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Advanced Finance (Fach) / Calculations (Lektion)

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  • gesucht: maximum fee that you could charge and investor to invest in your portfolio instead of the passive portfolio y>1  CML = rIndex-RF / σIndex  CAL = rp-RF-fee/σP y<1  CML = rIndex- RB / σIndex   CAL = rp-RB-fee /σP
  • gesucht: Expected return = 11% on the optimal CAL Optimal CAL = 11% = E(rC) = RF + E(rp)+RF/σP * σC
  • gesucht: Over- oder Underpriced Stock ∂ = Actual Return - Fair Return >0 => Underpriced Actual Return = (ST+ Dividend / S0) -1)
  • Better Stock Selector Alphas suchen! αi = E(ri) - (RF + (rM - RF ) • ßi) 
  • Price of Stock at the end of the year ri = RF + (RF-rm) • ßi => discount rate Price = S0 • (1+ri ) - D 
  • Optimal weights for minimum-variance portfolio and no RF w*1 = σ2B - CovS,B / σ2B + σ2S - 2*covS,B After: 1. Calculate E(ri) 2. Calculate Variance, Standard Deviation 3. Calculate Sharpe Ratios
  • Solve numerically the minimum variance weights with 2 risky stocks and a risk-free asset w*1 = (RS-RF)σ2B - (RB-RF)CovS,B / (RS-RF)σ2B +(RB-RF)σ2S -(RS-RF)(RB-RF) 2*covS,B
  • Gesucht y* yr* = rc - rF / rp-rF yσ* = σc/σp yA* = rp - rf / A • σ2
  • Correlation -1 and value of RF σP= l w1σ1 - w2σ2 l 1. w1 and w2 => weights for risk-free asset 2. Calculate E(ri) = expected rate of return from risk-free portfolio => risk-free rate
  • Perfect weights for a required standard deviation w1 = σC / σP 2. E(ri) = w1*rf + w2*rp
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  • Price of a Stock Price = Dividend / Discount Rate 
  • Calculate Net Present Value of a given Cash-Flow NPV + Maximum Beta ri = RF + (rm-RF)*ß NPV= -100 + ∑ (CF)/ (1+ri)T
  • No arbitrage ß ß1 = E(r1)+RF / E(r2) + RF
  • Maximum level of risk aversion to which one would prefer risky assets to T-Bills => w1 > w2 <=> w1 > 0.5 w1 = rM - RF / A • σ2 > 0.5
  • Does the CAPM hold or not? Given: E(ri) and ßi → Direct Comparison E(ri), σ, ri → SR berechnen E(ri), ßi, ri → SML E(ri), σi → ?? We don't know as CAPM only computed for systematic risk
  • Calculate the Correlation Coefficient cov = π1 • (r1- E(r1))(r2 - E(r2)) + π2 • (r1- E(r1))(r2 - E(r2))... pA,B = CovA,B / σAσB
  • For which level of risk Aversion will ne investor neither borrow nor lend (y=1) y=1 rStock-rBorrow / σ2 < A < rStock-RF / σ2 y = 1 for _ < A < _
  • gesucht: revised estimate revised estimate = old estimate + sum of products of unexpected change 1. find change Δ = unexpected change - usual return 2. revised estimate = current return + (ßi • Δ1 + ß2 • Δ2)
  • Expected retun-beta relationship ind a multi-factor economy LGS E(rA) = RF + ß1 • RP1 + ß2 • RP2 => ß1 + ß2 + RF-E(rA) E(rB) = RF + ß1 • RP1 + ß2 • RP2 => ß1 + ß2 + RF-E(rB) E(rp) = RF + ßp1 • (E(r1) - RF) + ßp2  • (E(r2) - RF)
  • What is the risk-free rate in a well diversified economy? One-factor model LGS 1 + ßA - E(rA) 1 + ßB - E(rB) =>   RF = erstes Ergebnis
  • Zero net investment: analyst profits when he invest 1 Mio and buys 1 Mio to eliminate market exporue 1 Mio • ( ri + (ßi • RM) - 1Mio • ( -ri + ( ßi •RM) => Zero-Net investment → Returni = 1 Mio  • 2ri σ = number of stocks 〈(  2 • 1Mio / Number of stocks  • σ)2 〉
  • APT: Calculate the variance of return of securities A,B,C σ2 = ß2 • σ2M + σindividual2 When inifinite number of securities => same mean, variance = σ2 = ß2 • σ2M 
  • standard deviation of idiosycratic risk √σ2/ number of stocks = σ2 (ei)
  • Annuity formulas A = PV • r/ 1- (1/1+r)n  A = FV • r/ (1+r)n-1
  • Compound semiannually V * (1+r/m)m•t
  • Compound continuously V • er•t
  • relative contribution ∑w1 • covi,p / σ2p = wi*ß

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