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Advanced Finance (Fach) / The Capital Asses Pricing Moel (Lektion)

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  • Capital Allocation between risky Asset and the Risk-Free-Asset <3
  • Assumptions of CAPM Market in equilibrium Competitive market Single-Period horizon All assets can be traded No Frictions Investors are Rational Information is symmetric Investers are mean-variance optimizers Perfectly divisible financial claims 
  • Utility l Why is the Utility function always convex? -u''(W)/u'(W) = k > 0 u'(W) = How marginal unit changes + u''(W) <0 = concave - l risk averse >0 convex l 
  • CAPM by Sharpe CAPM is more general model, developed by Sharpe  Consider a two asset portfolio one asset is market portfolio M, weight (1 − x) other asset is individual stock i, weight x   E(ri)=rf +(E(rm)−rf)βi 
  • Security Market Line E(ri)=rf +(E(rm)−rf)βi  SML only prices the systematic risk Is therefore valid for all investment objects.  Pricing relation for entire investment universe including inefficient portfolios Pricing relation for entire investment universe Including individual assets  clear price of risk: E(rm) − rf clear measure of risk: β  well-diversified investors value assets according to their contribution to portfolio risk  if asset i increases portf. risk E (ri ) > E (rp ) if asset i decreases portf. risk E (ri ) < E (rp )  expected risk premium proportional to β 
  • Security Market Line E(ri)=rf +(E(rm)−rf)βi  SML only prices the systematic risk Is therefore valid for all investment objects.  Pricing relation for entire investment universe including inefficient portfolios Pricing relation for entire investment universe Including individual assets  clear price of risk: E(rm) − rf clear measure of risk: β  well-diversified investors value assets according to their contribution to portfolio risk  if asset i increases portf. risk E (ri ) > E (rp ) if asset i decreases portf. risk E (ri ) < E (rp )  expected risk premium proportional to β 
  • The Capital Market Line CML prices all risks  Only valid when all risk is systematic risks = for efficient portfolios -> otherwise, CML uses ’wrong’ risk measure  efficient portfolios only differ in proportion M in it so all efficient portfolios are perfectly positively correlated: ρM,(1−x)M = 1  ---> if ρp,m =1⇒σpρp,m =σp and CML=SML ---> difference is correlation term, that is ignored in CML  E(rp)=rf +E(rm)−rf / σM * σp 
  • Uncertainty can be accounted for in 3 different ways: 1  Adjust discount rate to risk adjusted discount rate 2  Adjust cash flows to certainty equivalent cash flows 3  Adjust probabilities (expectations operator) from normal to risk neutral or equivalent martingale probabilities 
  • Sharpe Measure CML and SML relate expected return to risk  return-to-variability ratio = Risk premium peru nit of Risk  can be reformulated as ex post performance measures relate realized returns to observed risk  Sharpe uses slope of CML for this: SRp = rp − rf /σ􏰅p 
  • Sharpe ratios widely used to rank portfolios, funds or managers identify poorly diversified portfolios (too high σ􏰅p) identify funds that charged too high fees (rp too low) 
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  • Sharpe ratio can be adapted: measure the risk premium over other benchmark than rf - also known as the information ratio measure risk as semi-deviation (downward risk) - known as Sortino ratio 
  • Assumptions CAPM Financial markets are perfect and competitive: no taxes or transaction costs, all assets are marketable and perfectly divisible, no limitations on short selling and risk free borrowing and lending large numbers of buyers and sellers, none large enough to individually influence prices, all information simultaneously and costlessly available to all investors Investors maximize expected utility of their end wealth by choosing investments based on their mean-variance characteristics over a single holding period have homogeneous expectations w.r.t. returns (i.e. they observe same efficient frontier) 
  • important consequences if the following assumptions don't hold different borrowing and lending rates invalidate same risk-return trade-off for all (see picture) if investors see different frontiers, effect comparable to restriction, e.g. ethical and unethical investments (see picture) 
  • APT Summary and Formula Multi-factor model = more factors for which we have rsk than in the CAPM Ross  E(r) = ϒ0 + ϒ1b1 + ϒ2b2  ϒ0 = risk free rate ϒ1 = Risk Premium 1  b1 = sensitivity  Summarizing, Arbitrage Pricing Theory: Rests on different assumptions than CAPM Is more general than CAPM makes less restrictive assumptions allows more factors, more realistic Is less precise than CAPM does not give a volume of risk (what or even how many factors to use) does not give a price of risk (no expression for factor risk premia, have to be estimated empirically) has interesting applications in risk management, default prediction, etc. 
  • CAPM on estimation results 􏰅 rpt − rft = α􏰅 + Bp ( rmt − rft ) + ε􏰅pt  versus E(rp)=rf + ßi ( E(ri)- rf) α = Should be 0 because investors are only compentated for taking additional risk  ß = not sepcified but should be correlatied with market portfolio 􏰅rpt − rft = Excess return on day t  ε􏰅pt = Disturbance term = idiosyneratic risk = risk per unit 
  • Excess Return The difference in any particular period between the actual rate of return on a risky asset and the actual risk-free rate
  • APR: Annualizing using simple interest, Annual Percentage Rate APR = (1+EAR)T -1 / T)
  • Effective Annual Rate (EAR) EAR: Percentage increase in funds invested over a 1-year horizon  1+EAR = (1+r(T)) 1/T
  • Rates of return: Single period HPR = (P1-P0 + D) / P0
  • Arbitrage Pricing Theory Arbitrage pricing theory builds on such a multi-factor return generating process Distinguishes between the expected part of stock returns and unexpected part Unexpected part (risk) consists of systematic (or market) risk and unsystematic (or idiosyncratic) risk Market risk not expressed as covar with market but as sensitivity to (any) number of risk factors  APT’s equilibrium condition is: the absence of arbitrage opportunities Means if you make a well diversified portfolio (􏰃i xi εi = 0): 
  • APT and the thee no-arbitrage conditions 1  􏰃∑xi =0means: that requires no net investment -> sum portfolio weights is zero: 􏰃i xi = 0 2  􏰃∑xibik = 0 means: that involves no risks -> weighted sum of allbik is zero: 􏰃i xibik =0forallk 3  􏰃∑xiE(ri) = 0 means:then the expected return must be zero: 􏰃i xiE(ri)=0   Ususal form of APT  E(ri)=rf + 􏰃∑k=1􏰄bik (E(Fk)−rf)
  • Three important economic insights of the MVE Diversification pays: better (μp , σp )-combinations possible. All assets most likely ine¢ cient on their own. Once a portfolio is fully diversiÖed, higher expected returns can only be achieved by taking on higher variance (risk). The mean-variance frontier shifts outward when additional assets are added to the investment space. Covariance across assets within portfolio is key, not standard deviation of asset in isolation!! 

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