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DCF Valuation with Cash FlowCessation Risk

Atanu Saha and Burton Malkiel

The typical discounted cash fiow model used to value assetsopenly projects cash fiows for an initial set of years and thentypically assumes that the cashflows will grow at a constantrate into the indefinite future. In this paper, we discuss theimplications for valuation and the discount rate when oneassumes that cash fiows have a non-zero probability ofcessation throughout the valuation period. We demonstratethat, if one allows for even a small cessation probability, thena substantially higher discount rate is required in derivingthe present value of the expected stream of cash fiows.

•V . ^

•The typical discounted cash flow (DCF) model usedto value assets explicitly projects cash flows for an initialset of years and then usually assumes that the cash flowswill grow at a constant rate into the indefinite future. Inthis paper, we discuss the implications for valuation andthe discount rate when the critical assumption regardingthe existence of an infinite stream of cash flows of the DCFmodel is abandoned or modified. In many valuation settingsit is eminently reasonable to assume that cash flows have anon-zero probability of cessation throughout the valuationperiod. For example, this assumption would be appropriatein valuing firms like small biotechnology companies, start-up ventures, hedge funds, and individual projects within a

Atanu Saha is the Senior Vice President and Head of the NY Office ofCompass Lexecon in New York, NY. Burton Malkiel is the Chemical BankChairman's Professor of Economics, Emeritus and Senior Economist atPrinceton University in Princeton, NJ.

The authors are immensely grateful to the research support provided byCarina Chambarry, Erica Liang, Sangeeta Ahmed, Mary Shen, and othermembers of the Compass Lexecon staff. The authors are also indebted tothe helpful comments of the referee and Ramesh Rao (Editor).

firm, as well as in valuing distressed assets with high defaultrisk. In this paper, we demonstrate that, if one allows foreven a small probability that the cash flows will cease toexist at any point in time, then a substantially higher discountrate is required in deriving the present value of the expectedstream of cash flows.

I. Typical Equity DCF Models

DCF models to value common stocks date back atleast to Fisher (1961) and Williams (1938). The simplestform of the model was popularized by Gordon (1959). InGordon's (1959) model, the stream of cash flows, taken tobe dividends, increased each year at a rate of g. Thus, theprice of a common stock, P, is taken to be the discountedpresent value of the stream of future dividends, where Z) isthe dividend paid during the preceding year, and r denotesthe discount rate:

p =-(1)

Assuming that g < r and letting N—> oo, the above equationsimplifies to;

(2)

It is unrealistic to project infinite growth at a constant ratefor each common stock and each asset to be valued. Thisis especially so since there is generally a life cycle to everybusiness. A stock may produce extraordinary growth during

176

SAHA AND MALKIEL - DCF VALUATION WITH CASH FLOW CESSATION RISK

early stages of its life but eventually the growth subsides to alower rate. Hence, many equity valuation models explicitlyproject that cash flows will grow at an initial (high) ratefor some number of years and then subside to a (lower)sustainable rate into the indefinite future. Such a modelallows for g> r, for at least an initial time period.

For example, Malkiel (1963) assumes that two growthrates be used: an initial rate reflecting the early life of thecompany and then a second rate reflective of the sustainablegrowth rate of the overall economy. Finance texts such asBodie, Kane, and Marcus (2010) and Brealey and Myers(2003) have presented three-stage growth models. Inprinciple, the DCF could be implemented with multiple-stage models. The important point to emphasize is thatin all of the extant models of stock valuation, it is eitherexplicit or assumed that some growth continues well into thefijture. Even when that future is modeled by assuming someterminal value for the shares, the implicit assumption thatsupports the existence of a terminal value is that the eamingsstream continues infinitely. While the future stream may bevolatile and risky, the risk-adjusted discount rates used inmost valuation models do not account for cessation risk.

A. CAPM-based Discount Rates

A fundamental principle of valuation states that the worthof any asset, be it a stock, bond, real estate property, etc.,is the discounted present value of the stream of cash flowsthe asset is expected to produce. The discount rate is takento be the base risk-free rate (typically the rate of interest onsafe govemment securities) plus some risk premium, chosento reflect the riskiness of the cash flow stream that is to bediscounted. Investment texts such as Ross, Westerfield,and Jaffe (2009), Brealey and Myers (2003), and Bodie etal. (2010) recommend that the appropriate discount rate beselected by reference to the Capital Asset Pricing Models(CAPM).' Hence, the discount rate that should be used tovalue a common stock (or the stream of eamings fi"om aparticular corporation) will depend on the "systematic" risk(the relative volatility) of the eamings stream itself, whichaccording to the CAPM models, is captured by a stock's beta(ß).

B. Limitations of CAPM-based Discount Rates

Unfortunately, ß is infamously ditïïcult to measure.Indeed, Roll (1977) has argued that since the tme market

' Thomas and Gup (2010) acknowledge several market derived discountrates other than the CAPM. It suggests that "[t]he weighted average cost ofcapital (WACC) represents the appropriate discount rate to use to discountcash flows available to entire firm," (288). It also discusses the dividenddiscount model (DDM), which "takes the familiar Gordon constant growthmodel of Pu= D, / (k - g) and solves for the cost of equity, k = D,/ P^ +g," (289). "The DDM approach, however, is limited since many firms payeither no dividends or arbitrarily low dividends," (ibid).

177

portfolio is impossible to measure with any precision, it isimpossible to know what ß should be used in the calculation.Moreover, empirical tests of CAPM have often showninconclusive support for the theory (see, for example,Fama and French (1992)). Fama and French (1992) havesuggested that a three-factor model is better able to explainthe required rates of retum (and therefore the appropriaterisk-adjusted discount rates) to be used for common stocks.According to Fama and French (1992), stock rettims arerelated not only to ß but also to firm size (as measured byequity capitalizations) and some "value" metric (typicallythe relationship of market to book value).

Furthermore, some studies have shown that, in additionto market risk, idiosyncratic risk could matter in assetvaluations (see for example Xu and Malkiel (2003) andGoyal and Santa Clara (2003)). In the context of asset pricinga key manifestation of idiosyncratic risk is the asset's totalvolatility, not simply its volatility relative to a market index.Because the CAPM-based discount rate only accounts formarket risk, valuation models may greatly underestimate thediscount rate (or overstate the net present value (NPV)) insettings where the idiosyncratic risk of the cash flows matters.We will argue, this is especially so in cases where there is asignificant probability that the future stream of cash flowsmay completely cease; and this risk, in CAPM parlance, isa "non-diversifiable risk" (see Hall and Woodward (2010)for an excellent discussion on this issue). While the CAPM-based discount rate can provide a useful starting point, webelieve that an additional adjustment to the discount rate iswarranted to account for cash flow cessation probability, insettings where such a possibility is not immaterial. We willshow that to account for the likelihood that fixture cash flowsmay completely cease, the appropriate discount rate maybe much higher than suggested by popular models. Whileprevious studies (see for example Butler and Schachter(1989) and Danielson and Scott (2006)) have recognized theneed for a "risk-adjusted" discount rate, the contribution ofthis paper is to provide an analytically consistent approachand a closed-form solution for the appropriate discount rate.

In fact, valuation practitioners have long recognizedthis limitation of the CAPM framework. For example, thedirector of equity research at Momingstar writes:

"The most common reason that we tum to altemativevaluation tools is when a company's survival is at risk.We've found that relying on a DCF model for these typesof companies fiequently results in an overestimation ofthe company's worth, because implicit in a DCF is theassumption of a perpetuity, which generally comprises alarge chunk of the total intrinsic value."^

^ See Dorsey (2010).

Typical DCF models such asGordon's (1959) growth modeland multiple stage modelsusually use CAPM-baseddiscount rates, which do notaccount for cash flow cessationrisk.

178

Indeed, usage of high discount rates is common in manyindustries and assets characterized by relatively high riskof cash flow cessation. For example, a study undertakenby Sahlman and Scherlis (2009) at the Harvard BusinessSchool shows that venture capitalcompanies value their investmentsin target companies using a veryhigh discount rate, typically 35%or higher. The study argues thatventure capitalists use these highdiscount rates to account for thematerial possibility that a venturewill not succeed, and, as a result,that future cash flows would benegative. Similarly, investorsin biotechnology companies face the possibility that aninvestment could quickly lose all of its value if a drug doesnot perform well in clinical trials or is not approved byregulators. A study on valuing biotechnology firms findsthat biotechnology investors typically value investmentsusing discount rates of 25% and higher."

In many settings, the possibility of a substantial reductionin cash flows, as opposed to a complete cessation of theflows, may be more plausible. For example, in most mergersand acquisitions, the valuation of the transaction is based oncash flow projections of the combined entity, which ofteninclude estimates of synergistic revenue enhancements orcost reductions. However, it is widely documented in priorstudies that, in a vast majority of cases, only a fraction ofthe projected synergy benefits is actually achieved by themerged entity.' In settings such as these, the DCF valuationmodel would require introduction of the probability that,in any given year throughout the valuation period, only afraction of the projected cash flows would be realized. Thus,this more general model subsumes the cash flow cessation-based DCF model, the latter being a special case wherein the

' See Sahlman and Scherlis (2009). Also see Hall and Woodward (2010),p. 1168. They write: "Our data include 22,004 venture-backed companies,the great majority of al! such companies in the United States for the periodfrom the beginning of 1987 through the third quarter of 2008. Among theexit values used in the analysis, 2,015 are IPOs, 5,625 are acquisitions,and 3,352 are confirmed zero value exits. Of the remaining companies, wetreat those more than five years past their last rounds of venture tundingas having exited at some time with zero value; 4,220 companies fall intothis category... The remaining 6,792 companies have not yet achieved theirexit values." The data imply that more than 14,000 firms out of the 22,004venture-backed companies being studied (or 65% of total) had failed toachieve IPOs or acquisitions.

" See Frei and Leleux (2004).

' See Capron (1999); Devos, Kadapakkam, and Krishnamurthy (2009);Aganval, Barney, Foss, and Klein (2009); Zu Anyphausen-Aufsess,Koeppen, and Schweizer (2007).

JOURNAL OF APPLIED FINANCE - NO. 1, 2012

fraction (of the projected cash flows) is set to zero.In this paper, we first derive in an infinite horizon setting

the closed-form solutions for NPV, when cash flows have afinite probability of cessation at each period. Second, we

derive a simple formula for the cashflow cessation risk-adjusted discountrate. Third, we extend the cash flowcessation model to the more generalframework wherein there is a finiteprobability that only a fraction of thecash flows would be realized at everyperiod. Finally, we undertake anempirical analysis using a databaseof hedge funds to provide estimatesof cessation risk and the cessation

risk-adjusted discount rate in one industry.

II. Impact of Cash Flow Risk on Valuation:An Illustrative Example

Before laying out the analytical framework for thederivation of cessation risk-adjusted discount rate, it mightbe illustrative to begin with a simple example of a traditionalDCF model and then demonstrate the effect of cash flowcessation risk on NPV and the discount rate. We make thefollowing assumptions:

(a) At the beginning of the first year, the firm's cash flowis $ 100, and it grows at a constant rate of 10% per year forthe first six years and then at 3% per year from the seventhyear to the infinite future.

(b) The firm is debt-free, and its cost of capital is 15%,which is the firm's CAPM-based discount rate.

This foregoing set of assumptions is reflected in Panel Aof Table I. Under these assumptions, the NPV of the cashflows is seen to be $1,226.

In Panel B of Table I, holding all other assumptionsunchanged, we consider the scenario where the cash flowshave a 10% annual probability of ceasing to exist. Thus, inPanel B, the expected value of the cash flow at the end ofYear 2 is 90% of $100, or $90, which grows at a 10% rate to$99. Similarly, for Year 3, the expected value of the cashflow from the preceding year (i.e., $99) grows by 10% to$108.9; however, since there is a 10% chance of cessation,the expected value in Year 3 is $108.9x0.9 = $98. Similarcalculations are undertaken for the remaining years. Keepingall the other assumptions about cash flow growth rate anddiscount rate unchanged, the NPV of cash flows in thisscenario is found to be $623, down from $1226 in the casewhere the cash flows had no explicitly modeled cessationrisk (Panel A of Table I). Thus, introduction of only 10%annual cessation risk reduces the NPV by nearly 50%.

In Panel C of Table I, we examine what the discount rate

SAHA AND MALKIEL - DCF VALUATION WITH CASH FLOW CESSATION RISK 179

Table I. DCF Valuations under Alternative AssumptionsTable I presents examples of traditional and modified DCF models. In Panels A, B, and C, we demonstrate the effects of varying cash nowcessation risks on NPV and the discount rate.

Assumptions

Discount Rate

Cash Flow Growth Rate

Terminal Growth Rate

Cessation Probability

Projected Flow

Terminal Value

Present Value Factor

Present Value of Cash Flows

Total Present Value

Panel A. No Cessation

15.0%

10.0%

3.0%

0%

Year1

$100

1.00

$100

$1,226

Year 2

$110

0.87

$96

Panel B. 10% Probability of Cessation

Assumptions

Discount Rate




Projected Flow

Terminal Value



Total Present Value

15.0%

10.0%

3.0%

10%

Year1

$100

1.00

$100

$623

Panel C. Required Discount Rate to Make the

Assumptions

Discount Rate




Projected Flow

Terminal Value



Total Present Value

27.8%

10.0%

3.0%

0%

Year 1 Year 2

$100 $110

1.00 0.78

$100 $86

$623

Year 2

$99

0.87

$86

TwoNPVs

Year 3

$121

0.61

$74

Year 3

$121

0.76

$91

Year 3

$98

0.76

$74

Equal

Year 4

$133

0.48

$64

Year 4

$133

0.66

$88

Year 4

$97

0.66

$64

Year 5

$146

0.38

$55

Years$146

0.57

$84

Years$96

0.57

$55

Year 6

$161

$669

0.29

$244

Year 6

$161

$1,382

0.50

$767

Year 6

$95

$395

0.50

$244

180

needs to be within the traditional DCF framework (thatassumes no cessation risk of cash flows) to produce a NPVof $623, which is the NPV if the cash flows have a 10%probability of cessation. Panel C of Table 1 shows thatthe discount rate has to rise from 15% to 27.8% within thetraditional DCF framework to yield the equivalent NPV of$623.

Comparison of Panels A and C of Table 1 shows that thekey difference lies in the discount rates; the undiscountedcash flows are exactly the same for Years 1 through 6. InPanel C, the terminal value is lower because the discountrate is higher. The results in Table I show that the NPV of acash flow stream that has a 10% chance of cessation in anygiven year, and is discounted at a 15% rate, is equivalentto the NPV of the same cash flows (without any explicitmodeling of cessation risk) discounted at a higher rate of27.8%, holding everything else constant.^ As demonstratedthrough this simple example, even a small chance of cashflow cessation implies a significantly lower NPV andcorrespondingly implies a higher discount rate.

While the simple numeric example contained in Table I isillustrative, a key issue left unexplained in the example washow the revised discount rate (of 27.8%) was computed inPanel C, based on which the NPVs in Panels B and C werefound to be equal. To address this question we lay out thegeneral analytical framework for DCF modeling with cashflow cessation risk.

III. DCF Model with Cash Flow CessationRisk: Analytical Framework

In the analysis that follows, we assume, without loss ofgenerality, that the cash flow at the beginning of the valuationperiod is $ 1.

A. NPV in Traditional DCF

We adopt the following notations: g = terminal growthrate; r = discount rate. Then NPV of the stream of cash flowis:

1+gY (1+g)l + rj {r-g)' (3)

This is the familiar Gordon (1959) model result discussedearlier.


B. NPV in DCF with Possible Cessation ofCash Flows

Assume in addition to above, that at each period there is afinite probability (denoted by d) that the cash flow will ceaseto exist. Therefore, at any period, the probability of the

continuation of cash flow is: 0 < (l -1/) < 1. For example, inthe second period, the probability of cash flow continuationis {\-df; in the third period it is {\-df, and so on. However,at no period in the future is cessation certain (i.e., {\-dfasymptotically approaches zero as / increases but is neverequal to zero, unless d=\). In this fi-amework, the NPV canbe computed as:

(4)g)

From the comparison of (3) and (4), K nests V^ as aspecial case when d=0. Also, if i/ = 1, then V^ = 0; that is, ifcessation is certain, NPV of the cash flows is zero.

C. Cessation Probability-adjusted DiscountRate

One can solve for the cessation probability-adjusteddiscount rate (denoted by r*) within the traditional DCFfi-amework by setting V equal to V and solving for r*:

(5)

to get:

r =d + r

[l-d] (6)

Note: (a) r* is independent ofg, and, (b) r = r' when d=0.Thus, r* indicates the discount rate one should be using

within the traditional DCF framework to calculate NPV, ifone recognizes a finite probability of cash flow cessation.Using the assumptions regarding the values of r and d,contained in Panel B of Table I, for the above expressionfor r* yields:

' Thomas and Atra (2010) discuss the "LifeCycle" model which "determinesthe uniform market derived discount rate across sectors by equatingcash flows produced by the model to current prices. The system adjustsindividual securities' values for risk, such as leverage, by modifying theexpected cash flows instead of the discount rate," (299).

27.78%= (i-.i) ,

which matches the discount rate contained in Panel C of


Table II. Implied Discount Rates for Various Cessation ProbabilitiesTable II presents the resulting risk-adjusted discount rates (r*) for various cessation probabilities and unadjusted discount rates. The tabledemonstrates the sensitivity of r* to cash flow cessation risk.

Unadjusted Discount Rate

8%

10%

12%

15%

20%

0%

8.0%

10.0%

12.0%

15.0%

20.0%

5%

13.7%

15.8%

17.9%

21.1%

26.3%

Cessation

10%

20.0%

22.2%

24.4%

27.8%

33.3%

Probability

15%

27.1%

29.4%

31.8%

35.3%

41.2%

20%

35.0%

37.5%

40.0%

43.8%

50.0%

25%

44.0%

46.7%

49.3%

53.3%

60.0%

Table I.To illustrate the sensitivity of the discount rate to cessation

probabilities, we present in Table II various values of r* forvarious cessation probabilities and unadjusted discount rates.For example, based on the assumptions of a 10% cessationrisk as in Table I, the implied discount rate corresponding tothe unadjusted discount rate of 15% is found to be 27.8%,and the figure is highlighted in bold in Table II.

D. Two Different Growth Rates

In typical DCF models, the cash flow growth rate duringthe explicitly modeled years is different than the one in theterminal period (as was done in Table I). We demonstratebelow that even if the explicitly modeled years' growth rateis different from the terminal growth rate, the key results setout above hold. We assume that:

• There are n explicitly modeled years;

• The growth rate in the explicitly modeled years is

denoted by g , and the terminal growth rate is denoted,

as before, by g, and that g ^ g .

Then the NPV in the traditional DCF framework wherecash flows are discounted using a discount rate of r isdenoted by:

(7)

The NPV of the expected cash flows fi-om the DCF modelwith d cessation probability is denoted by:

Now the equivalent result can be stated as follows:

Result

The proofs of this as well as other results are presented inthe Appendix.

E. Cash Flow Diminution Risk

Up to this point, we have assumed that at each period thecash flow has a finite probability of cessation. In this section,we generalize the framework to introduce the possibility thatcash flows have a finite probability of being reduced by afraction. Thus, complete cessation is a special case of thisgeneralized framework when this fraction is 100%.

Specifically, keeping all the notation unchanged, letus further assume that at any period the cash flow has aprobability of i/of being reduced by/percent; that is, at anyperiod the cash flow has c/probability of realizing a value of(l-f). Thus, cessation is implied when/=l.

Under these assumptions we state the following result:

(10)

From the comparison of (10) and (4), V^ nests V^ as aspecial case when/= 1.

As before, one can solve for the cash flow diminutionprobability-adjusted discount rate (denoted by r" ) withinthe traditional DCF firamework by setting V^ equal to K andsolving for r":

Result2: K(g,r**) = F(g ,J , / , r ) ,where :

'(8) (11)

182 JOURNAL OF APPLIED FINANCE - NO. 1, 2012

Table III. Implied Discount Rates for Various Cash Flow Reduction ValuesTable III presents the resulting risk-adjusted discount rates (r") for various diminution risks and unadjusted discount rates. The tabledemonstrates the sensitivity of/-"to cash flow diminution risk.

UnadjustedDiscount Rate

8%

10%

12%

15%

20%

0%

8.0%

10.0%

12.0%

15.0%

20.0%

Cash Flow Reduction

20%

10.2%

12.2%

14.3%

17.3%

22.4%

% (with

40%

12.5%

14.6%

16.7%

19.8%

25.0%

Probabiiityof 10%)

60%

14.9%

17.0%

19.1%

22.3%

27.7%

80%

17.4%

19.6%

21.7%

25.0%

30.4%

100%

20.0%

22.2%

24.4%

27.8%

33.3%

From Result 2, /* collapses to r' when / = 1, andr**collapses to r when either i /o r /= 0.

Finally, the generalized framework can accommodate twodifferent growth rates; this result is formalized below.

Result : (12)

To illustrate the sensitivity of the discount rate to cashflow diminution risks, we present in Table III values of r"forvarious diminution rates and unadjusted discount rates. Ineach case, we have fixed the value of d (the probability ofcash flow reduction) to 10%.

IV. An Application: Valuing a Hedge Fund

A. Hedge Funds' Cessation Risk

A key question that remains is how one can estimate thecessation probability, d, in practical valuation settings. Inthis section, we use data from two sources, (a) the HedgeFund Research Database (HFR) and (b) the TASS Database(TASS), maintained by Lipper/Thomson Reuters to estimatehedge funds' mortality risk.'' This estimation provides anillustration of the method of estimating d. We then use thisestimated cessation probability in calculating a risk-adjusteddiscount rate for the valuation of hedge funds.

We begin by describing the methodology employed tocompile the data set used to calculate hedge fund attritionrates. In order to exclude potential duplicate hedge fundsfound within the TASS and HFR databases, we separatelyidentified and removed hedge funds for which there was acomparable hedge fund that (a) had the same fund manager,(b) had identical strategies, (c) had a similar name, (d) startedand, where applicable, ended their retum series on the samedates, and (e) the retums had a minimum correlation of

' Both databases report historical retums for hedge fonds that either haveclosed ("dead") or are still active ("alive"). We combined both the "dead"and "alive" funds from these databases into a single database.

0.990. We also removed hedge funds with stagnant retumsand funds with less than six months' worth of data.*' Thesescreens led to the removal of approximately 13% of thefunds in the TASS database and approximately 8% of thefunds in the HFR database.

We then proceeded to identify duplicate fiinds across bothdatabases through the identification of potential duplicateñjnds by relying on the fund name, the managing firm, andthe correlation between their retums. This procedure led usto conclude that approximately 43% of the funds within theHFR database are potential duplicate funds also found in theTASS database. Upon deleting these funds from the HFRdatabase and combining both databases, we created a masterhedge fund database. As a final filter, we further refined oursample by excluding from the combined database any hedgefund with less than twelve months of reported retums.

We used the combined data set to calculate hedge fimdattrition rates over time.'" For the purpose of this paper, wehave considered two indicators of fund cessation. The firstconsiders the stoppage of reporting of retums by the fundas cessation. However, because fiands may stop reportingretums to databases for reasons other than financial distress,we considered an altemative indicator of cessation; underthis altemative, a fund that stops reporting retums andexperiences a negative cumulative retum for the last threemonths prior to stoppage of reporting is considered to haveceased operations.

' Stagnant retum was defined as a retum that did not change for a period ofthree or more months.

' For the HFR database we also excluded "shell" fonds, which are fondsthat are designed to mimic an existing fund. A fond was considered to be a"shell" fond if (a) it was managed by HFR, (b) its retums were at least 0.990correlated with an existing fund, and (c) the date frame for which the "shell"fund reports retums were encompassed by the fund they were designed tomimic.

'" In order to avoid potential biases due to backfilling, we deleted fi-om ourdatabase any retums reported prior to the date the hedge fond was originallyadded to either the TASS or HFR database.


Table IV. Hedge Fund Attrition RatesTable IV presents hedge funds' attrition rates under two indicators of cessation. In Panel A, stoppage of reporting is considered cessation;in Panel B, stoppage of reporting and negative retums for the last three months is considered cessation.

Year

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

2008

2009

Mean

Median

Total Funds268

1,057

1,260

1,403

1,476

1,582

2,088

2,549

3,102

3,836

4,609

4,963

5,329

4,984

Count27

110

237

210

321

248

220

303

340

480

633

846

1367

598

Panel A

Percent (%)

10.1

10.4

18.8

15.0

21.7

15.7

10.5

11.9

11.0

12.5

13.7

17.0

25.7

12.2

14.7

13.1

Count

14

58

147

106

182

132

131

127

162

215

224

360

1048

322

Panel BPercent (%)

5.2

5.5

11.7

7.6

12.3

8.3

6.3

5.0

5.2

5.6

4.9

7.3

19.7

6.6

7.9

6.4

Table IV presents hedge funds' attritions rates underthese two indicators which are shown in Panel A and B,respectively. Not unexpectedly, the attrition rates in PanelA are higher than those in Panel B. For the purpose of thecalculation of hedge funds' cessation risk-adjusted discountrate, we use the attritions rates in Panel B.

B. Hedge Funds' Cessation Risk-adjustedDiscount Rate

As Table FV shows, the hedge fund industry has had a longhistory of high cessation risk. For example, prior to the recentfinancial crisis, 1998 (the year Long Term Capital imploded)was marked by high attrition rate. Only two years later, in2000, with the demise of the intemet bubble, hedge fundattrition rate exceeded that of 1998. In the period between2001 and 2007, attrition rates remained in the 5%-8% range.However, in 2008, in the midst of the financial crisis, thehedge fiind industry experienced an attrition rate of nearly20%, a record high during the 1996-2009 sample period."

Table IV data are generally consistent with hedge failure

" The attrition rate is 19.7% if one considers stoppage of reporting andnegative retums for the last 3 months as cessation. The attrition rate goesup to nearly 26% if one considers stoppage of reporting as cessation.

rate estimates found in other studies. According to datareleased by Hedge Fund Research (HFR), the hedge fundindustry had a record 1,471 liquidations in 2008, and thistrend continued into 2009.'^ In November 2008, a MorganStanley analyst commented: "the hedge fund industry mayshrink as much as 45%" by the end of December 2008 "to$1.1 trillion from its peak of $1.9 trillion in June because ofinvestor redemptions and market losses."" The upshot ofthe data contained in Table IV (which are supported by otherstudies) is that the hedge fund industry is prone to a fairlyhigh failure rate. As a result, any valuation analysis withinthis industry has to account for the material probability ofcessation risk.

Below we provide an illustrative example of calculatingthe cessation risk-adjusted discount rate using data ontwo publicly traded hedge fund firms: Och-Ziff CapitalManagement (OZM) and Fortress Investment Group LLC(FIG).

For OZM, as a starting point, we use its CAPM-based

" See Bloomberg article, Kenneth Heinz, "Hedge Fund IndustryConsolidation Slows; Impact of Financial Crisis Widespread," HFR,9/17/2009, 12:47:22.

" See Bloomberg article. Bob Ivry and Gillian Wee, "Fortress HaltsDrawbridge Global Fund Withdrawals," Bloomberg News, 12/3/2008,16:57:04.

184

weighted average cost of capital (WACC) of 14.2%."'This WACC, in terms of the notations in our paper, is theunadjusted discount rate, r, for OZM. We assume OZM hasa cessation probability (i.e., d) of 6.4%, which is the medianhedge fund attrition rate from 1996 to 2009 as indicated inPanel B of Table IV. Thus, OZM's cessation risk-adjusteddiscount rate ( /) , using:

is found to be 22.0%, which is nearly 800 basis points higherthan the CAPM-based discount rate.

Similarly, for FIG, its CAPM-based discount rate is foundto be 22.6%.'' Again, assuming FIG has the same cessationprobability of 6.4%, its cessation risk-adjusted discount rateis estimated to be 30.9%. The difference between CAPM-based and the cessation risk-adjusted discount rate is also

'•' The 14.2% discount rate is calculated based on a 15.3% CAPM-basedcost of equity and a 4.4% cost of debt, weighted by 89.7% (equity-to-capital) and 10.3% (debt-to-capital), respectively. The CAPM-based cost ofequity is calculated based on a beta of 1.6 (using weekly share price retumfrom inception to 3/18/2011 and S&P 500 Index as the market benchmark),long-horizon equity risk premium of 6.7% from 1926 to 2009, and 20-yearUS Treasury bond yield of 4.6% as of 12/31/2009. Data are obtained fromBloomberg and Ibbotson SBBI 2010 Classic Yearbook.

" The 22.6% discount rate is calculated based on a 24,7% CAPM-basedcost of equity and a 1.8% cost of debt, weighted by 90.6% (equity-to-capital) and 9.4% (debt-to-capital), respectively. The CAPM-based cost ofequity is calculated based on a beta of 3.0 (using weekly share price retumfrom inception to 3/18/2011 and S&P 500 Index as the market benchmark),long-horizon equity risk premium of 6.7% from 1926 to 2009, and 20-yearUS Treasury bond yield of 4.6% as of 12/31/2009. Data are obtained fromBloomberg and Ibbotson SBBI 2010 Classic Yearbook.


found to be around 800 basis points in this case.

V. Concluding Comments

The discount rate is a key input in any discounted cash flowvaluation model. The existing literature on valuation hasgenerally proposed the usage of the CAPM-based discountrate in computing the present value of a stream of future cashflows. In this study, we have attempted to demonstrate thatin many valuation settings, the CAPM-based rate—whileproviding a useful starting point—generally understates thetrue discount rate because the CAPM framework fails to fullyincorporate the cessation risk of the cash flows. Illustrativeexamples show that the cessation risk-adjusted discountrate can be considerably higher than the CAPM-based rate;the difference between the two depends, of course, on theestimated magnitude of cessation risk. We have argued thatin several industries such risks are material and cannot beignored for any valuation exercise. In the aftermath of therecent global financial crisis, during which many large, well-capitalized and diversified firms ceased to exist, the need toproperly account for cash flow cessation risk in valuationmodels can hardly be overstated.

For future research, it would be worthwhile to explorethe analytical framework where a firm's cash flow cessationprobability declines each year instead of staying constant (aswe have assumed in this paper). This framework might berelevant in industries where, as firms mature, their risk offailure declines over time. This analytical fi-amework willundoubtedly require a far more complicated solution forthe NPV of cash flows. Furthermore, estimating the rateof decline of failure risk over time using data on start-upcompanies might also pose unique challenges. •


Appendix

Proof of Result 1:

For any explicitly modeled year, 7, for equivalence to hold,the following must be true:

= 0.

Solving this equation yields:

d + rr =-

(2)

argument as above except replace g by g; that is it followsfrom above that the following must be true:

This completes the proof.

(1) Proof of Result 2:

For any explicitly modeled year,/, for equivalence to hold,the following must be tme:

Since it holds for thej* year, it must be tme that it holds tme I h -t- r " j Ifor all explicitly modeled n years. Hence, it must true thatthe sum of the cash flows for the n years must be equal, that Solving this equation yields:is:

d-f + rr ^ .

(3) Proof of Result 3:

Now tuming to terminal value (TV), we can follow the same The proof follows from Results 1 and 2.

• (5)

(6)

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