Finance FHS - Key Formulas

Chapter 1: Week 0 - Returns, Risk, and Two-Asset Portfolios

1.1 Returns of an asset across states

Suppose an asset has state-contingent simple returns:

upper R_1, upper R_2, ..., upper R_{upper S}

in states 1, 2, ..., S occurring with probabilities:

lower p_1, lower p_2, ..., lower p_{upper S}

lower p_{lower s} >= 0, sum_{lower s=1}^{upper S} lower p_{lower s} = 1

1.1.1 Expected (mean) return

E[upper R] = sum_{lower s=1}^{upper S} lower p_{lower s} * upper R_{lower s}

1.2 Variance and standard deviation

1.2.1 Variance (several states of world form)

Var(upper R) = sum_{lower s=1}^{upper S} lower p_{lower s} * (upper R_{lower s} - E[upper R])^2

1.2.2 Variance (computational form)

Var(upper R) = E[upper R^2] - (E[upper R])^2

where:

E[upper R^2] = sum_{lower s=1}^{upper S} lower p_{lower s} * upper R_{lower s}^2

and (E[upper R])^2 is the mean return previously calculated, squared.

1.2.3 Standard deviation

sigma_{upper R} = sqrt(Var(upper R))

1.3 Two assets: weighted average returns and variance

Let assets A and B have random returns upper R_A and upper R_B. Let the portfolio weight in A be lower w, so weight in B is 1 - lower w. The portfolio return is:

upper R_lower p = lower w * upper R_upper A + (1 - lower w) * upper R_upper B

1.3.1 Portfolio expected return (weighted average)

E[upper R_lower p] = lower w * E[upper R_upper A] + (1 - lower w) * E[upper R_upper B]

1.3.2 Covariance and correlation

Cov(upper R_upper A, upper R_upper B) = E[(upper R_upper A - E[upper R_upper A]) * (upper R_upper B - E[upper R_upper B])]

Corr(upper R_upper A, upper R_upper B) = rho_{upper A upper B} = Cov(upper R_upper A, upper R_upper B) / (sigma_upper A * sigma_upper B)

1.3.3 Portfolio variance (two-asset)

Write sigma_upper A^2 = Var(upper R_upper A), sigma_upper B^2 = Var(upper R_upper B), and sigma_{upper A upper B} = Cov(upper R_upper A, upper R_upper B). Then:

Var(upper R_lower p) = lower w^2 * sigma_upper A^2 + (1 - lower w)^2 * sigma_upper B^2 + 2 * lower w * (1 - lower w) * sigma_{upper A upper B}

Equivalently using correlation:

Var(upper R_lower p) = lower w^2 * sigma_upper A^2 + (1 - lower w)^2 * sigma_upper B^2 + 2 * lower w * (1 - lower w) * rho_{upper A upper B} * sigma_upper A * sigma_upper B

It's with the correlation we see the benefits of diversification (try with cases of rho_{upper A upper B} = {-1; 0; +1}). Perfect correlations are the cases with rho_{upper A upper B} = {-1; +1}.

1.3.4 Portfolio standard deviation

sigma_lower p = sqrt(Var(upper R_lower p))

1.4 Present Value, Annuities, Perpetuities, and IRR

1.4.1 Present value of a cash flow stream

Let cash flows be upper C_1, upper C_2, ..., upper C_{upper T} paid at times 1, 2, ..., T, and let the discount rate at period t be lower r_lower t:

PV = sum_{lower t=1}^{upper T} upper C_lower t / (1 + lower r)^lower t

1.4.2 Annuity (flat cash-flows)

An ordinary annuity pays a constant amount upper C at t = 1, 2, ..., T:

PV_{annuity} = upper C * sum_{lower t=1}^{upper T} 1 / (1 + lower r)^lower t = (upper C / lower r) * (1 - (1 / (1 + lower r)^{upper T}))

1.4.3 Growing annuity

A growing annuity pays upper C_1 at t=1, then grows at constant rate lower g: upper C_lower t = upper C_1 * (1 + lower g)^{lower t - 1}. For lower r != lower g:

PV_{growing annuity} = sum_{lower t=1}^{upper T} (upper C_1 * (1 + lower g)^{lower t - 1}) / (1 + lower r)^lower t = (upper C_1 / (lower r - lower g)) * (1 - ((1 + lower g) / (1 + lower r))^{upper T})

1.4.4 Perpetuity (flat cash-flows)

A perpetuity pays a constant amount upper C forever starting at t=1. The PV answer will be in t=0 terms:

PV_{perpetuity} = sum_{lower t=1}^{infinity} upper C / (1 + lower r)^lower t = upper C / lower r

1.4.5 Growing perpetuity (Gordon growth)

(requires lower r > lower g).

A growing perpetuity pays upper C_1 at t=1 and grows at rate lower g forever. The PV answer will be in t=0 terms. upper C_lower t = upper C_1 * (1 + lower g)^{lower t - 1}. If lower r > lower g:

PV_{growing perpetuity} = upper C_1 / (lower r - lower g)

1.5 Project Decision Criteria Formulas

1.5.1 Net Present Value

The Net Present Value (NPV) is the present value of all future cash-flows (outflows or inflows) net of any outflow today (t=0):

NPV(lower r) = upper C_0 + sum_{lower t=1}^{upper T} upper C_lower t / (1 + lower r)^lower t

1.5.2 Internal Rate of Return (IRR)

The IRR is the discount rate lower r^* that sets the net present value (NPV) to zero. General cash flows: For initial outlay upper C_0 at t=0 and subsequent cash flows upper C_1, ..., upper C_{upper T}:

0 = NPV(lower r^*) = upper C_0 + sum_{lower t=1}^{upper T} upper C_lower t / (1 + lower r^*)^lower t

Investment convention. Often upper C_0 < 0 (cost today) and upper C_lower t > 0 for lower t >= 1 (future inflows). The IRR is typically found numerically. Excel has an IRR function.

1.5.3 Discounted Payback Period

The discounted payback period is the earliest time T such that the present value of expected future cash flows equals or surpasses the initial investment:

sum_{lower j=1}^{upper T} E[upper C upper F_{lower t + lower j}] / (1 + lower r)^lower j >= Initial Investment

This gives the earliest time you "break even" in discounted terms, but it ignores any cash flows occurring after time T.

Chapter 2: Week 1 - Asset Pricing Introduction towards CAPM

2.1 Returns

2.1.1 Return from prices

If price at time t is upper P_lower t and at t+1 is upper P_{lower t+1}:

upper R_{lower t+1} = (upper P_{lower t+1} - upper P_lower t) / upper P_lower t = (upper P_{lower t+1} / upper P_lower t) - 1

2.1.2 Total Returns of an asset

If price at time t is upper P_lower t, at t+1 is upper P_{lower t+1}, and cash-flow or dividend payments in t+1 is upper D_{lower t+1}:

upper R_{lower t+1} = (upper P_{lower t+1} + upper D_{lower t+1} - upper P_lower t) / upper P_lower t = (upper P_{lower t+1} + upper D_{lower t+1}) / upper P_lower t - 1

2.1.3 Dividend Discount Model (One-Period Pricing)

The price of a share at time t equals the present value of the expected dividend next period plus the expected resale price:

upper P_lower t = E[upper D_{lower t+1} + upper P_{lower t+1}] / (1 + lower r)

Where upper P_lower t is the share price at time t. upper D_{lower t+1} is the dividend paid at time t+1. E[] denotes expectations (i.e., all t+1 terms are forecasts made at time t). lower r is the required rate of return on the asset.

Rearranging the pricing equation gives the required gross return:

1 + lower r = E[upper D_{lower t+1} + upper P_{lower t+1}] / upper P_lower t

Reminder: lower r reflects both the time value of money and compensation for risk.

2.2 Portfolio Diversification

2.2.1 Portfolio Moments

This subsection is for general formulas, obviously in the previous chapter you have the 2-asset versions of returns and variance you are expected to be able to use in an exam.

Expected portfolio return. For a portfolio with weights lower w_lower i on assets i = 1, ..., N:

E[lower r_lower p] = sum_{lower i=1}^{upper N} lower w_lower i * E[lower r_lower i]

What portfolio variance depends on. Portfolio variance depends on: the variances of individual asset returns, sigma_lower i^2, and the correlations rho_{lower i lower j} between each pair of assets.

Portfolio variance:

sigma_lower p^2 = sum_{lower i=1}^{upper N} lower w_lower i^2 * sigma_lower i^2 + sum_{lower i=1}^{upper N} sum_{lower j=1}^{upper N} lower w_lower i * lower w_lower j * rho_{lower i lower j} * sigma_lower i * sigma_lower j

2.2.2 Risk Types

The total risk of an asset can be separated into two components. Risks that affect many/most/virtually all risky assets, that is Systematic Risk, and risks that affect only one/few risky assets, that is Idiosyncratic Risk.

Total risk = Systematic Risk + Idiosyncratic Risk.

2.2.3 Combining a Risk-Free Asset with a Risky Portfolio

This section explains the equations of the Capital Market Line (CML). Suppose you combine a risk-free asset with return lower r_lower f and a risky asset (or risky portfolio) with return lower r_upper X. Let lower w be the weight invested in the risky asset X, and 1 - lower w the weight invested in the risk-free asset.

Then the combined portfolio return is:

lower r_lower p = (1 - lower w) * lower r_lower f + lower w * lower r_upper X

Expected return:

E[lower r_lower p] = E[lower r_lower f] + lower w * (E[lower r_upper X] - E[lower r_lower f])

(If lower r_lower f is truly risk-free and known, then E[lower r_lower f] = lower r_lower f).

Standard deviation:

sigma(lower r_lower p) = lower w * sigma(lower r_upper X)

(Because variance of the risk-free asset is 0).

2.3 Sharpe Ratio and the Tangency Portfolio

The Sharpe ratio measures the portfolio's excess expected return per unit of risk (risk measured by standard deviation).

2.3.1 Sharpe ratio

For a portfolio p with expected return E[lower r_lower p], standard deviation sigma(lower r_lower p), and risk-free rate lower r_lower f:

Sharpe(lower p) = (E[lower r_lower p] - lower r_lower f) / sigma(lower r_lower p)

2.3.2 Tangency portfolio

The tangency portfolio is the risky portfolio that maximizes the Sharpe ratio (equivalently, it maximizes the slope of the Capital Allocation Line, it is the point of tangency between the efficient frontier and the line extending from the risk-free point on the y-axis of the return-variance space):

lower p^* = arg max_lower p (E[lower r_lower p] - lower r_lower f) / sigma(lower r_lower p)

Chapter 3: Week 2 - CAPM

3.1 Capital Asset Pricing Model (CAPM)

The CAPM links an asset's expected return to its exposure to market (systematic) risk, measured by Beta. It predicts that only market risk is rewarded in expected returns.

E[lower r_lower i] = lower r_lower f + beta_lower i * (E[lower r_upper M] - lower r_lower f)

where lower r_lower f is the risk-free rate, lower r_upper M is the market return, and E[lower r_upper M] - lower r_lower f is the market risk premium.

3.1.1 Beta and two ways to calculate it

Covariance-variance definition. Beta is the sensitivity of asset i to the market:

beta_lower i = Cov(lower r_lower i, lower r_upper M) / Var(lower r_upper M) = rho_{lower i upper M} * (sigma_lower i / sigma_upper M)

From CAPM, you can also calculate Beta as:

beta_lower i = (lower r_lower i - lower r_lower f) / (lower r_upper M - lower r_lower f)

3.2 Ordinary Least Square (OLS) Model Basics: For CAPM testing understanding

A simple linear regression relates a dependent variable y to an independent variable x via:

lower y = alpha + beta * lower x + epsilon

lower y: dependent variable (the data you want to explain).
lower x: independent (explanatory) variable (also called regressor, predictor, covariate).
epsilon: error term (unobservable), capturing all determinants of y other than x. Basically, noise.
alpha: intercept parameter; beta: slope parameter.

lower x and lower y are observable random variables; epsilon is not observable (it's assumed/allocated in the statistics). Goal: Estimate parameters alpha and beta that best describe the relationship between x and y.

3.2.1 Assumptions on the Error Term

Zero mean error. Under this condition E[epsilon] = 0. alpha = E[lower y] - beta * E[lower x].

Zero conditional mean (exogeneity). E[epsilon | lower x] = 0.

This implies the error term is uncorrelated with x: Cov(lower x, epsilon) = 0.

Intuitively, regardless of whether x is low or high, the average error is the same. If the linear model is a good approximation of the true relationship, this assumption is plausible; if not, the model may be mis-specified and estimation can be biased.

3.2.2 Ordinary Least Squares (OLS) Estimation

Given sample observations {(lower x_lower i, lower y_lower i)} from i=1 to n, OLS chooses alpha_hat and beta_hat to minimize the sum of squared residuals:

(alpha_hat, beta_hat) = arg min_{alpha, beta} sum_{lower i=1}^{lower n} (lower y_lower i - alpha - beta * lower x_lower i)^2

Yields Estimates:

beta_hat = Cov(lower x, lower y) / Var(lower x)

alpha_hat = lower y_bar - beta_hat * lower x_bar

The estimates alpha_hat and beta_hat depend on the sample. With small samples, or samples drawn from a narrow sub-population (e.g., a sector-specific ETF), estimates can be imprecise and may not generalize well.

3.2.3 Omitted Variable Bias

A major concern is omitted variables. Suppose there is a variable z that affects y and is correlated with x. If z is omitted from the regression, then typically E[epsilon | lower x] != 0, so beta_hat may capture not only the effect of x on y, but also variation in y due to z.

Controlling for z. Include z as an additional regressor to better isolate the relationship between x and y:

lower y = alpha + beta * lower x + gamma * lower z + lower u

Under appropriate conditions (notably E[lower u | lower x, lower z] = 0), beta_hat in the multiple regression provides a better estimate of the link between x and y.

3.3 Estimating and Testing CAPM: Time-Series vs Cross-Section

For the exam, understanding the logic and mechanisms being calculated and tested are important, not the regression formulae/notations. In empirical CAPM work, Beta is typically estimated using time-series data (variation over time), and then the CAPM prediction (that Beta is priced) is tested using cross-sectional (one point in time) variation in expected returns.

3.3.1 Step 1: Time-series regression to estimate beta_lower i (first pass)

For each asset i, estimate beta_lower i from a time-series regression of excess returns:

lower r_{lower i, lower t} - lower r_{lower f, lower t} = alpha_lower i + beta_lower i * (lower r_{upper M, lower t} - lower r_{lower f, lower t}) + epsilon_{lower i, lower t}

t = 1, ..., T. where beta_hat_lower i is the OLS slope estimate of the asset's market exposure.

3.3.2 Step 2: Cross-sectional test of whether Beta is priced (second pass)

The CAPM implies that higher-Beta assets should have higher expected excess returns, with the price of risk equal to the market risk premium.

Fama-MacBeth (two-pass) approach. Using the estimated beta_hat_lower i from Step 1, run cross-sectional regressions (typically each period t):

lower r_{lower i, lower t} - lower r_{lower f, lower t} = gamma_{0, lower t} + gamma_{1, lower t} * beta_hat_lower i + lower u_{lower i, lower t}, i=1,...,N

and then average the estimated slopes over time:

gamma_bar_1 = (1 / upper T) * sum_{lower t=1}^{upper T} gamma_{1, lower t}

Under the CAPM, gamma_0 should be close to zero.

Other cross-sectional tests. In Step 2, you can run cross-sectional regressions that include additional candidate predictors beyond the market beta estimated from the time-series step. Examples include firm size (market capitalization), profitability, valuation ratios, and other characteristics. Collect these additional explanatory variables in a vector upper X_lower i.

A general cross-sectional specification is:

lower r_lower i = lower a + lower b * beta_lower i + lower c * upper X_lower i + upper E_lower i

where upper X_lower i contains other potential predictors of returns for asset i, and c is a vector of slope coefficients. The CAPM implies the cross-sectional restrictions:

lower a = lower r_lower f, lower b = E[lower r_upper M] - lower r_lower f, lower c = 0

i.e., the intercept equals the risk-free rate (in levels), the slope on beta_lower i equals the market risk premium, and no additional variables in upper X_lower i should help predict expected returns.

3.4 Beta Properties in Portfolio

The beta of a portfolio is the weighted average of the betas of the assets in the portfolio:

beta_lower p = sum_{lower i=1}^{upper N} lower w_lower i * beta_lower i

where sum_{lower i=1}^{upper N} lower w_lower i = 1.

Two-asset example. With two assets A and B, weights w and 1-w, and betas beta_upper A and beta_upper B:

beta_lower p = lower w * beta_upper A + (1 - lower w) * beta_upper B

3.5 CAPM and risk adjustment: Return on equity vs return on firm (whole asset)

Discounted cash flow (DCF) pricing. A generic asset (or project) price at time t can be written as the present value of expected future cash flows:

upper P_lower t = sum_{lower j=1}^{infinity} E[upper C upper F_{lower t + lower j}] / (1 + lower r)^lower j

where E[upper C upper F_{lower t + lower j}] are expected cash flows (e.g., bond coupons, dividends, project cash flows) and r is the appropriate risk-adjusted discount rate for those cash flows.

Estimating a risk-adjusted required return via CAPM. For an equity claim (or a project with equity-like risk), estimate its beta and use CAPM to obtain a "fair" required return:

lower r_upper E = lower r_lower f + beta_upper E * (E[lower r_upper M] - lower r_lower f)

Intuition: a suitably chosen traded portfolio with similar systematic risk is an outside option, so the project (or equity) must offer a comparable expected return.

3.5.1 Equity valuation vs firm valuation

If valuing equity via dividends, discount expected dividends using the required return on equity, lower r_upper E:

upper P_lower t = sum_{lower j=1}^{infinity} E[upper D upper I upper V_{lower t + lower j}] / (1 + lower r_upper E)^lower j

Firm valuation. When valuing the firm (whole firm: debt + equity), the appropriate discount rate is generally the firm's overall required return lower r_{firm} not the required return on equity:

upper V_{firm} = sum_{lower t=1}^{infinity} E[upper C upper F_{lower t}^{firm}] / (1 + lower r_{firm})^lower t

generally lower r_{firm} != lower r_upper E. (Here upper C upper F_{lower t}^{firm} denotes cash flows to the firm, e.g. free cash flow to the firm, depending on the valuation setup).

3.5.2 Fixed leverage: WACC as the firm discount rate

If the firm's debt-to-equity ratio is fixed (i.e., leverage is maintained at a target), a common choice is the weighted average cost of capital (WACC):

lower r_{firm} = WACC = (upper E / upper V) * lower r_upper E + (upper D / upper V) * lower r_upper D

where upper V = upper D + upper E.

E is the market value of equity, D is the market value of debt, and lower r_upper D is the required return on debt.

We will seem other other formulas to estimate the firm required return based on other capital structure assumptions in future lectures.