 0.35 0.00 0.0074 61065 TO Y 0.06 0.48 0.00 0.0075 64186 CI 0.20 0.28 0.00 0.0076 64282 LTD 0.16 0.42 0.00 0.0077 64311 NSC ° 0.02 0.34 0.00 0.0078 65138 ONE 0.10 0.37 0.00 0.0079 65875 VZ 0.11 0.29 0.00 0.0080 66093 SBC 0.12 0.29 0.00 0.0081 66157 USB 0.12 0.37 0.00 0.0082 66181 HD 0.33 0.32 0.05 0.0583 66800 AIG 0.26 0.26 0.05 0.0584 69032 MWD 0.34 0.47 0.00 0.0085 70519 C 0.35 0.36 0.03 0.0586 75034 BHI 0.11 0.41 0.00 0.0087 75104 VIA 0.21 0.37 0.00 0.0088 76090 HET 0.09 0.39 0.00 0.0089 82775 HIG 0.24 0.37 0.01 0.0090 83332 LU 0.10 0.54 0.00 0.00


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simugramsystems 23

Table A5. P ortfolio Allo cation from S&P 100Requiring at Least 5% Return for Lower 20 Percentile

Maximizing Mean or Medianwith Max 5% in An y Stock.

id permno ticker par alloc npar alloc par alloc nparallocmaxmean maxmean maxmedian maxmedian

1 10104 ORCL 0.05 0.05 0.05 0.052 10107 MSFT 0.05 0.05 0.05 0.053 10145 HON 0.00 0.00 0.00 0.004 10147 EMC 0.05 0.05 0.05 0.055 10401 T 0.00 0.00 0.00 0.006 10890 UIS 0.05 0.05 0.03 0.057 11308 KO 0.00 0.00 0.00 0.008 11703 DD 0.00 0.00 0.00 0.009 11754 EK 0.00 0.00 0.00 0.0010 11850 XOM 0.00 0.00 0.00 0.0011 12052 GD 0.00 0.00 0.00 0.0012 12060 GE 0.00 0.00 0.00 0.0013 12079 GM 0.00 0.00 0.00 0.0014 12490 IBM 0.05 0.05 0.05 0.0515 13100 MAY 0.00 0.00 0.00 0.0016 13856 PEP 0.00 0.00 0.00 0.0017 13901 MO 0.00 0.00 0.00 0.0018 14008 AMGN 0.05 0.05 0.05 0.0519 14277 SLB 0.00 0.00 0.00 0.0020 14322 S 0.00 0.00 0.00 0.0021 15560 RSH 0.05 0.05 0.05 0.0022 15579 TXN 0.05 0.05 0.04 0.0523 16424 G 0.00 0.00 0.00 0.0024 17830 UTX 0.00 0.00 0.00 0.0025 18163 PG 0.00 0.00 0.00 0.0026 18382 PHA 0.00 0.00 0.00 0.0027 18411 SO 0.00 0.00 0.00 0.0028 18729 CL 0.00 0.00 0.03 0.0029 19393 BMY 0.00 0.00 0.00 0.0030 19561 BA 0.00 0.00 0.00 0.0031 20220 BDK 0.00 0.00 0.00 0.0032 20626 DOW 0.00 0.00 0.00 0.0033 21573 IP 0.00 0.00 0.00 0.0034 21776 EX C 0.00 0.00 0.00 0.0035 21936 PFE 0.00 0.05 0.05 0.00


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simugramsystems 24

T able A.5 (con tin ued). P ortfolio Allo cation from S&P 100Requiring at Least 5% Return for Lo w er 20 P ercen tile

Maximizing Mean or Medianwith Max 5% in An y Sto ck.

id p ermno tic ker par allo c npar allo c par allo c npar allo cmax mean max mean max median max median

36 22111 JNJ 0.00 0.00 0.00 0.0037 22592 MMM 0.00 0.00 0.00 0.0038 22752 MRK 0.00 0.00 0.00 0.0039 22840 SLE 0.00 0.00 0.00 0.0040 23077 HNZ 0.00 0.00 0.00 0.0041 23819 HAL 0.00 0.00 0.00 0.0042 24010 ETR 0.00 0.00 0.00 0.0043 24046 CCU 0.05 0.05 0.00 0.0544 24109 AEP 0.00 0.00 0.00 0.0045 24643 AA 0.00 0.00 0.00 0.0046 24942 RTN 0.00 0.00 0.00 0.0047 25320 CPB 0.00 0.00 0.00 0.0048 26112 D AL 0.00 0.00 0.00 0.0049 26403 DIS 0.00 0.00 0.00 0.0050 27828 HWP 0.00 0.00 0.00 0.0051 27887 BAX 0.00 0.00 0.00 0.0052 27983 XRX 0.00 0.00 0.00 0.0053 38156 WMB 0.00 0.00 0.00 0.0054 38703 WF C 0.00 0.00 0.00 0.0055 39917 WY 0.00 0.00 0.00 0.0056 40125 CSC 0.00 0.00 0.00 0.0057 40416 AVP 0.05 0.00 0.05 0.0558 42024 BCC 0.00 0.00 0.00 0.0059 43123 ATI 0.00 0.00 0.00 0.0060 43449 MCD 0.00 0.00 0.00 0.0061 45356 TYC 0.05 0.05 0.05 0.0562 47896 JPM 0.00 0.00 0.00 0.0063 50227 BNI 0.00 0.00 0.00 0.0064 51377 NSM 0.05 0.05 0.05 0.0565 52919 MER 0.05 0.05 0.05 0.0566 55976 WMT 0.05 0.05 0.05 0.0567 58640 NT 0.00 0.00 0.00 0.0068 59176 AXP 0.00 0.00 0.00 0.0069 59184 BUD 0.00 0.00 0.00 0.0070 59328 INTC 0.05 0.05 0.05 0.0571 59408 BA C 0.00 0.00 0.00 0.0072 60097 MDT 0.05 0.05 0.05 0.0573 60628 FD X 0.00 0.00 0.00 0.0074 61065 TO Y 0.00 0.00 0.00 0.0075 64186 CI 0.00 0.00 0.00 0.0076 64282 L TD 0.00 0.00 0.00 0.0077 64311 NSC 0.00 0.00 0.00 0.0078 65138 ONE 0.00 0.00 0.00 0.0079 65875 VZ 0.00 0.00 0.00 0.0080 66093 SBC 0.00 0.00 0.00 0.0081 66157 USB 0.00 0.00 0.00 0.0082 66181 HD 0.05 0.05 0.05 0.0583 66800 AIG 0.05 0.05 0.05 0.0584 69032 MWD 0.05 0.05 0.05 0.0585 70519 C 0.05 0.05 0.05 0.0586 75034 BHI 0.00 0.00 0.00 0.0087 75104 VIA 0.00 0.00 0.00 0.0088 76090 HET 0.00 0.00 0.00 0.0089 82775 HIG 0.00 0.00 0.00 0.0590 83332 LU 0.00 0.00 0.00 0.00


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1. The main weapons of the investor are diversification and time.

2. Looking at the mean and variance Is not enough.

3.We have created a computer intensive forecasting paradigm which enables us readily and intuitively to consider questions of risk and growth simultaneously.

4. The set of algorithms can be readily learned by the financialprofessional. In connection with other information, it providesvaluable information for advising modifications to portfolios.

Conclusions


Documents

1 The Simugram: A Holistic Paradigm for Risk Analysis, James R. Thompson, Noah Harding Professor of Statistics Rice University Copyright 2003, Simugram