Financial economics

Overview

Financial economics is the branch of economics characterized by a "concentration on monetary activities", in which "money of one type or another is likely to appear on both sides of a trade".

Its concern is thus the interrelation of financial variables, such as share prices, interest rates and exchange rates, as opposed to those concerning the real economy. It has two main areas of focus: asset pricing and corporate finance; the first being the perspective of providers of capital, i.e. investors, and the second of users of capital. It thus provides the theoretical underpinning for much of finance. The subject is concerned with "the allocation and deployment of economic resources, both spatially and across time, in an uncertain environment". It therefore centers on decision making under uncertainty in the context of the financial markets, and the resultant economic and financial models and principles, and is concerned with deriving testable or policy implications from acceptable assumptions. It thus also includes a formal study of the financial markets themselves, especially market microstructure and market regulation. It is built on the foundations of microeconomics and decision theory. Financial econometrics is the branch of financial economics that uses econometric techniques to parameterise the relationships identified. Mathematical finance is related in that it will derive and extend the mathematical or numerical models suggested by financial economics. Whereas financial economics has a primarily microeconomic focus, monetary economics is primarily macroeconomic in nature.

Underlying economics

Financial economics studies how rational investors would apply decision theory to investment management. The subject is thus built on the foundations of microeconomics and derives several key results for the application of decision making under uncertainty to the financial markets. The underlying economic logic yields the fundamental theorem of asset pricing, which gives the conditions for arbitrage-free asset pricing. The various "fundamental" valuation formulae result directly.

Present value, expectation and utility

Underlying all of financial economics are the concepts of present value and expectation. Calculating their present value,

X

s j

/

r

{\displaystyle X_{sj}/r}

in the first formula, allows the decision maker to aggregate the cashflows (or other returns) to be produced by the asset in the future to a single value at the date in question, and to thus more readily compare two opportunities; this concept is then the starting point for financial decision making. (Note that here, "

r

{\displaystyle r}

" represents a generic (or arbitrary) discount rate applied to the cash flows, whereas in the valuation formulae, the risk-free rate is applied once these have been "adjusted" for their riskiness; see below.) An immediate extension is to combine probabilities with present value, leading to the expected value criterion which sets asset value as a function of the sizes of the expected payouts and the probabilities of their occurrence,

X

s

{\displaystyle X_{s}}

and

p

s

{\displaystyle p_{s}}

respectively. This decision method, however, fails to consider risk aversion. In other words, since individuals receive greater utility from an extra dollar when they are poor and less utility when comparatively rich, the approach is therefore to "adjust" the weight assigned to the various outcomes, i.e. "states", correspondingly:

Y

s

Arbitrage-free pricing and equilibrium

The concepts of arbitrage-free, "rational", pricing and equilibrium are then coupled with the above to derive various of the "classical" (or "neo-classical") financial economics models. Rational pricing is the assumption that asset prices (and hence asset pricing models) will reflect the arbitrage-free price of the asset, as any deviation from this price will be arbitraged away: the "law of one price". This assumption is useful in pricing fixed income securities, particularly bonds, and is fundamental to the pricing of derivative instruments. Economic equilibrium is a state in which economic forces such as supply and demand are balanced, and in the absence of external influences these equilibrium values of economic variables will not change. General equilibrium deals with the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that a set of prices exists that will result in an overall equilibrium. (This is in contrast to partial equilibrium, which only analyzes single markets.) The two concepts are linked as follows: where market prices are complete and do not allow profitable arbitrage, i.e. they comprise an arbitrage-free market, then these prices are also said to constitute an "arbitrage equilibrium". Intuitively, this may be seen by considering that where an arbitrage opportunity does exist, then prices can be expected to change, and they are therefore not in equilibrium. An arbitrage equilibrium is thus a precondition for a general economic equilibrium. "Complete" here means that there is a price for every asset in every possible state of the world,

s

{\displaystyle s}

, and that the complete set of possible bets on future states-of-the-world can therefore be constructed with existing assets (assuming no friction): essentially solving simultaneously for n (risk-neutral) probabilities,

q

s

{\displaystyle q_{s}}

, given n prices. For a simplified example see Rational pricing § Risk neutral valuation, where the economy has only two possible states – up and down – and where

q

u p

{\displaystyle q_{up}}

and

q

d o w n

{\displaystyle q_{down}}

(=

1 −

q

u p

{\displaystyle 1-q_{up}}

State prices

With the above relationship established, the further specialized Arrow–Debreu model may be derived.

This result suggests that, under certain economic conditions, there must be a set of prices such that aggregate supplies will equal aggregate demands for every commodity in the economy. The Arrow–Debreu model applies to economies with maximally complete markets, in which there exists a market for every time period and forward prices for every commodity at all time periods. A direct extension, then, is the concept of a state price security, also called an Arrow–Debreu security, a contract that agrees to pay one unit of a numeraire (a currency or a commodity) if a particular state occurs ("up" and "down" in the simplified example above) at a particular time in the future and pays zero numeraire in all the other states. The price of this security is the state price

π

s

{\displaystyle \pi _{s}}

of this particular state of the world; the collection of these is also referred to as a "Risk Neutral Density". In the above example, the state prices,

π

u p

{\displaystyle \pi _{up}}

,

π

d o w n

{\displaystyle \pi _{down}}

would equate to the present values of

$

q

u p

{\displaystyle \$q_{up}}

and

$

Stochastic discount factor

The stochastic discount factor (SDF) — also called the pricing kernel — provides an alternative but closely related approach to valuation.

In many asset-pricing models it is considered more fundamental than state prices, since it can be derived directly from investor preferences and utility maximization, and thus forms the conceptual foundation of much of modern financial economics. Although the true SDF is (generally) unobservable, practitioners often work with models that implicitly approximate it - this link will be highlighted in the sections that follow.

Under the SDF framework, the price of an asset is obtained by discounting its future payoff by the stochastic factor

m ~

{\displaystyle {\tilde {m}}}

, and taking the conditional expectation

E [

X

s

m ~

s

]

{\textstyle E[X_{s}{\tilde {m}}_{s}]}

;

the third equation above. In equilibrium models, this factor corresponds to the intertemporal marginal rate of substitution in consumption,

where its formulation - as aside - highlights the economic mechanism behind pricing: because investors are typically assumed to be risk-averse, marginal utility is high in unfavorable states of the world (such as recessions or low consumption); as a result, the SDF takes larger values in these states, making assets that pay off during bad economic conditions more valuable. In this way the SDF links the arbitrage-free pricing framework to underlying economic preferences and consumption risk. The SDF is then closely related to the term

Y

s

{\displaystyle Y_{s}}

, in the first formula above, which represents the state-dependent adjustment for risk aversion. In the stochastic discount factor formulation, this adjustment appears directly in the pricing kernel, which weights payoffs according to their marginal utility across states. It is similarly related to the relationship between risk-neutral and physical probabilities - in discrete time,

Resultant models

Applying the above economic concepts, we can derive various economic- and financial models and principles. As outlined, the two areas of focus are Asset Pricing and Corporate Finance, the first being the perspective of providers of capital, the second of users of capital. Here, and for (almost) all other financial economics models, the questions addressed are typically framed in terms of "time, uncertainty, options, and information", as will be seen below.

Time: money now is traded for money in the future. Uncertainty (or risk): The amount of money to be transferred in the future is uncertain. Options: one party to the transaction can make a decision at a later time that will affect subsequent transfers of money. Information: knowledge of the future can reduce, or possibly eliminate, the uncertainty associated with future monetary value (FMV). Applying this framework, with the above concepts, leads to the required models. This derivation begins with the assumption of "no uncertainty" and is then expanded to incorporate the other considerations. (This division sometimes denoted "deterministic" and "random", or "stochastic".)

Certainty

The starting point here is "Investment under certainty", and usually framed in the context of a corporation. The Fisher separation theorem, asserts that the objective of the corporation will be the maximization of its present value, regardless of the preferences of its shareholders. Related is the Modigliani–Miller theorem, which shows that, under certain conditions, the value of a firm is unaffected by how that firm is financed, and depends neither on its dividend policy nor its decision to raise capital by issuing stock or selling debt. The proof here proceeds using arbitrage arguments, and acts as a benchmark for evaluating the effects of factors outside the model that do affect value.

The mechanism for determining (corporate) value is provided by John Burr Williams' The Theory of Investment Value, which proposes that the value of an asset should be calculated using "evaluation by the rule of present worth". Thus, for a common stock, the "intrinsic", long-term worth is the present value of its future net cashflows, in the form of dividends; in the corporate context, "free cash flow" as aside. What remains to be determined is the appropriate discount rate. Later developments show that, "rationally", i.e. in the formal sense, the appropriate discount rate here will (should) depend on the asset's riskiness relative to the overall market, as opposed to its owners' preferences; see below. Net present value (NPV), the result of this discounted cashflow valuation, is the direct extension of these ideas typically applied to Corporate Finance decisioning. For other results, as well as specific models developed here, see the list of "Equity valuation" topics under Outline of finance § Discounted cash flow valuation.

Bond valuation, in that cashflows (coupons and return of principal, or "Face value") are deterministic, may proceed in the same fashion. An immediate extension, Arbitrage-free bond pricing, discounts each cashflow at the market derived rate – i.e. at each coupon's corresponding zero rate, and of equivalent credit worthiness – as opposed to an overall rate; see Rational pricing § Fixed-income securities. In many treatments bond valuation precedes equity valuation, under which cashflows (dividends) are not "known" per se. Williams and onward allow for forecasting as to these – based on historic ratios or published dividend policy – and cashflows are then treated as essentially deterministic; see below under § Corporate finance theory. For both stocks and bonds, "under certainty, with the focus on cash flows from securities over time," valuation based on a term structure of interest rates is in fact consistent with arbitrage-free pricing. Indeed, a corollary of the above is that "the law of one price implies the existence of a discount factor". Corresponding to these:

∑

s

π

s

= 1

/

r

{\textstyle \sum _{s}\pi _{s}=1/r}

; and equivalently, for the above "numeraire bond",

B

0

=

∑

s

π

s

{\textstyle B_{0}=\sum _{s}\pi _{s}}

Uncertainty

For "choice under uncertainty" the twin assumptions of rationality and market efficiency, as more closely defined, lead to: modern portfolio theory (MPT) with its capital asset pricing model (CAPM) – an equilibrium-based result; and to the Black–Scholes–Merton theory (BSM; often, simply Black–Scholes) for option pricing – an arbitrage-free result. As above, the (intuitive) link between these, is that the latter derivative prices are calculated such that they are arbitrage-free with respect to the more fundamental, equilibrium determined, securities prices; see Asset pricing § Interrelationship. Briefly, and intuitively – and consistent with § Arbitrage-free pricing and equilibrium above – the relationship between rationality and efficiency is as follows. Given the ability to profit from private information, self-interested traders are motivated to acquire and act on their private information. In doing so, traders contribute to more and more "correct", i.e. efficient, prices: the efficient-market hypothesis, or EMH. Thus, if prices of financial assets are (broadly) efficient, then deviations from these (equilibrium) values could not last for long. (See earnings response coefficient.) The EMH (implicitly) assumes that average expectations constitute an "optimal forecast", i.e. prices using all available information are identical to the best guess of the future: the assumption of rational expectations. The EMH does allow that when faced with new information, some investors may overreact and some may underreact, but what is required, however, is that investors' reactions follow a normal distribution – so that the net effect on market prices cannot be reliably exploited to make an abnormal profit. In the competitive limit, then, market prices will reflect all available information and prices can only move in response to news: the random walk hypothesis. This news, of course, could be "good" or "bad", minor or, less common, major; and these moves are then, correspondingly, normally distributed; with the price therefore following a log-normal distribution.

Under these conditions, investors can then be assumed to act rationally: their investment decision must be calculated or a loss is sure to follow; correspondingly, where an arbitrage opportunity presents itself, then arbitrageurs will exploit it, reinforcing this equilibrium. Here, as under the certainty-case above, the specific assumption as to pricing is that prices are calculated as the present value of expected future dividends, as based on currently available information. What is required though, is a theory for determining the appropriate discount rate, i.e. "required return", given this uncertainty: this is provided by the MPT and its CAPM. Relatedly, rationality – in the sense of arbitrage-exploitation – gives rise to Black–Scholes; option values here ultimately consistent with the CAPM. In general, then, while portfolio theory studies how investors should balance risk and return when investing in many assets or securities, the CAPM is more focused, describing how, in equilibrium, markets set the prices of assets in relation to how risky they are.

This result will be independent of the investor's level of risk aversion and assumed utility function, thus providing a readily determined discount rate for corporate finance decision makers as above, and for other investors. The argument proceeds as follows: If one can construct an efficient frontier – i.e. each combination of assets offering the best possible expected level of return for its level of risk, see diagram – then mean-variance efficient portfolios can be formed simply as a combination of holdings of the risk-free asset and the "market portfolio" (the Mutual fund separation theorem), with the combinations here plotting as the capital market line, or CML. Then, given this CML, the required return on a risky security will be independent of the investor's utility function, and solely determined by its covariance ("beta") with aggregate, i.e. market, risk. This is because investors here can then maximize utility through leverage as opposed to stock selection; see Separation property (finance), Markowitz model § Choosing the best portfolio and CML diagram aside. As can be seen in the formula aside, this result is consistent with the preceding, equaling the riskless return plus an adjustment for risk. A more modern, direct, derivation is as described at the bottom of this section; which can be generalized to derive other equilibrium-pricing models.

Black–Scholes provides a mathematical model of a financial market containing derivative instruments, and the resultant formula for the price of European-styled options.

The model is expressed as the Black–Scholes equation, a partial differential equation describing the changing price of the option over time; it is derived assuming log-normal, geometric Brownian motion (see Brownian model of financial markets). The key financial insight behind the model is that one can perfectly hedge the option by buying and selling the underlying asset in just the right way and consequently "eliminate risk", absenting the risk adjustment from the pricing (

V

{\displaystyle V}

, the value, or price, of the option, grows at

r

{\displaystyle r}

, the risk-free rate). This hedge, in turn, implies that there is only one right price – in an arbitrage-free sense – for the option. And this price is returned by the Black–Scholes option pricing formula. (The formula, and hence the price, is consistent with the equation, as the formula is the solution to the equation.) Since the formula is without reference to the share's expected return, Black–Scholes inheres risk neutrality; intuitively consistent with the "elimination of risk" here, and mathematically consistent with § Arbitrage-free pricing and equilibrium above. Relatedly, therefore, the pricing formula may also be derived directly via risk neutral expectation. Itô's lemma provides the underlying mathematics, and, with Itô calculus more generally, remains fundamental in quantitative finance.

As implied by the Fundamental Theorem, the two major results are consistent. Here, the Black-Scholes equation can alternatively be derived from the CAPM, and the price obtained from the Black–Scholes model is thus consistent with the assumptions of the CAPM. The Black–Scholes theory, although built on Arbitrage-free pricing, is therefore consistent with the equilibrium based capital asset pricing. Both models, in turn, are ultimately consistent with the Arrow–Debreu theory, and can be derived via state-pricing – essentially, by expanding the above fundamental equations – further explaining, and if required demonstrating, this consistency. Here, the CAPM is derived by linking

Y

{\displaystyle Y}

, risk aversion, to overall market return, and setting the return on security

j

{\displaystyle j}

as

X

j

Extensions

More recent work further generalizes and extends these models. As regards asset pricing, developments in equilibrium-based pricing are discussed under "Portfolio theory" below, while "Derivative pricing" relates to risk-neutral, i.e. arbitrage-free, pricing. As regards the use of capital, "Corporate finance theory" relates, mainly, to the application of these models.

Portfolio theory

Much of the development here relates to required return, i.e. pricing, extending the basic CAPM. Multi-factor models such as the Fama–French three-factor model and the Carhart four-factor model, propose factors other than market return as relevant in pricing. The intertemporal CAPM and consumption-based CAPM similarly extend the model. With intertemporal portfolio choice, the investor now repeatedly optimizes her portfolio; while the inclusion of consumption (in the economic sense) then incorporates all sources of wealth, and not just market-based investments, into the investor's calculation of required return. (Factor-pricing models such as these may be derived by assuming the SDF to be linear in observable risk factors.) Whereas the above extend the CAPM, the single-index model (SIM) is a more simple model. It assumes, only, a correlation between security and market returns, without (numerous) other economic assumptions. It is useful in that it simplifies the estimation of correlation between securities, significantly reducing the inputs for building the correlation matrix required for portfolio optimization there. The arbitrage pricing theory (APT) similarly differs as regards its assumptions. APT "gives up the notion that there is one right portfolio for everyone in the world, and ...replaces it with an explanatory model of what drives asset returns". It returns the required (expected) return of a financial asset as a linear function of various macro-economic factors, and assumes that arbitrage should bring incorrectly priced assets back into line. The linear factor model structure of the APT is used as the basis for many of the commercial risk and fund management systems employed by asset managers. Here, managers apply various of the abovementioned multi-factor models - often bespoke extensions - such that their portfolio has the desired exposure ("tilt") to macroeconomic, market and / or fundamental risk factors; respectively: Macro-, Factor-, and Style Funds. Research here, and application (e.g. "Smart Beta"), is ongoing, and over the years "hundreds of factors attempt to explain the cross-section of expected returns";

with this "factor zoo",

there is a risk of data mining, and researchers have proposed various criteria for establishing significance. At the same time, "classic" mean-variance optimization — i.e. building an efficient portfolio as described above — is still widely used by Asset Allocation Funds. Here, given issues noted with this approach, the application is typically in combination with other techniques. The Black–Litterman model is often employed. Black–Litterman departs from the original Markowitz model approach: it starts with an equilibrium assumption, as for the latter, but this is then modified to take into account the "views" (i.e., the specific opinions about asset returns) of the investor in question to arrive at a bespoke asset allocation. A further modification often used relates to the subsequent optimization: under (Tail) risk parity, the focus is on the allocation of risk, rather than the allocation of capital. Other developments re portfolio optimization include the following. Where factors additional to volatility are considered (kurtosis, skew...) then multiple-criteria decision analysis can be applied; here deriving a Pareto efficient portfolio. The universal portfolio algorithm identifies the "growth optimal portfolio" per the Kelly criterion, applying information theory to asset selection. Behavioral portfolio theory recognizes that investors have varied aims and create an investment portfolio that meets a broad range of goals. Copulas have lately been applied here; recently this is the case also for genetic algorithms and machine learning, more generally; see below. The Treynor–Black model utilizes the outlined SIM structure in portfolio optimization; here, essentially, one holds a passively invested index fund, with (some) high alpha securities overweighted.

Derivative pricing

In pricing derivatives, the binomial options pricing model provides a discretized version of Black–Scholes, useful for the valuation of American styled options. Discretized models of this type are built – at least implicitly – using state-prices (as above); relatedly, a large number of researchers have used options to extract state-prices for a variety of other applications in financial economics. For path dependent derivatives, Monte Carlo methods for option pricing are employed; here the modelling is in continuous time, but similarly uses risk neutral expected value. Various other numeric techniques have also been developed. The theoretical framework too has been extended such that martingale pricing is now the standard approach.

Drawing on these techniques, models for various other underlyings and applications have also been developed, all based on the same logic (using "contingent claim analysis"). Real options valuation allows that option holders can influence the option's underlying; models for employee stock option valuation explicitly assume non-rationality on the part of option holders; Credit derivatives allow that payment obligations or delivery requirements might not be honored. Exotic derivatives are now routinely valued. Multi-asset underlyers are handled via simulation or copula based analysis. Similarly, the various short-rate models allow for an extension of these techniques to fixed income- and interest rate derivatives. (The Vasicek and CIR models are equilibrium-based, while Ho–Lee and subsequent models are based on arbitrage-free pricing.) The more general HJM Framework describes the dynamics of the full forward-rate curve – as opposed to working with short rates – and is then more widely applied. The valuation of the underlying instrument – additional to its derivatives – is relatedly extended, particularly for hybrid securities, where credit risk is combined with uncertainty re future rates; see Bond valuation § Stochastic calculus approach and Lattice model (finance) § Hybrid securities.

Following the Crash of 1987, equity options traded in American markets began to exhibit what is known as a "volatility smile"; that is, for a given expiration, options whose strike price differs substantially from the underlying asset's price command higher prices, and thus implied volatilities, than what is suggested by BSM. (The pattern differs across various markets.) Modelling the volatility smile is an active area of research, and developments here – as well as implications re the standard theory – are discussed in the next section. After the 2008 financial crisis, a further development: as outlined, (over the counter) derivative pricing had relied on the BSM risk neutral pricing framework, under the assumptions of funding at the risk free rate and the ability to perfectly replicate cashflows so as to fully hedge. This, in turn, is built on the assumption of a credit-risk-free environment – called into question during the crisis. Addressing this, therefore, issues such as counterparty credit risk, funding costs and costs of capital are now additionally considered when pricing, and a credit valuation adjustment, or CVA – and potentially other valuation adjustments, collectively xVA – is generally added to the risk-neutral derivative value. The standard economic arguments can be extended to incorporate these various adjustments. A related, and perhaps more fundamental change, is that discounting is now on the Overnight Index Swap (OIS) curve, as opposed to LIBOR as used previously. This is because post-crisis, the overnight rate is considered a better proxy for the "risk-free rate". (Also, practically, the interest paid on cash collateral is usually the overnight rate; OIS discounting is then, sometimes, referred to as "CSA discounting".) Swap pricing – and, therefore, yield curve construction – is further modified: previously, swaps were valued off a single "self discounting" interest rate curve; whereas post crisis, to accommodate OIS discounting, valuation is now under a "multi-curve framework" where "forecast curves" are constructed for each floating-leg LIBOR tenor, with discounting on the common OIS curve.