apt versus capm1. evidence on capm theory and arbitrage pricing theory (apt ) 2. analysis of capm...

APT versus CAPM:An analysis with implications forprivate equity valuation practice

© 2020 Malcolm McLelland

Malcolm McLelland, Ph.D.McLelland + Palazzi | Financial economics

APT versus CAPM: An analysis with implications for private equity valuation


Outline

1. Evidence on CAPM theory and arbitrage pricing theory (APT )2. Analysis of CAPM theory versus APT3. Example: Using APT to price USD inflation risk4. On pricing (and risk hedging in) non-traded assets using APT5. Example: Using APT to estimate international cost of capital6. Why APT is likely to be optimal relative to CAPM7. Implications for private equity valuation

1. Evidence on CAPM theory and APT


“The [original] version of the CAPM … has never been an empirical success. In the earlyempirical work, the Black (1972) version of the model, which can accommodate a flattertradeoff of average return for market beta, has some success. But in the late 1970s, researchbegins to uncover variables like size, various price ratios and momentum that add to theexplanation of average returns provided by beta. The problems are serious enough toinvalidate most applications of CAPM.” (emphasis added)

Well-known financial economists Fama and French state in their 2006 article “The Capital Asset Pricing Model: Theory and Evidence” (Journal of Economic Perspectives), p. 43:

What would explain the empirical results? If capital markets (1) are not in continuous CAPM equilibrium and (2) also price risks other than the CAPM capital market risk premium, then:

with other risk factors,

0 0 1 0 0

0 1 0 0

E , ,i m i i i im

i i i i im

R R R R R R

R R eR R

population conditional unrestricted population modelexpectation function x

restricted empirical model

x δ x

0

leastsquares

estimation ofrestricted model

under thecondition of

CAPM betaE , =estimates aredownwardly

biased

ˆE( )

i m

i i

R R

δ x 0

1. Evidence on CAPM theory and APT


Chen, Roll, and Ross (CRR, 1986) find empirical evidence inconsistent with CAPM and consistent with APT using US equity return data from 1958 through 1984, which can be summarized as:

change in unexpectedmonthlyexpected unexpected change inequityinflation inflation yield curvereturn

conditional mean effects of changes ininterest rates

ˆ ˆ ˆˆ ˆi i iDEI DEI iUI UI iUTS UTSR x x x

unexpected monthlychange in risky industrialbond premium growth

conditional mean conditional and yield curve slope effects of mean

changes incorporate risk

ˆ ˆiURP URP iMP MPx x

effects

of changes inproduction

This empirical result seems to have led to the interpretation of APT as simply “CAPM with additionalrisk factor premiums estimated using Fama-MacBeth (1973) regressions” (which is not exactly accurate).

2. Analysis of CAPM theory versus APT


CAPM assumptions …

CAPM-1: Investors’ capital market interactions result in an equilibrium where they have equivalent beliefs about asset return expectations, variances, and covariances for all assets contained in the market portfolio;

CAPM-2: a unique, observable risk-free asset with a known risk-free yield to maturity exists; and

CAPM-3: investors’ market interactions set relative prices, wi, such that expected aggregate return variation in the global market portfolio is minimized subject to the constraints of (i) relative asset prices representing market portfolio value-weights sum to one, and (ii) value-weighted expected asset returns sum to the expected global market portfolio return. From these assumptions, it follows that …

, , 0

G 0 00 0

global capi

0 1

risk-free aggregate market price asset price of risky assets

minimize

subject to 1 and ( ) (

/ ( ... ... )

i j

N

i j iji j

N NiG

i i i i iG Gi i GG

i i i N

ww w

w w R R R R R R

w V V V V V

global capitaltal

market portfolio, market riskpremium

) 0,1,2,... ,

G

i N



The meaning of CAPM assumptions …

CAPM-1 CAPM assumes an equilibrium in investor beliefs (simultaneously with CAPM-2 and CAPM-3).

CAPM-2 CAPM assumes a unique, observable debt security with zero default and inflation risk exists.

CAPM-3 CAPM assumes investors have sufficient information and ability to solve the following constrained optimization problem:

,G

, 0 0 0

0 1

risk-free aggregate market price asset price of risky assets

minimize subject to 1 and

/ ( ... ... )i j

N N N

i j ij i i ii j i i

i i i N

ww w w w R R

w V V V V V

CAPM does not explain or predict how investors observeor develop information needed to obtain the solution.

how to estimate?

how to estimate?

how to observe?

how to observe?

0 0

iGi iG G

GG

R R R R

CAPM is based on assumptions that are unobservable and untestable; and

is inconsistent with empirical evidence. It arguably follows that CAPM cannot reasonably be used to price, hedge, or

diversify capital market risks.



APT assumptions …

APT-1: No risk arbitrage opportunities exist in expectation; and

APT-2: investors prefer greater expected returns for accepting greater risk and, therefore, have the objective of maximizing expected value; and, so, minimize the expected return of the investment portfolio via arbitrage.

APT-3,4: Not required for the general theory. (3) Sufficient assets (including derivatives) exist in the market with positive or negative correlation to portfolio risk exposures to hedge the risks in expectation; and (4) random risk factors do not have significant effects on asset returns on average over time. It follows that …

asset return risk sensitivities to risk factors

0risk-minimzing

portfolio no-arbitragepor

0

no-arbitrage portfolioreturn and risk prices

(1 ) ( ) ( )

t t

βx

w

R α βx

0 1 β w λ

R βλ

0

0

expected no-arbitragetfolio return for asset with

risk sensitivities return andrisk prices

i

i i

i

R

βλ

β λ

Risk depends on the asset, not on theory.

Optimality depends on a selected risk portfolio,not a theoretical global

portfolio.

Risk factor prices are extracted from market data; they are not a (theoretically)

observable risk premium.

Expected return of portfolio with no sensitivity to

risks x

Market risk premiums for asset

with a sensitivity of 1 to risk k = 1,2,…,m

In contrast to CAPM, APT explains and predicts howto arbitrage an arbitrary set of risks and presents theasset pricing implications of such arbitrage activity.

3. Example: Using APT to price USD inflation risk


Asset pricing problem. Noting that capital markets price risks rather than assets, and that risks are hedged and priced in the global rather than a local capital market, suppose we want to estimate the (global) price of 1-year USD inflation risk.

Risk sensitivities. 1-year USD inflation risk is likely most directly—though not completely—priced in the USmarket for 1-year maturity U.S. Treasury Bills (UST1). Two securities in the global market that can be used tohedge USD inflation risk are 1-year maturity German government bonds (DB1) and the iShares MSCI All CountryWorld Index ETF (ACWI). DB1 and ACWI monthly holding returns have risk sensitivities to monthly holdingreturns on UST1 between 2014.01 and 2018.12 of …

DB1 DB1 DB1UST1UST1 UST1

AWCI AWCI AWCIUST1UST1 UST1

.00019 .15055

.00938 8.8101tt t

tt t

xR xxR x

Risk-minimizing portfolio is the solution to a system of 2 equations with 2 unknowns (1 + #risks):

AWCI AWCI DB1DB1 AWCI DB1UST1 UST1 UST1

DB1 DB1 AWCI AWCI AWCI DB1 DB1 AWCIUST1 UST1 UST1 UST1 UST1

1.017.017

/10 /

w w ww w w

3. Example: Using APT to price USD inflation risk


No-arbitrage risk pricing. The average market risk premium for 1-year USD inflation risk—the inverse price of an asset with sensitivity of 1 to the risk—can be estimated as …

DB1DB1UST1 0 0AWCIAWCIUST1 USD1 USD1

mean returns summation(i.e., inverse factors (1s)average price and risk of total risks) sensitivities

1 .150551 8.8101

11

RR

0

USD1

1

market price of optimal DB1-AWCI portfolio risks independent of USD1market price of USD1 risk

.000128 .00004

.005134 .00058

Because the risk prices were estimated from monthly return data, they are converted to an annual basis as follows:

{DB1,AWCI} 0 USD1.00004 .00058

exp 12 1 exp 1.00048 .00696i iR

It follows that the expected no-arbitrage annual return to USD1 risk (the global risk price) is …

USD1 exp .00048R USD10 1

.00696 1 .0070 (0.7%)

4. On pricing (and risk hedging in) non-traded assets using APT


Privately-held assets not traded in organized markets are termed non-traded assets. Publicly-held assets tradedin organized markets are termed traded assets. Although not all assets trade in capital markets, most risks do;including seemingly unobservable risks (McLelland 2019, Chapter 5). To understand how to apply APT to price andhedge risks in non-traded assets, consider the relationship between economic profit and equity market return:

1

11expected market price is, by the

definition of economic profit, equalto plus economic profit

minus dividends and otherequity distributions

it

it it

it

it it itit it it it

it iP

P

DIV

P DIVP P DIVP P

1 1

economic profit as a proportion of expected traded assetbeginning expected market price is market return and

equivalent to the expected asset return economic profi

( )( ) it i

it it it it

R RP

ββ

1

profit return

traded asset profit isproportional to expectedt depend asset market returnon risk sensitivities

( ) ( )it i it it iP R β β

1

economic profit and risk relationship

asset return and risk relat

( , ) ( , )

( , )

( , )

it i i it i i

it i i i i t

it i i i i t

CF

R

ASSUMPTION: cash flow isproportional to economic profit

β β

β β x

β β x

risk-free componentsof cash flows andeconomic profitare proportional

1

risk sensitivities of cash flows, economic

profit, and assetionship returns are equivalent

i it i

i i

P

β β

Asset pricing theory applies equally to non-traded assets under one, simple, arguably reasonable assumption:

Under the assumption that asset cash flow is proportional to economic profit, it follows that asset pricing theory—APT included—applies

equally to non-traded assets (e.g., private equity), although at least 1 + #risks = n traded assets with

risk sensitivities are needed to price risks.

5. Example: Using APT to estimate international cost of capital


Asset pricing problem. Suppose we are interested in estimating the long-run ICC for a 10-year investmenthorizon on a 100% equity interest in a (hypothetical) privately-held manufacturer of jet aircraft componentsbased in Brazil: Componentes de Aeronaves Brasileiros S.A. (CAB). Discussions with executives suggest CABhas only three fundamental sources of risk over profits and cash flows: the risk that airlines cancel aircraftpurchase contracts; USDBRL foreign currency exchange risk; and global aggregate economic consumption risk.Discussions and analysis suggest three observable risk factor proxies: (1) monthly change in US aircraftdeliveries (AD), (2) monthly change in USDBRL, and (3) monthly NYSE equity market index return (NYA).

Risk sensitivities. Because there are m = 3 risks to be priced, n = 1 + m = 4 assets are selected that are believed tomost directly and completely price and hedge the risks: Embraer equity (EMBR3), Boeing equity (BA), S&P 500Index Exchange Traded Fund (SPY), and 5-year U.S. Treasury Bills (UST5). Estimated risk sensitivities are ...

(.597) ( .001) ( .001) ( .001)

(.005) ( .001) ( .001) ( .001)

( .001) ( .001) ( .001) ( .001)

(.45

EMBR

BA

SPY

UST5

.0035 .1124 .0839 .7721

.0120 .0275 .0624 1.0851

.0056 .0035 .0178 .9264

.0007

t

t

t

t

t

RRRR

R

AD USDBRL NYA

2) ( .001) ( .001) ( .001)

2EMBR

2AD BA

2USDBRL SPY

2NYA UST5

estimated mean-centered-squarisk factors

.147

.385

.938

.040.0125 .0703 .0085

1

t

t

t

R

Rx R

x Rx R

α β β β

redstatistics

asset return risk sensitivities to risk factors

t t

βx

R α βx



Risk-minimizing portfolio is the solution to a system of 4 equations with 4 unknowns (1 + #risks):

EMBR EMBR

BA BA

SPY SPY

UST5 UST5

( 1 )

.0581.1124 .0839 .7721

.8895.0275 .0624 1.08511.0799.0035 .0178 .92641.1323.0125 .0703 .0085

1 10 10 10 1

w ww ww ww w

0 1 β w

risk-minimizing portfolio

No-arbitrage risk pricing. The average market risk premiums can be estimated as …

0 0

AD

USDBRL

NYA

historicalexpectedreturns

.0041 .1124 .0839 .7721

.0131 .0275 .0624 1.0851

.0064 .0035 .0178 .9264

.0006 .0125 .0703 .0085

1111

λ1 β

R

0

AD

USDBRL

NYA

no-arbitrage portfolioreturn, , and risk

factor market premiums

.0057.0697.0594.0017



Private equity interest risk sensitivities of CAB were estimated using monthly operating profit data obtained from the company’s internal financial statements:

International cost of capital for the CAB private equity interest, resulting from above estimates, is …

( .001) ( .001) ( .001) (.219) (.269)CAB 1 CAB 2

CAB profit less prior period, CAB risk factor sensitivity estimateslagged risk factor effects and -values with s

.820 .7480 .4396 .4397 .2373ln lnt t

p

2CAB

AD

USDBRL

NYAample size 192and -squared statistic of .342

1

t

t

tnR R

xxx

CAB.AD AD CAB.USDBRL USDBRL CAB.NYA NYACAB 0

.0057 .4396( .0697) .4397( .0594) .2373(.0017)

no-arbitrage AD riskportfolio exposure

return premium

.0684 .3677 .

exp[ 12( )] 1

exp(

R

USDBRL risk NYA risk annualized

exposure exposure ICCdiscount premium estimate

3134 .0048 .136) 1

6. Why APT is likely to be optimal relative to CAPM


— CAPM is based largely on unobservable / untestable assumptions and does not explain or predicthow investors actually determine the (equilibrium) price of any arbitrary asset; it rather explains andpredicts how an expected asset price is determined assuming other prices are in equilibrium… but the empirical evidence is largely inconsistent with CAPM assumptions and theory.

— Interpreting APT as “CAPM with additional risk factor premiums estimated using Fama-MacBeth (1973) regressions” seems to suggest the necessity of large capital asset data sets, whichis incorrect and understates the usefulness of APT: As shown in the applied APT examples, if the assumptions of APT are met, then only n assets = 1 + #risks are required to apply APT.

— In contrast to CAPM, APT is a theory stated directly in terms of how one would optimallyarbitrage or hedge—in expectation—any arbitrary set of risks based on (stable, reliable) estimates of (1) asset sensitivities to risk factors and (2) expected asset returns.

— Because (1) APT applies equally to privately-held capital assets, and (2) it is well-accepted that privately-held asset (e.g., private equity interest) returns are sensitive to risks other than CAPM aggregate capital market risk, it follows that (3) APT can be used to appropriately hedge the risk exposures, in expectation, and price / value such non-traded assets.

7. Implications for private equity valuation


— There is little justification for using CAPM methods in private equity valuation.

— APT-based methods for private equity valuation are based on simpler, morerobust theory and evidence.

— The optimality of the no-arbitrage risk hedging and valuation under APT can be demonstrated empirically, which is particularly useful in private equity andM&A transaction negotiations, and in various commercial litigation settings.

Questions? Please feel free to contact …

Malcolm [email protected]

apt versus capm1. evidence on capm theory and arbitrage pricing theory (apt ) 2. analysis of capm...

Documents