Models of Stock Returns–A Comparison

Kon, Stanley J., “Models of Stock Returns–A Comparison,” The Journal of Finance, Vol 39, No 1 (1984), 147-165.

Purpose:  To explain the observed kurtosis (fat tails) and positive skewness in the distribution of stock returns.

Findings:  The discrete mixture of normal distributions proposed in this paper explains these moments better than existing models, including the Student-t.

Motivation:  Mean-variance portfolio theory, option pricing models, and empirical tests of capital asset pricing models and efficient markets make assumptions about the distribution of stock returns.  It has been shown that stock returns cannot be approximated using a single normal distribution, but the normal is just what the theoretical models assume.

Methods:  Returns may be driven by multiple normal distributions–one for random shocks, one for firm-specific information, one for macroeconomic information, etc.  In other words, it is not necessary that each observation of stock return be drawn from the same distribution as all others.  This paper uses a mix of up to five normal distributions and likelihood ratio tests to model the returns of the 30 Dow Jones stocks.

  • Let  r_t = \alpha_i + u_{it}, where r_t is the observed return for period t, and u_{it} is normally distributed with mean zero and variance \sigma_i^2.
  • Let \underline{\gamma}_i be a normal distribution with mean \alpha_i and mean \sigma_i^2.
  • Let \lambda_i = T_i/T be the observations associated with information set I_i over total observations, or the proportion of total observations that are associated with information set I_i.
  • Let \underline{\theta} = \{\alpha_i, ..., \alpha_N, \sigma_i^2, ..., \sigma_N^2, \lambda_i, ..., \lambda_{N-1}\} be a vector of parameters.
  • Let \underline{r} be the vector of returns.
  • The vector of parameters, given a vector of returns, can be found by solving

max \ell(\underline{\theta}/\underline{r}) = \Pi_{t=1}^T \left[\sum_{i=1}^N \lambda_i p(r_t|\underline{\gamma_i})\right].


  • All 30 Dow Jones stocks can be explained by a mixture of 2, 3, or 4 normals.
  • Returns of the S&P500, value-weighted, and equal-weighted market indices were also explained by mixtures of normals.
  • The mixture of normals describes 27 of the 30 stocks, and all three indices, better than does the student-t distribution, which also has fat tails and has been used to model stock returns.
  • Differences in the means of the normal distributions can explain the skewness of observed stock returns.
  • Differences in the variance of the normals can explain the kurtosis of stock returns.
  • Stock returns do appear to be normally distributed, but the distribution parameters are time-varying, and the timing of parameter shifts also varies across stocks.

Dynamic Agency and the q Theory of Investment

DeMarzo, Peter M., Michael J. Fishman, Zhiguo He, and Neng Wang, 2012, “Dynamic Agency and the q Theory of Investment,” The Journal of Finance, Vol. 67, No. 6 (2012), 2295-2340.

Purpose:  To introduce an agency problem into the standard q theory of investment; to show that cash flow is not the best predictor of investment.

Motivation:  A large body of literature uses cash flow to predict firm investment levels.  This paper argues that a better proxy is “financial slack,” which is directly related to the agency problem.


  • Productivity is a Brownian motion, and the agent controls the drift but not the volatility.
  • w = W/K is the agent’s total expected payoff per unit of capital, and must be high enough to incentivize the agent to maximize productivity.
  • The level of w depends on λ, σ, and historical firm profitability.
    • λ is a measure of the extent of the agency problem.
    • σ is the volatility of firm productivity.
    • Past productivity raises or lowers w, and the agent loses his job when w = 0.
    • A portion of w is deferred, giving the agent a stake in continued firm success.
  • Investor’s expected payoff per unit of capital, p(w), is a function of how much they pay the agent.
  • Average q is total firm value per unit of capital stock, or qa = p(w) + w.
  • Marginal q, or qm = p(w)wp’(w).
    • Firms invest when marginal q is less than 1, so investment is a function of w. It follows that investment depends upon λ, σ, and past firm performance.
  • “Financial slack” equals w/λ, and is the largest productivity shock the firm can suffer without changing agents.
  • The agent accumulates cash and available credit equal to the firm’s “financial slack,” then distributes excess income to shareholders.


  • Financial slack is a better predictor of investment than cash flow.
  • Average q is higher than marginal q because an increase in capital stock K reduces w, and hence reduces the agent’s [historically determined] incentives to maximize productivity.
  • Financial slack and profitability are substitutes in determining average q.
  • When the firm is profitable, w rises, the agent’s incentives grow, and return on investment increases.
    • Investment is serially correlated.
  • The cost of incentivizing the agent leads to underinvestment in every state of the world.

Asset commonality, debt maturity and systemic risk

Allen, Franklin, Ana Babus, and Elena Carletti, 2012, “Asset commonality, debt maturity and systemic risk,” Journal of Financial Economics 104 (2012), 519-534.

Purpose:  To model a connection between the commonality of financial institutions’ (banks’) asset portfolios and the maturity of their debts.

Motivation:  The financial crisis starting in 2007 brought attention to the systemic risks stemming from linkages between the portfolios of financial institutions.  This paper investigates how debt maturity might cause such linkages to become a systemic risk.


  • Long-term debt is not related to the riskiness of overlaps between institutions’ asset portfolios.
  • Clustered asset structures (where portfolios only overlap within a cluster) produce more systemic risk, and usually lead to lower total welfare.
  • Short-term debt can transfer insolvency between banks.
  • The use of short-term debt and the link between short-term debt and systemic risk, depend upon the structure of overlaps between banks’ portfolios (asset structure).

Model:  Six banks invest in six risky projects, exchange shares in their own project with others, and finance their investment with either short- or long-term debt in period 0.

  • The six banks exchange shares in each other’s projects to reduce risk, and do so in two patterns.
    • “Clustered” – two groups of three banks each exchange assets solely within the group, so that all banks within each group have identical portfolios.
      • In this structure, there is greater information spillover within groups, so trouble at one bank is most likely to also cause investors in the others not to roll over. Thus, there are more liquidations than in the unclustered group.
    • “Unclustered” – each bank exchanges only with the two neighboring banks, and no two banks have identical portfolios.
  • When banks use short-term debt, investors decide at the end of period 1 whether to roll over the banks’ debt based on a signal indicating whether at least one bank will fail.
  • Solvency or insolvency is determined at the end of period 2.
  • Investors’ failure to roll over causes early bank liquidation and is the source of systemic failure.


  • When banks use short-term debt and have overlapping asset portfolios, the structure of the overlap is an important factor in determining systemic risk.

The Cash Flow Sensitivity of Cash

Almeida, Heitor, Murillo Campello, and Michael S. Weisbach, 2004, “The Cash Flow Sensitivity of Cash,” The Journal of Finance 59 (4), 1777-1804.

Purpose:  This paper develops a model of a firm’s demand for liquidity, and uses the model to obtain a new measurement of how important financial constraints are in affecting corporate policy.

Theory:  For financially constrained firms, or for firms that expect financial constraints in the future, higher cash flows should lead to more cash on the balance sheet.  For unconstrained firms that do not expect future constraints, the level of cash flows and the amount of cash on the balance sheet should be unrelated.  If this “cash flow sensitivity of cash” can be shown to be correlated with proxies for financial constraint, it will be a useful indicator of the influence of such constraints on firm behavior.

Motivation:  The effects of financial constraints on firm behavior and firms’ financial management are two topics usually studied separately, though they are closely linked.  Most previous literature regarding this link discusses the connection between financial constraints and firm investment demand.  There is a robust debate around this, with no clear consensus on what the empirical evidence means.  One complication with the literature is that current investment demand is correlated with future investment demand (since current investments impact cash flow).  Cash balances are not inherently linked to cash flows, and so are unrelated to future investment demand.  Therefore, the link between cash balances and financial constraints is less problematic to measure.

Empirical Evidence:  Using a large sample of manufacturing firms from 1971-2000, divide firms into subgroups on proxies for financial constraint.  Measure the cash flow sensitivity of cash for each group.


  • Firms exhibiting proxies for financial constraint hold 15% of assets as cash, compared to only 8-9% for firms that are not under [proxied] financial constraint
  • For 4 of the 5 financial constraint proxies, the firms expected to be constrained have the higher cash sensitivity to cash flows
  • Cash flow sensitivity of cash is a useful method for identifying financially constrained firms

Measuring investment distortions arising from stockholder-bondholder conflicts

Parrino, Robert, and Michael S. Weisbach, 1999, “Measuring investment distortions arising from stockholder-bondholder conflicts,” The Journal of Financial Economics 53, 3-42.

Purpose:  This paper calculates the expected wealth transfer between stock- and bondholders occurring when a firm begins a new project.  It also estimates how stockholder-bondholder conflict impacts investment decisions, and whether it can explain cross-sectional variation in capital structure.


  • There will be underinvestment when the firm is faced with safe projects.
    • Stockholders demand a higher return than the CAPM rate.
      • This effect is stronger for high-leverage firms.
    • Safe projects with low returns benefit bondholders at the expense of equity holders.
  • There will be overinvestment when the firm is faced with risky projects.
    • Stockholders are willing to gamble, and will even invest in negative NPV projects if the potential payoff is high.
      • This effect is also stronger for high-leverage firms.
    • Risky projects with negative expected (but high potential) payoffs benefit equity holders at the expense of bondholders.
  • Longer debt duration is vulnerable to larger agency problems
  • Lower marginal tax rates lead to slightly larger distortions
  • The stockholder-bondholder conflict does exist and there do seem to be empirical investment distortions, but these distortions are too small to be useful in explaining most firms’ capital structure decisions.
    • The distortion is only 0.14% for a firm with 20% debt-to-capital ratio, compared with a 3% noise factor in measuring cost of capital


Motivation:  Managers seek to maximize shareholder value, not necessarily firm value.  The “underinvestment problem” is when managers avoid a positive NPV project that would increase firm value but would lower stockholder value.  The “overinvestment problem” is when managers undertake a negative NPV project that lowers firm value but raises stockholder value.  These agency problems have been widely discussed in the literature for decades, but there is no consensus on their magnitude or on how important they really are.

Data/Methods:  Use numerical simulations to estimate the impact of debt on the investment decisions of a levered firm whose managers seek only to maximize stockholder value.  Compute the stockholder-bondholder wealth transfer accompanying projects with known characteristics.

  • Compustat data for firms from 1981-1995
  • Monte Carlo:  Assume a firm with known cash flows following a random walk without drift for 30 years, after which cash flows are static.  Assume a project financed entirely with equity, whose cash flows similarly follow a random walk without drift for 30 years.  Assume a correlation of 0.5 between firm and project cash flows, run the simulation 5,000 times, and compute the ex ante value of debt and equity each time.
    • Value of debt is the sum of discounted future cash flows to bondholders.
    • Value of equity is the discounted cash flows to stockholders plus a terminal value.

Conclusions:  Distortions in stockholder-bondholder required investment returns vary along several dimensions.  However, for the typical firm they are much smaller than the noise in cost-of-capital measurement.  The effect exists but is too small to explain cross-sectional variation in capital structure.

The Cost of Capital, Corporation Finance, and the Theory of Investment

Modigliani, Franco, and Merton Miller, 1958, “The Cost of Capital, Corporation Finance, and the Theory of Investment,” The American Economic Review, Vol. 48, No. 3, 261-297.

Purpose:  This paper proposes a theory explaining how a firm’s stock price (market value) is impacted by managers’ capital structure decisions.

Motivation:  Previous work regarding the cost of capital treats assets as having known income streams, and adjusts for uncertainty simply by subtracting “risk discounts” from the expected rates of return.  This treatment of risk is inadequate.  A market-value approach—where the cost of capital is the return on an investment that does not affect a firm’s stock price—has promise, but we need a theory describing the impact of a firm’s capital structure on its market value.


  • Model 1:  Corporations can only issue common equity
    • Assume perfect markets, no agency problems, and an economy where all assets are owned by corporations that can only finance operations with common equity.  Therefore, the rate of return on one share equals expected return to the share divided by share price.
      • Since there are no agency problems, retained earnings are the same as cash dividends.
    • Assume classes of corporations where each share’s expected return is perfectly correlated with all others in the class.  Then there is one rate of return for each class, and all shares in a class are perfect substitutes (up to a scale factor).
  • Model 2:  Corporations can issue bonds in addition to common equity
    • Assume bonds trade in perfect markets and all corporations and households have a perfect credit rating.  Then all bonds are perfect substitutes (up to a scale factor), and have the same expected rate of return.
    • Market value is independent of capital structure.
      • If an individual values leverage, he can take it on himself by borrowing money to buy more stock in an unlevered company.
    • The expected rate of return on stock in a levered company is the rate of return on a pure-equity company from its same class, plus a premium equal to the debt-to-equity ratio times the spread between the class-specific equity return and the [universal] cost of debt.
  • Model 3:  Corporate interest payments are tax-deductible
    • The debt-vs-equity consideration is important for overall corporate liquidity management due to taxes, timing, market sentiment, investor tax profiles, etc.
    • However, all that matters in project financing is the cost of capital.  A preference for one type of financing over another does not make a project more or less profitable.

Empirical Evidence and Conclusion:

  • Using data from the only two relevant studies:
    • There is no significant relationship between leverage and cost of capital.
    • As leverage increases, expected return to equity increases.
  • The amount of leverage can be important over the life of a corporation, but is irrelevant in determining the profitability of a project.

Agency Costs of Free Cash Flow, Corporate Finance, and Takeovers

Jensen, Michael C., 1986, “Agency Costs of Free Cash Flow, Corporate Finance, and Takeovers,” The American Economic Review 76 (2), 323-329.

Purpose:  This paper develops a theory linking debt, agency costs of free cash flow, and corporate takeovers.

Theory:  Managers seek to maximize their own influence, and not necessarily shareholder return.  They also have discretion over free cash flow, which represents an agency problem for firms with high free cash flows and low growth prospects.  Debt can be used to limit managers’ discretion over cash flows, and so increases in leverage can create value for shareholders beyond the tax implications.

  • Debt holders can force firm reorganization without bankruptcy more quickly and easily than equity holders.  This means higher-leveraged firms tend to be leaner and better managed.
  • Takeover targets should include firms with poor earnings and poor management, or firms with excellent earnings that management does not use to create value.
  • Lack of growth opportunities frequently leads to the undertaking of value-destroying projects.  Firms pursuing diversification and firms in industries with overcapacity often fit in this category.
  • These firms ought to see more takeovers, threats of takeovers, and subsequent debt increases.

Empirical Evidence:

  • Evidence from LBOs and going-private transactions
    • Most firms that are taken private are those with low growth and high potential for free cash flows—hence, high agency costs.  Strip financing (all bond-holders hold equal proportions of debt in each tier of seniority) reduces conflict of interest among bond holders, which gives them more power over the firm.
    • Very few of these transactions have gone into bankruptcy, suggesting that debt holders can force management efficiency and limit suboptimal investments.
  • Evidence from the oil industry
    • During the 1970s, oil prices increased and optimal capacity decreased, so that oil firms were both highly profitable and destined to shrink.  Oil managers continued investing in exploration & discovery projects that returned less than the cost of capital.
    • Around this time, oil companies began to merge and restructure under threat of takeover.  They increased leverage and cut value-destroying investments.


  • Managers of firms with low growth opportunities can’t be trusted with high cash flow.
    • Leverage can be used to limit the free cash flow agency problem.
  • Firms with high cash flows and low growth opportunities should be prime takeover targets.
  • Acquisitions financed with debt and cash should create more value than similar transactions done with equity.