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Finance Notes Principles of Finance Notes

Fm213 Formula Sheet Notes

Updated Fm213 Formula Sheet Notes

Principles of Finance Notes

Principles of Finance

Approximately 16 pages

This summarises the complete set of formulae you will need for the FM213 course....

The following is a more accessible plain text extract of the PDF sample above, taken from our Principles of Finance Notes. Due to the challenges of extracting text from PDFs, it will have odd formatting:

FM213 Formula Sheet Statistics πΆπ‘œπ‘£(𝑋, π‘Œ) = 𝜎!,# = 𝜌!,# 𝜎! 𝜎# General calculation notes β€’ In order to construct a zero-risk, zero-cost, positive income in perpetuity portfolio the investor needs to make sure that his total cash flow today is zero and the future cash flows are risk free. β€’ β€’ Risk-free debt: Ξ²$ = 0, r$ = r% Unlevered (all equity) firm: Ξ²& = Ξ²' , r& = r' r = r% + Ξ²(r( βˆ’ r% ) CAPM r& = r% + Ξ²& (r( βˆ’ r%) r' = r% + Ξ²' (r( βˆ’ r% ) Gordon's growth model P) = DIV* rβˆ’g Present value calculations PV formula Perpetuity W/o growth 𝑃𝑉+ = 8,.* Annuity 𝐢 𝐢 = , (1 + π‘Ÿ) π‘Ÿ 𝑃𝑉/ = 𝑃𝑉* βˆ’ 𝑃𝑉0 = = w/ growth𝑃𝑉+ = 8 ,.* 𝐢 𝐢(1 + 𝑔),2* = (1 + π‘Ÿ), π‘Ÿβˆ’π‘” 𝐢 1 <1 βˆ’ = (1 + π‘Ÿ)1 π‘Ÿ 𝑃𝑉/ = Requires r > g, otherwise CF grows more quickly than discount factor will = 𝐢 𝐢 1 βˆ’ 1 π‘Ÿ (1 + π‘Ÿ) π‘Ÿ 1+𝑔 1 𝐢 𝐢 βˆ’? @ 1+π‘Ÿ π‘Ÿβˆ’π‘” π‘Ÿβˆ’π‘” 1+𝑔 1 𝐢 @ B A1 βˆ’ ? 1+π‘Ÿ π‘Ÿβˆ’π‘” cause the sum to be infinite Converting rates with different payment frequencies Continuous discounting Real interest rates Approximate real interest rates (1 + π‘Ÿ3 )*0 = C1 + π‘Ÿ4 D + (1 + π‘Ÿ6 )0 = 1 + π‘Ÿ7 5 𝑃𝑉 = 𝑋𝑒 281 (1 + 𝑖) = 𝑖 β‰ˆπ‘Ÿβˆ’πœ‹ 1+π‘Ÿ 1+Ο€ Stocks and bonds Bond price and YTM P= F cF cF cF cF + + +. . + =8 (1 + y)9 (1 + y): (1 + y)9 1 + y (1 + y)0 9 :.* cF 1 F = ?1 βˆ’ @+ (1 + y)9 (1 + y)9 y Semi-annual coupon bond Expected return on a share P= r= cF cF cF +F 2 2 2 + +. . + 0 y y 09 O1 + P O1 + yP O1 + 2P 2 2 E: [D:;* + P:;* βˆ’ P: ] E: (D:;* ) E: (P:;* βˆ’ P: ) = + P: P: P: Expected return = CF at t+1/current price = expected div. yield + expected capital gain (expected %+ in the share price) Share price, assuming perpetualP: = 8 <.* E: (D:;< ) (1 + rT)< stream of dividends Share price, assuming perpetualP: = 8 <.* D D = (1 + rT)< rT stream of constant dividends Gordon growth formulaP: = 8 <.* P) = Return on Equity Earning's growth D(1 + g)<2* D:;* = (1 + rT)< rT βˆ’ g (1 βˆ’ plowback ratio) Γ— EPS* rT βˆ’ ROE Γ— plowback ratio ROE* = EPS* BV) (Equity) per share g = ROE Γ— Plowback ratio If ROE and plowback ratio are constant, then earning's growth = dividend's growth Dividend's growth Present Value of 𝑔 = π‘Ÿ= βˆ’ 𝑑𝑖𝑣. 𝑦𝑖𝑒𝑙𝑑 = π‘Ÿ= βˆ’ β€’ Growth Opportunities (PVGO) β€’ β€’ β€’ P= '>? @ + PVGO PVGO: = 𝑑𝑖𝑣 𝐸𝑃𝑆 (*2BCDEFGHI @G:?!"# @M Without growth, the price will equal the earnings for next period capitalised at r '>? > = r(1 βˆ’ >PQO > ) PVGO - while is the stock price higher when a company plows back its earnings. The PVGO comes from the fact that the firm is retaining earnings that are generating a return of 20% (the ROE) while the discount rate is only 10%. Thus the value of the firm rises. Valuing government bonds Macaulay duration (negative elasticity) Shortcoming of duration: use of a linear 1 C< D = 8i (1 + y)< P I <.* approximation for a non-linear convex relationship between bond prices and yields DR = Modified duration (volatility) D 1+y %Ξ”P = Spot rate and forward interest rate Ξ”P β‰ˆ βˆ’DR Ξ”y P (1 + r0 )0 = (1 + r* )(1 + f0 ) (1 + rS )S = (1 + rS2* )S2* (1 + fS ) = (1 + r*)(1 + f0 ) … (1 + fS ) f0 = (1 + r0 )0 βˆ’1 (1 + r* ) 𝑓T = Inferring one- and two-year forward rates with CF for 2 bonds with different CF (1 + π‘ŸT )T βˆ’1 (1 + π‘ŸT2* )T2* P = C* d* + C0 d0 , d* = 1 1 , d0 = (1 + r* )(1 + f* ) 1 + r* streams Interpretation of forward rates: 1. no-arbitrage one-year rate agreed today for a loan to start in one year's time 2. Pure Expectation Hypothesis (the (constant) premium is zero, T = 0, for all time-periods and all bonds, regardless of time-to-maturity): forward rate is the market forecast of the 1-year rate in one year's time Risk, return and cost of capital variance S v )0 Οƒ = 8 P< (R < βˆ’ R 0 <.* Sample variance Expected return of portfolio Portfolio variance s0 = 1 v)0 8(R < βˆ’ R Kβˆ’1 U <.* S E(R > ) = x* Β΅* + x0 Β΅0 + . . . + xS Β΅S = 8 x< Β΅< Two asset p/f: E(R > ) = x* Β΅* + x0 Β΅0 S S <.* S Var(R > ) = 8 8 x< xV ρ<,V Οƒ< ΟƒV = 8 x<0 Οƒ0< + 8 x< xV ρ<,V Οƒ< ΟƒV <.* V.* <.* ) = x*0 Οƒ*0 + x00 Οƒ00 + 2x* x0 ρ*,0 Οƒ* Οƒ0 = x*0 Οƒ*0 + x00 Οƒ00 + 2x* x0 cov(x* , x0 ) Covariance π‘π‘œπ‘£(𝑋, π‘Œ) = Beta Ξ²< = βˆ‘(𝑋, βˆ’ 𝑋~)(π‘Œ, βˆ’ π‘Œ~) π‘βˆ’1 Cov(R < , R > ) 𝜌,,+ Οƒ, Οƒ+ Οƒ< = = ρ<,> Οƒ> Var(R > ) Οƒ0+ Beta of a Beta of the portfolio is the weighted average of betas of individual stocks portfolio Ξ²> = 8 x< Ξ²< S <.* Portfolio variance Οƒ0, = Ξ²0, Οƒ0+ + Οƒ0X, 0 = 𝜌,Y Οƒ0, 0 Οƒ + ΟƒX, Οƒ0+ + = Cρ,Y Οƒ, D + Οƒ0X, , 0 < ρ < 1 0 variance = Systematic/aggregate/market-wide risk + idiosyncratic risk/firm-specific risk Difference between variance and beta as a measure of risk Variance Beta β€’ measures the total risk (both unique risk and market β€’ a measure of market risk - measures the risk) of a security sensitivity of the security returns to changes in market returns β€’ a measure of stand-alone risk β€’ In a well-diversified portfolio, unique risks tend to β€’ useful in the context of a well-diversified cancel each other out and only the market risk remains. portfolio. β€’ Market portfolio has a beta of one. Portfolio theory and CAPM Feasible set solution equation π‘šπ‘–π‘› Οƒ0+ 𝑠. 𝑑. βˆ‘π‘₯, = 1, 𝐸(𝑅+ ) = 𝐸 Z# ,Z$ ,...,Z% CML 𝑅+ = 𝑅\ + 𝐸(𝑅3 ) βˆ’ 𝑅\ Γ— Οƒ+ = 𝑅\ + 𝑆𝑅3 Γ— Οƒ+ 𝜎3 𝑆𝑅3 = sharpe ratio of the market CAPM ECR V D = R % + Ξ²V (E(R R ) βˆ’ R % ) If a firm has zero systematic risk, 𝑅] = 𝑅\ 𝛽] = 𝑃) = πΆπ‘œπ‘£C𝑅] , 𝑅^ D π‘‰π‘Žπ‘Ÿ(𝑅^ ) 𝐷* 𝐷* = π‘Ÿ βˆ’ 𝑔 (𝑅\ + βŠ𝐸(𝑅^ ) βˆ’ 𝑅\ D βˆ’ 𝑔 Excess return 𝐸C𝑅] D βˆ’ 𝑅\ = 𝛼] + 𝛽] C𝐸(𝑅^ ) βˆ’ 𝑅\ D APT 𝐸C𝑅] D = 𝑅\ + 𝛽*,] C𝑅_* βˆ’ 𝑅\ D + 𝛽0,] (𝑅_0 βˆ’ 𝑅\ ) Farma-French 3 factor model Sharpe ratio 𝐸(Ξ±) = 0 E(R V ) βˆ’ R % = Ξ²R,V (E(R R ) βˆ’ R % ) + Ξ²?,V (E(R ?R` )) + Ξ²P,V (E(R aR` )) 𝑠hπ‘Žπ‘Ÿπ‘π‘’ π‘Ÿπ‘Žπ‘‘π‘–π‘œ = 𝑒π‘₯𝑐𝑒𝑠𝑠 π‘Ÿπ‘’π‘‘π‘’π‘Ÿπ‘› π‘£π‘œπ‘™π‘Žπ‘‘π‘–π‘™π‘–π‘‘π‘¦ Sharpe ratio = risk/return trade-off Under the CAPM, we would expect the market portfolio to dominate as it delivers the best possible Sharpe Ratio. β€’ in a CAPM world: the market portfolio is an efficient portfolio β€’ A combination of the market portfolio and the risk-free asset (sometimes called a 'leveraged position' in the market) has o highest expected return of any portfolio for a given volatility o lowest volatility for a given expected return. Impacts of risk preferences on portfolio selection β€’ The portfolio with a higher Sharpe ratio is preferred by someone who is willing to take a little risk in order to earn more return β€’ The variance minimising portfolio is appropriate for someone who cares only about experiencing the smallest possible risk. β€’ The MVP investor invests most of her cash in the safe asset, while the EWP(equally weighted portfolio) investor invests more of the cash in the riskier asset as it has greater mean return β€’ As soon as the investor starts to worry about risk, then diversification plays a role. Forwards and futures Long payoff S9 βˆ’ F Short payoff F βˆ’ S9 Making to market β€’ Gain = margin balance - initial margin β€’ Loss = total amount in margin calls β€’ In total, the investor pays quantity * market price of the asset + loss in margin calls - gain Covered interest rate parity 𝐹\ = b 𝑆C1 + π‘Ÿ\_c8d,ef D 1 C1 + π‘Ÿ\gc3d6h,i D 1

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