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ACTS 4302/ACTS 4309 IFM FORMULA SUMMARY IFM Lesson 1: Introduction to Derivatives

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ACTS 4302/ACTS 4309 IFM FORMULA SUMMARY IFM Lesson 1: Introduction to Derivatives ACTS 4302/ACTS 4309 IFM FORMULA SUMMARY IFM Lesson 1: Introduction to derivatives Measures of market size and activity: 1. Trading volume; 2. Market value; 3. Notional value; 4. Open interest. Derivatives serve the following purposes: 1. Risk management; 2. Speculation; 3. Reduced transaction costs; 4. Regulatory arbitrage. 2 IFM Lesson 2: Project Analysis 1. The Net Present Value (NPV) of a project is: 1 X FCFn NPV = ; (1 + r)n n=0 where FCFn is the free cash flow at time n, r is the cost of capital. 2. If FCC's are k at time 1 and grow at constant rate g < r, then k NPV = r − g 3. Sample mean: n 1 X R¯ = R n i i=0 4. Sample variance, biased estimate: n 1 X 2 σ^2 = R − R¯ B n i i=0 5. Sample variance, unbiased estimate: n 1 X 2 σ^2 = R − R¯ UB n − 1 i i=0 6. Sample confidence interval using the unbiased estimate for the sample variance: 1 1 R¯ − p · σ^ ; R¯ + p · σ^ n UB n UB 7. Downside semi-variance: 2 h 2i σSV = E min 0; (R − µ) 8. Sample downside semi-variance: n 1 X 2 σ^2 = min 0; R − R¯ SV n i i=0 9. Value-at-risk of a random variable X at security level α: −1 V aRα(X) = FX (α) 10. Tail value-at-risk of a random variable X at security level α for downside risk: 1 Z V aRα(X) T V aRα(X) = E [XjX < V aRα(X)] = xf(x) dx α −∞ 11. Tail value-at-risk of a random variable X at security level α for downside risk: 1 Z 1 T V aRα(X) = E [XjX > V aRα(X)] = xf(x) dx 1 − α V aRα(X) 3 IFM Lesson 3: Monte Carlo Simulation Monte Carlo methods generate pseudorandom numbers from a distribution. 1. For a continuous distribution, if F (x) is a cumulative distribution function and u1; u2; : : : ; un are −1 randomly generated uniform numbers between 0 and 1, then xi = F (ui); i = 1; 2; : : : ; n will be pseudorandom numbers generated from the distribution F . 2. For a discrete distribution, if p0; p1; : : : ; pn;::: are probabilities P r(N = k) = pk, and u1; u2; : : : ; un are randomly generated uniform numbers between 0 and 1, then for each u, the corresponding k is generated based on the following table: u k 0 ≤ u < p0 0 p0 ≤ u < p0 + p1 1 ::: ::: Pk−1 Pk i=0 pi ≤ u < i=0 pi k ::: ::: 4 IFM Lesson 4: Efficient Markets Hypothesis (EMH) Based on EMH, stock price takes into account: 1. Weak form of market efficiency states that stock price takes into account past stock prices. 2. Semi-strong form of market efficiency states that stock price takes into account past stock prices and public/easily accessible information. 3. Strong form of market efficiency states that stock price takes into account past stock prices, public/easily accessible information and private/hard-to-get information. Evidence for EMH: 1. Evidence for weak form of market efficiency reveals itself in stock prices following random walk. 2. Evidence for semi-strong form of market efficiency reveals itself in takeover announcements being immediately reflected in stock price. 3. Evidence for strong form of market efficiency reveals itself in the majority of the fund managers not being able to beat the market. 5 IFM Lesson 5: Mean-Variance Portfolio Theory 1. Variance on return on portfolio is: n n X X V ar(R) = xixjCov(Ri;Rj) i=1 j=1 2. Variance on return on equally-weighted portfolio is: 1 n − 1 V ar(R) = AveV ar + AveCov n n 3. Volatility of return on portfolio in terms of its components: n X σP = xi · ρP;i · σi i=1 4. Sharpe Ratio: α − r φ = f σ 6 IFM Lesson 6: Capital Asset Pricing Model (CAPM) 1. Beta of investment i with respect to portfolio P : P Corr(RP ;Ri)SD(Ri) Cov(RP ;Ri) βi = = SD(RP ) V ar(RP ) 2. Required return for investment i with respect to portfolio P : P E [Ri] − rf > φP · SD(Ri) · Corr(RP ;Ri) = βi (E [RP ] − rf ) 3. Beta: Corr(RMkt;Ri)SD(Ri) Cov(RMkt;Ri) βi = = SD(RMkt) V ar(RMkt) 4. Beta of portfolio as a function of betas of its assets: n X βP = xi · βi i=1 5. Capital Market Line is the line relating expected return of efficient portfolio to its volatility. 6. Security Market Line is the line relating expected return of investment to beta. 7 IFM Lesson 7: Cost of Capital 1. Fundamental approach to calculating market return: if P0 is the value of the market, then Div1 RMkt = + g P0 2. Debt cost of capital: the relationship between bond's yield and its expected return, rd = y − pL; where y is the annual effective yield to maturity on the bond, p is the annual probability of default, and L is the proportion of debt that is lost when a default occurs. 3. The unlevered or the asset cost of capital is calculated the weighted average of the equity and debt costs of capital: E D r = · r + · r U E + D E E + D D 4. In the presence of cash C, the unlevered or the asset cost of capital is calculated the weighted average of the equity and debt costs of capital and the risk-free rate: E D C r = · r + · r − · r U E + D − C E E + D − C D E + D − C f 5. The unlevered beta is the weighted average of the equity and debt betas: E D β = · β + · β U E + D E E + D D 8 IFM Lesson 8: Behavioral Finance and Multifactor Models 1. Arbitrage Pricing Theory (APT): X Fn E[Rs] − rf = n = 1Nβs (E[RFn ] − rf ) 2. APT with self-financing portfolios: X Fn E[Rs] = n = 1Nβs E[RFn ]; Fn where βs is the beta of the stock with respect to the portfolio Fn: Fn Corr(RFn ;Rs)SD(Rs) Cov(RFn ;Rs) βs = = SD(RFn ) V ar(RFn ) 3. Fama-French-Carhart (FFC) factor specification: Mkt SMB HML PR1YR E[Rs] = rf + βs (E[RMkt] − rf ) + βs E[RSMB] + βs E[RHML] + βs E[RPR1YR] 9 IFM Lesson 9: Capital Structure 1. Definition of perfect market: (1) Competitive prices are available to all. (2) Transactions are efficient. (3) Capital structure provides no information. 2. Modigliani and Miller Propositions: I. Capital structure does not affect firm value. II. Cost of equity capital rises with leverage: D r = r + (r − r ) E U E U D 3. Equity beta: D β = β + (β − β ) E U E U D 10 IFM Lesson 10: The Effect of Taxes on Capital Structure 1. V L = V U + PV (Interest Tax Shield) 2. Present value of interest tax shield for permanent debt: Dτc, where τc is the company's tax rate. 3. Pre-tax weighted average cost of capital: E D r = · r + · r pre−tax W ACC E + D E E + D D 4. After-tax weighted average cost of capital or weighted average cost of capital (WACC): E D D r = · r + · r (1 − τ ) = r − · r · τ W ACC E + D E E + D D c pre−tax W ACC E + D D c 5. If debt-equity ratio is constant, the interest tax shield is V L − V U , where V L is computed at the WACC and V U is computed at the pre-tax WACC. 11 IFM Lesson 11: Other Factors Affecting Optimal Debt-Equity Ratio 1. Indirect costs of bankruptcy: 1) Loss of customers 2) Loss of suppliers 3) Loss of employees 4) Loss of receivables 5) Fire sale of assets 6) Insufficient liquidation 7) Costs to creditors 2. Trade-off theory: V L = V U +PV (Interest Tax Shield)−PV (F inancialDistressCosts)+PV (Agencybenefits)−PV (Agencycosts) 3. Asset substitution problem: companies in distress substitute risky assets for non-risky assets. 4. Debt overhang: companies do not make positive-NPV investments because only creditors will benefit. 5. Approximate required NPV for equity holders to benefit: NPV β D > D · ; where I βE E D is debt, E is equity, I is the amount invested, βD and βE are the betas for debt and equity. 6. Leverage ratchet effect: presence of debt leads to issuing more debt. 7. Agency benefits: 1) Control of company in fewer hands. 2) Management has greater share of equity, discouraging waste. 3) No empire building. 4) Management more likely to be fired in financial distress. 5) Financial distress may lead to wage concessions. 6) More incentive to compete. 8. Credibility principle: actions speak louder than words, when the words are in self-interest. 9. Adverse selection: sellers with private information sell the least desirable items. 10. Lemons principle: buyers discount price when seller has private information. 11. Pecking order hypothesis: management prefers to finance first with retained earnings, then with debt, and only finally with equity. 12 IFM Lesson 12: Equity Financing I. Sources of Equity for Private Companies: 1. All companies start out as private. 2. Founders are individuals who invest their own money to start a company. 3. Angel investors are individuals who supplement the funds of the founders. 4. Convertible notes are the certificates that angel investors or angel groups get instead of a stock since at the beginning it is difficult to estimate the value of the company. 5. Venture capital firms are the companies that help companies raise equity when they start to expand.
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