Asset pricing

Asset pricing is a field of Financial economics that studies how the prices and expected returns of financial assets are determined. It examines the relationship between an asset's expected cash flows, risk, uncertainty, investor preferences, and market conditions.

Asset-pricing theory is used to analyze securities and financial claims such as stocks, bonds, derivatives, and other investment assets.

Basic concept

The value of a financial asset can be related to the present value of its expected future cash flows.

A simplified representation is:

$P_{0} = \sum_{t = 1}^{n} \frac{E (C F_{t})}{(1 + r)^{t}}$

where:

$P_{0}$ = current asset price;
$E (C F_{t})$ = expected cash flow at time $t$ ;
$r$ = appropriate discount rate;
$t$ = time period;
$n$ = number of periods.

The appropriate discount rate may incorporate the risk-free rate and one or more risk premiums.

Expected return

Asset-pricing models frequently relate expected returns to the compensation required for bearing risk.

A simplified relationship is:

$Expected Return = Risk-Free Rate + Risk Premium$

For an asset $i$ :

$E (R_{i}) = R_{f} + {Risk Premium}_{i}$

where:

$E (R_{i})$ = expected return of asset $i$ ;
$R_{f}$ = risk-free rate.

The specific form of the risk premium depends on the asset-pricing model being used.

Capital Asset Pricing Model

The Capital asset pricing model (CAPM) is one of the best-known asset-pricing models.

It relates an asset's expected return to its systematic risk:

$E (R_{i}) = R_{f} + β_{i} [E (R_{m}) - R_{f}]$

where:

$E (R_{i})$ = expected return of asset $i$ ;
$R_{f}$ = risk-free rate;
$β_{i}$ = beta of asset $i$ ;
$E (R_{m})$ = expected market return;
$E (R_{m}) - R_{f}$ = expected market risk premium.

Under CAPM, investors are compensated for systematic risk rather than diversifiable asset-specific risk.

Arbitrage pricing theory

Arbitrage pricing theory (APT) provides a multifactor approach to asset pricing.

A simplified APT representation is:

$E (R_{i}) = R_{f} + \sum_{k = 1}^{K} β_{i k} λ_{k}$

where:

$E (R_{i})$ = expected return of asset $i$ ;
$R_{f}$ = risk-free rate;
$β_{i k}$ = sensitivity of asset $i$ to risk factor $k$ ;
$λ_{k}$ = risk premium associated with factor $k$ ;
$K$ = number of systematic risk factors.

Unlike the single-factor CAPM framework, APT allows multiple systematic factors to influence expected returns.

Factor models

Modern asset pricing frequently uses multifactor models to explain differences in expected asset returns.

A general factor-model representation is:

$R_{i} = α_{i} + \sum_{k = 1}^{K} β_{i k} F_{k} + ϵ_{i}$

where:

$R_{i}$ = return of asset $i$ ;
$α_{i}$ = intercept;
$β_{i k}$ = sensitivity of asset $i$ to factor $k$ ;
$F_{k}$ = return associated with factor $k$ ;
$ϵ_{i}$ = asset-specific component.

Factor models can be used to analyze the relationship between asset returns and systematic economic or financial factors.

Consumption-based asset pricing

Consumption-based asset pricing relates asset prices to investors' consumption decisions and marginal utility.

A simplified pricing relationship can be represented as:

$P_{t} = E_{t} [M_{t + 1} X_{t + 1}]$

where:

$P_{t}$ = asset price at time $t$ ;
$M_{t + 1}$ = stochastic discount factor;
$X_{t + 1}$ = future payoff;
$E_{t}$ = expectation conditional on information available at time $t$ .

The stochastic discount factor represents the economic value assigned to future payoffs under uncertainty.

No-arbitrage pricing

A fundamental principle in asset pricing is the Law of one price, under which identical assets with identical cash flows should generally have the same price in an arbitrage-free market.

Arbitrage involves attempting to obtain a riskless profit from price inconsistencies.

A simplified no-arbitrage condition is:

$P_{A} = P_{B}$

when assets $A$ and $B$ provide identical future cash flows under identical conditions.

If:

$P_{A} \neq P_{B}$

an arbitrage opportunity may exist, subject to transaction costs, market frictions, and other constraints.

Risk and return

Asset pricing distinguishes between different forms of investment risk.

Systematic risk is associated with broad market or economic factors.

Unsystematic risk is specific to an individual asset or issuer and can potentially be reduced through diversification.

A simplified representation is:

$Total Risk = Systematic Risk + Unsystematic Risk$

Different asset-pricing models make different assumptions about which risks should be compensated through expected returns.

Risk premium

The risk premium represents the additional expected return associated with bearing investment risk.

A simplified expression is:

${Risk Premium}_{i} = E (R_{i}) - R_{f}$

Under CAPM:

${Risk Premium}_{i} = β_{i} [E (R_{m}) - R_{f}]$

The required risk premium can vary according to the asset, investor, market, and model used.

Bond pricing

Asset pricing also applies to fixed-income securities.

A simplified bond-pricing equation is:

$P = \sum_{t = 1}^{n} \frac{C_{t}}{(1 + y)^{t}} + \frac{F}{(1 + y)^{n}}$

where:

$P$ = bond price;
$C_{t}$ = coupon payment at time $t$ ;
$F$ = face value;
$y$ = yield or discount rate;
$n$ = number of periods.

The required yield can incorporate a risk-free interest rate and appropriate risk premiums.

Equity pricing

Equity valuation can involve discounting expected future dividends or other expected cash flows.

A simplified dividend-discount relationship is:

$P_{0} = \sum_{t = 1}^{\infty} \frac{E (D_{t})}{(1 + r)^{t}}$

where:

$P_{0}$ = current stock price;
$E (D_{t})$ = expected dividend at time $t$ ;
$r$ = required rate of return.

Specific equity valuation models can incorporate dividend growth, earnings, free cash flow, or other measures of expected future economic benefits.

Market efficiency

Asset pricing is closely related to the Efficient-market hypothesis, which examines the extent to which asset prices reflect available information.

If markets efficiently incorporate available information, consistently identifying securities that are substantially mispriced can be difficult after accounting for risk, transaction costs, and other constraints.

Behavioral asset pricing

Behavioral finance examines how psychological factors can affect investor decisions and asset prices.

Behavioral approaches to asset pricing may consider factors such as:

investor sentiment;
overconfidence;
loss aversion;
limited attention;
herd behavior;
other systematic behavioral biases.

These approaches provide alternative explanations for some observed market behavior that may not be fully explained by traditional asset-pricing models.

Empirical asset pricing

Empirical asset pricing uses historical and observed market data to evaluate theoretical models and identify factors associated with differences in asset returns.

Researchers may examine:

market capitalization;
valuation ratios;
profitability;
investment;
momentum;
liquidity;
interest rates;
macroeconomic variables.

The relationship between an explanatory factor and asset returns does not necessarily establish a causal relationship.

Asset pricing and portfolio management

Asset-pricing models are used in portfolio management to estimate expected returns, assess risk exposures, construct portfolios, and evaluate investment performance.

Risk-adjusted performance measures include:

$Sharpe Ratio = \frac{R_{p} - R_{f}}{σ (R_{p})}$

and:

$Treynor Ratio = \frac{R_{p} - R_{f}}{β_{p}}$

These measures evaluate realized investment returns relative to different measures of risk.

Asset pricing and corporate finance

Asset pricing has important applications in corporate finance.

For example, CAPM can be used to estimate the cost of equity:

$K_{e} = R_{f} + β_{e} [E (R_{m}) - R_{f}]$

The estimated cost of equity can then be incorporated into the weighted average cost of capital and used in investment appraisal and valuation.

Importance

Asset pricing provides a theoretical and quantitative framework for understanding how financial assets are valued and how expected returns relate to risk and uncertainty. Its models are used in investment analysis, portfolio management, corporate finance, financial valuation, and academic research.

Limitations

Asset-pricing models rely on assumptions that may not fully reflect actual financial markets.

Limitations can include:

uncertainty in expected-return estimates;
difficulty measuring the true market portfolio;
changing relationships between risk factors and returns;
transaction costs and taxes;
market frictions;
information asymmetry;
behavioral influences;
model risk.

Different asset-pricing models can therefore produce different estimates of expected returns or asset values.

Basic concept

Expected return

Capital Asset Pricing Model

Arbitrage pricing theory

Factor models

Consumption-based asset pricing

No-arbitrage pricing

Risk and return

Risk premium

Bond pricing

Equity pricing

Market efficiency

Behavioral asset pricing

Empirical asset pricing

Asset pricing and portfolio management

Asset pricing and corporate finance

Importance

Limitations

See also