Introducing Graham Risk

as a Counter-Cyclical Alternative

to Variance

©LINKS Analytics BV 2011

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Introducing Graham Risk

as a Counter-Cyclical Alternative to Variance

Taron Ganjalyan, Bob Debi-Tewari

Fourth Draft

April 2011

Variance is inversely related to asset returns, which creates a cyclical

overexposure to systemic risks. This paper develops the concept of

Graham Risk (GR) as a counter-cyclical alternative to variance by

introducing short-term imperfect rational expectations of investors.

Background

One of the most damaging financial crises in the recent history of capital markets

prompted a major reassessment of risk management theory and practice. Plenty has been

written both against and in defense of variance and VaR as a risk measure, however,

actual advances in the subject area have been relatively limited. This paper will refer to a

selection of well written summaries of key concepts in modern risk management and

their criticism. A large part of the paper’s focus, however, will be on entirely new

material - a fundamentally new framework to manage risk.

Satyajit Das (2007) summarizes key concepts of modern risk management and

associated problems. A more formal paper by Douady and Taleb (2010) explains the

concept of statistical undecidability, which is particularly relevant for assessing the value

of efforts going into modeling the tails of a statistical distribution. Finally, Grantier

(2009) summarizes modern portfolio theory from Markowitz to Sharpe in a broader

context of behavioral finance and value investing.

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Variance-based risk measures: cyclical by design

While there are several drawbacks with modern risk management practice, the most

inconvenient of all is the one that actually causes losses – the cyclical nature of all

variance/covariance-based risk measures. In a recovery phase of the economic cycle,

asset prices rise at a steady pace, while volatilities and correlations fall. The more money

is ploughed into the assets, the lower the volatility and correlations, and the higher the

prices. The simple intuitive consequence of this, according to modern risk management

practice, is that the higher the price of an asset, the lower the risk of investing in that

asset.

Figure 1: VIX and S&P 500 - an illustration of increasing price and falling risk

Ever since variance-based risk measures entered active service in financial institutions,

fund managers had intuitive difficulty with variance as a risk measure. This was partly

due to the non-linear mathematical structure of variance – squaring the return differences

makes absolutely no fundamental sense to a business analyst. But the biggest concern of

particularly value managers is that variance points at low risks exactly when value

management style implies high risks. To illustrate this, the shaded area in Figure 1

corresponds to the period of 2003-2007, when stock prices continued to rise while the

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VIX index (a proxy for market-wide volatility) continued to fall, which resulted in lower

and lower risk numbers.

Variance as a measure of risk, in fact, resulted in a vicious circle of systemic risk

accumulation. Lower variance readings resulted in higher allocation to stocks and

equity-linked instruments, which in turn created a sustained excess supply of money

chasing the same investments, driving the variance and VaR numbers even lower.

The vicious circle has a built-in self destruction mechanism. Higher asset prices can only

be justified by higher investors’ expectations with respect to cash flows. Given the fact

that the process is self-sustaining, at some point prices reach levels that are sustainable

only if absolutely every heightened cash flow expectation materializes. Inevitably, there

comes a moment when something goes wrong (oil in 1970s, Russian and Asian debt in

1998, technology in 2000, sub-prime mortgages in 2007). Here again, variance as a risk

measure punishes investors with a vengeance. Due to the nature of the construct small

changes in returns result in amplified changes in the resulting measure. Sudden increases

in risk (falling prices) mean that fund managers have to sell. Overnight, a non-risky

position becomes extremely risky. Selling pressure coming in unison drives prices

abruptly down, causing further hikes in volatility, increasing risk and yet again creating

more selling pressure.

This cyclical nature of variance alone renders this measure hazardous in practical use.

Covariance (or correlation) adds an additional level of discomfort. In a cyclical recovery,

correlations are high and the benefits of diversification are exaggerated. Unfortunately,

correlation too is inversely related to stock market returns, which means in a market

correction correlations sharply increase thus removing the additional diversification

benefits from the VaR calculation. This effect is well documented and acknowledged by

risk practitioners. Kroner and Ng(1998) demonstrate that the existing popular time

varying covariance models do not capture the empirical behavior of the covariance.

Hong, Tu and Zhou (2003) apply a model-free approach to demonstrating the

asymmetric correlation effect across various Fama-French size and value portfolios.

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Figure 2: S&P 500 and 1-year correlation of daily returns of S&P500 and MSCI World Index

The development of empirically sound variance-covariance models is still the main

focus of academic community. Stochastic volatility, correlation of returns, variance and

dispersion have been included in the state-of-the-art models. An allowance has been

made for extreme returns by utilizing jump diffusion processes. Practical applicability of

these models in the structured products market is conceivable, however, even with these

advances variance as a risk measure falls short of institutional investors’ demand of

capital preservation. The key issue is not addressed – variance as a risk measure

forces investors to buy expensive and sell cheap.

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Value Investing and Risk

The prime cause of the risk-return divide between the risk managers and fund managers

is the definition of risk. As it was demonstrated, variance or volatility is an imperfect

measure of asset risk originating from outside the realm of investment management.

There were investment risk measures more native to investment management, and one of

these measures was formulated by Benjamin Graham (2003 revised edition, first

published in 1949). Graham postulated that investment risk is the difference between the

price and the intrinsic value of the security. Lower prices compared to the value mean

that investors have a certain cushion and can absorb erosion of value before their

investment results in a loss. In contrast with variance, this measure of risk (the Graham

measure) is counter-cyclical by design.

Figure 3: The Graham risk measure

Unfortunately, this line of reasoning was overwhelmed by the arrival of mathematical

precision offered by the modern portfolio theory. And yet, implementation of the

Graham risk in the modern institutional asset management context would deliver a

counter-cyclical risk measure that is aligned with the front line of the investment

management industry – the fund manager.

Implementation of Graham Risk requires a set of market-wide and individual asset

related assumptions. The key assumption implied by Graham Risk is that individual

assets have clearing market prices above or below the fair value. Extending Graham

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Risk application to markets implies acceptance of sustained periods of over- and under-

pricing of markets as a whole.

There is plenty of scattered evidence to suggest that the efficient market hypothesis

requires revisiting1. For practical applications, it is sufficient to assume that investors

have imperfect rational short-term expectations. A formal representation of Graham Risk

is presented in Appendix 1.

Implementing Graham Risk: introducing Cumulative Revenue Surprise

The most obvious way to estimate Graham Risk is to use median analyst estimate data.

Such data is readily available from a number of commercial sources. For a number of

reasons, however, analyst estimates are not the best instrument to gauge Graham Risk.

First, biases and inconsistencies in analyst estimates are well documented: analyst

estimates have a positive bias2, which can be estimated using publicly available

information3.

Second, Graham Risk is based on marginal expectations of investors, while available

data sources focus on sell-side analysts, who are not investors and can be driven by

institutional incentives other than investment returns. Third, analyst expectations are

heavily managed by corporate management – guidance issued by companies is a

common practice. Finally, it is notoriously difficult to trace changes in analyst

expectations due to the inconsistencies in databases.

In order to estimate GR, one has to take into account exaggerated expectations

extrapolated by investors. Expectations based on observable news are extrapolated along

supply chains, across markets and competitors. This process has been coined “read

across” by analysts. It is, therefore, necessary to register all relevant news flow in the

network surrounding the company - its competitors, suppliers, customers.

This paper utilizes the Cumulative Revenue Surprise (CRS) measure in the LINKS4

database as a gauge for Graham Risk. The database covers over 2500 companies

1 as early as Jensen (1978)

2 Abarbanell (1991), Brown, Foster and Noreen (1985)

3 Klein (1990), Das, Levine, and Sivaramakrishnan (1998)

4 LINKS database is developed and maintained by LINKS Analytics B.V. As of October 2010, the database is not publicly

available.

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globally. News flow such as revenue surprises and major corporate releases is traced

along the network of economic relationship between the companies.

CRS, as defined in LINKS, is the sum of all additional revenue implied by the news flow

along the supply chains of a company. For instance, a large surprise announced by an

automobile manufacturer is traced to the suppliers based on business rules defined in the

database. The “read across” numbers from all announcements are added at any point in

time based on a sliding window.

LINKS CRS of an asset expands in line with positive announcements by clients,

suppliers, competitors. These announcements are observed by investors and expectations

are built based on the economic relationships between companies. In this analysis, we

assume changes in revenue expectations translate into commensurate expectations with

respect to cash flows. A more comprehensive approach would focus on the cash flow

expectations directly.

Figure 4: LINKS CRS - a proxy for Graham Risk and the VIX index

LINKS CRS, as a proxy for Graham Risk, has a number of advantages over VaR:

It is counter-cyclical: shaded regions in Figure 4 indicate the many occasions

when Graham risk would suggest risks accumulation, while VIX is low. Most

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notably, Graham risk signaled building up risks from mid-2007, while the

volatility reacted only during the turmoil. Following Graham Risk, as a risk

measure, would have resulted in alignment between the risk manager and

portfolio manager – the fund manager would have cut the risks drastically in

early 2007 and would be prompted to add allocations amidst the crisis of 2008.

Graham Risk is linear. This can be observed in Figure 4. In the period of

2003-2006, when realized volatilities were low, implied volatilities (VIX) were

within a very narrow range. In this period systemic risk built up over several

years is unnoticeable simply because the magnitude of change in VIX is not

constant. The Graham Risk, in contrast, changed substantially in that period, first

indicating falling system-wide risks until 2005 and subsequently accumulating

risks before 2007.

Graham Risk in this implementation reflects true economic links and not

return correlations. Owing to its construction, return covariance depends

heavily on instantaneous reactions to gauge relationships. If two underlying

assets have clear economic links with a certain time lag, consistently gauging

this relationship with covariance is impossible. LINKS CRS, on the other hand,

is based on the underlying economic relationships, which means risk spill-over

between assets can be gauged without relying on immediate reactions.

There are no perfect risk metrics for universal use by all types of investors. In the

analysis that will follow the attractiveness of GR over VaR for risk budgeting purposes

is valid for large institutional and high-net-worth investors with reasonably diversified

portfolios and a combination of high- and low-risk assets.

LINKS CRS and VaR are used to allocate assets between high-risk (S&P 500) and low-

risk (cash) asset classes. LINKS CRS data is available from 2002 onwards, which limits

the time series that can be used for this analysis to eight years. The time period is quite

relevant, since it includes one of the major financial crises in the past 100 years and a

few smaller market contractions.

It is assumed that the portfolio is launched with an indifferent (50/50) allocation between

risky and risk-free asset. Risk-free asset in practice can be interpreted as an allocation to

a bond portfolio, money market instruments or introduction of an equity hedge portfolio.

Each month an allocation decision is made based on a LINKS CRS and VaR limit on the

whole portfolio. Risk budgets are fully utilized in every month; short-selling and

leverage constraints are applied to replicate large institutional/HNW portfolios.

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In this analysis, a standard delta-normal 1-month VaR estimate has been used with a

99% confidence level. A similar analysis has been carried out using Monte-Carlo

simulation yielding very similar results. While more complex VaR estimates may, in

theory, yield marginally better results, the gap between Graham Risk and VaR cannot be

closed simply due to the more appropriate construction of the metric. The results of the

analysis are presented in Table 1.

A typical risk-controlled asset allocation outcome is represented by Figure 5.

Figure 5: Risk asset allocation at 5% VaR and 0.05% LINKS CRS

Constrained by the LINKS CRS risk budget, a portfolio’s equity allocation would vary

between 4% and100% in case of the most risk averse portfolio and 38% and100% in

case of the most risk seeking portfolio. Equity portfolio allocation with a VaR constraint

would vary between 7% and 22% and 66% and 100% for the risk averse and risk

seeking portfolios respectively. It is immediately evident that LINKS CRS is more

selective in terms of equity exposure. While the VaR-based risk budget prompts a

gradual increase in risky asset weight to 100% throughout the 2002-2007 period, LINKS

CRS varies equity exposure in that period between 25% and 40%.

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In the base case of 5% VaR limit and 0.05% LINKS CRS limit the LINKS CRS

constrained portfolio has a volatility of 6.2% vs. 9.5%, and what is more important, a

maximum monthly loss of 3.5% vs. 13.8% in case of VaR. Returns of LINKS CRS

portfolio are significantly better due to the large risky asset allocation in distressed

situations, which in practice can be constrained by imposing a risky asset allocation

bandwidth.

While CRS as a proxy for Graham risk can be used as a risk measure on a stand-alone

basis, it can also be translated into probabilities. A precautionary note here - one of the

unintended consequences of widespread use of VaR is the perception of precise risk

control. Translating CRS into probabilities of loss can, in practice, have similar

consequences.

Furthermore, probability estimation assumes yet again certain distribution functions just

like VaR and are bound to be inaccurate at extremes. However, since GR is a counter-

cyclical measure of risk, estimating the tails of return distribution is no longer critical,

since the accumulation of risks is signaled well before tail events occur. APPENDIX 2

describes a simple methodology to translate CRS into probability.

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Conclusion

Plenty has been written over time about the limitations of variance and VaR as a risk

measure. The most damaging feature of VaR as a risk measure is its cyclical nature.

Variance and VaR are usually falling in a cyclical rebound in the economy, when

expectations, and with them likelihood of failure, increase. During these periods, VaR-

based risk budgets allow for overexposure to building up expectations in the economy.

Once a correction in asset prices occurs, owing to the convexity of variance, VaR-based

risk budgets exaggerate the risks and cause large sell-offs, thus generating a further

pressure on variance and creating a vicious circle.

An alternative conceptual definition of investment risk was proposed by Benjamin

Graham in his seminal work - “The Intelligent Investor” (1949). This paper develops the

concept of Graham Risk (GR) further by introducing short-term imperfect rational

expectations of investors and allowing for exaggerated reactions. The resulting risk

measure – is a function of change in implied cash flow expectations for an asset

and the discount rate.

LINKS CRS is introduced as a proxy for implied cash flow expectations and Graham

Risk, and it is demonstrated that in contrast with variance-based risk measures, Graham

risk is counter-cyclical and linear, which in practice translates into more intuitive and

balanced risk management.

This effort is the first step in devising and implementing Graham Risk in the modern

investment management context. GR has plenty of caveats and we are far from the

suggestion that this is an all-encompassing risk metric. There are at least two major areas

of further research – the distress risk and portfolio construction using GR. The former is

critical, since GR assumes going concern of the underlying companies, while the latter is

of practical importance to day-to-day portfolio management.

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Table1: VaR and LINKS CRS based asset allocation outcomes

Risk budgets LINKS CRS 0.01% 0.02% 0.03% 0.04% 0.05% 0.06% 0.07% 0.08% 0.09% 0.10%

VaR 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

End Value: LINKS CRS 114 126 133 140 145 149 152 153 152 151

VaR 106 106 106 106 105 103 101 104 108 113

Average monthly return: LINKS CRS 0.13% 0.23% 0.29% 0.35% 0.38% 0.41% 0.43% 0.44% 0.43% 0.42%

VaR 0.06% 0.06% 0.06% 0.06% 0.05% 0.03% 0.01% 0.04% 0.08% 0.12%

Lowest monthly return: LINKS CRS -2.0% -2.0% -2.1% -2.8% -3.5% -4.3% -5.0% -5.8% -6.6% -7.4%

VaR -2.5% -5.1% -7.8% -

10.8% -

13.8% -

17.1% -

20.5% -

20.5% -

20.5% -

20.5%

Realized volatility: LINKS CRS 2.3% 3.6% 4.5% 5.4% 6.2% 7.0% 7.8% 8.3% 8.9% 9.4%

VaR 2.3% 4.0% 5.8% 7.6% 9.5% 11.2% 12.6% 13.2% 13.8% 14.5%

Minimum weight or risk asset: LINKS CRS 4% 8% 11% 15% 19% 23% 27% 30% 34% 38%

VaR 7% 13% 20% 26% 33% 39% 46% 53% 59% 66%

Maximum weight of risk asset: LINKS CRS 100% 100% 100% 100% 100% 100% 100% 100% 100% 100%

VaR 22% 43% 65% 87% 100% 100% 100% 100% 100% 100%

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References

Abarbanell, Jeffrey S., 1991, Do Analysts’ Earnings Forecasts Incorporate Information

in Prior Stock Price Changes?, Journal of Accounting and Economics 14, 147–165.

Brown, Philip, George Foster, and Eric Noreen, 1985, Security Analyst Multi-Year

Earnings Forecasts and the Capital Market (American Accounting Association, Sarasota,

FL).

Das, Satyajit, 2007, Perfect Storms – Beautiful & True Lies In Risk Management

Das, Somnath, Carolyn B. Levine, and K. Sivaramakrishnan, 1998, Earnings

predictability and bias in analysts’ earnings forecasts, Accounting Review 73, 277–294.

Douady, Raphael and Taleb, Nassim Nicholas, Statistical Undecidability (October 12,

2010). Available at SSRN: http://ssrn.com/abstract=1691165

Grantier, Bruce, Benjamin Graham and Risk (2009)

Graham, Benjamin, 2003, The Intelligent Investor Revised edition. Harper Collins

Hong, Yongmiao, Tu, Jun and Zhou, Guofu, 2003, Asymmetric Correlation of Stock

Returns: Statistical Tests and Economic Evaluation

Jensen, Michael C., Some Anomalous Evidence Regarding Market Efficiency. Journal

of Financial Economics, Vol. 6, Nos. 2/3, pp. 95-101, 1978. Available at SSRN:

http://ssrn.com/abstract=244159 or doi:10.2139/ssrn.244159

Kaplan, Steven N., Ruback, Richard S.,1995, The Valuation of Cash Flow Forecasts: An

Empirical Analysis Vol. 50, No. 4, pp. 1059-1093

Karolyi, G. Andrew, Stulz , René M., 1996, Why Do Markets Move Together? An

Investigation of U.S.-Japan Stock Return Comovements The Journal of Finance Vol. 51,

No. 3, pp. 951-986

Klein, April, 1990, A direct test of the cognitive bias theory of share price reversals,

Journal of Accounting and Economics 13, 155–166.

Kroner, K.F., Ng, V.K., 1998. Modeling asymmetric co-movements of asset returns.

Review of Financial Studies 11, 817–844.

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APPENDIX 1: Formal Representation of Graham Risk

Graham Risk as a concept firmly resides in the field of corporate finance and valuation.

The formal representation of the measure is:

(I)

where E(V) is the expected value of the asset, while P is the price. It is interesting to

note that based on the efficient market hypothesis, Graham Risk is always zero for an

institutional investor with diversified portfolios. Any difference between value and price

is quickly arbitraged away.

While price is observable, the expected value needs to be estimated:

(II)

where t is the period, E(CFn) is the expected cash flow in period n and r is the discount

rate. Since the present analysis is outside the realm of capital asset pricing model

(CAPM), the relevant discount rate will be discussed separately.

A standard application of two-stage discounted cash flow analysis yields

(III)

This is a standard two-stage discounted cash flow model with a terminal value calculated

based on the Gordon growth model. It is useful to think about the first part of the value

as the explicit forecast horizon, where investors have explicit views on expected cash

flows. In practice, this is typically the street analyst forecast horizon – typically 2-3

years forward.

The second term uses rT and gT – terminal values for the discount factor and growth

respectively. These values are relatively stable and reflect industry-level growth

prospects in the longer-term. Copeland, Koller and Murrin (1995) assert that the terminal

value constitutes bulk of expected value of a company (in many industries up to 100%).

A crucial assumption in this implementation of Gordon Risk framework is that investors

do have rational expectations with respect to the long-term prospects of the industry,

while they tend to over- or under-react to the near-term information flow. This means

that asset prices can be represented as:

(IV)

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where EI is the set of irrational expectations of investors. Substituting equations (III)

and (IV) in (I) yields

(V)

For practical applications, marginal increase in risk is far more important than absolute

levels. Decisions in period t depend entirely on the changes in environmental variables

from t-1 to t. Marginal Graham Risk in t would be:

(VI)

Finally, since we are interested in over-reactions, assuming = 0, i.e. rational

expectations do not change, would only mean that investors would err on the side of

caution. This yields:

(VII)

It will be demonstrated that this last form of risk measure can be used effectively for

gauging asset risks. It is interesting to note that is inversely related to r – the

discount rate.

There is a crucial difference between traditional CAPM handling of a discount rate and

the implications of Graham Risk application. Expected cash flows in CAPM are

discounted at a rate corresponding to the uncertainty surrounding these cash flows. The

latter is usually derived from the market prices. CAPM, therefore, assumes that

uncertainties of these cash flows are fully captured by the variance of the underlying

stock price.

In contrast, the Graham Risk framework assumes that markets can exaggerate the true

uncertainty surrounding cash flows. The true risk, in this framework, stems from the

excessive expectations. Stocks that have naturally more uncertain cash flows can be less

risky than “safer” cash flow stocks, if the built-up expectations are lower.

Since the risk of a stock is captured in the numerator of the risk measure, the discount

rate should reflect merely the long-term returns in alternative asset classes. A

benchmark, in this instance could be the long-term risk free rates. An interesting

consequence is that increasing interest rates would lower risks of investing in risky

assets. This is quite intuitive, since increasing nominal interest rates signal higher

inflation, higher cash flows and better business environment.

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APPENDIX 2: Probability Translation of Graham Risk

A Logit model has been used to estimate the probability of loss that is greater than µ

given a CRS reading of over c:

(VIII)

where Rt is the return in period t.

Figure 6: Probability of loss at various CRS levels

Once again it should be emphasized that the extreme values of probability are

inaccurate, and in practice – irrelevant. The critical contribution of Graham Risk is to

move from cyclical risk management to counter-cyclical risk management.

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