FINS2624 Lecture Notes - Lecture 5: Sharpe Ratio, Capital Asset Pricing Model, Efficient Frontier

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16 May 2018
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Lecture 7: SIM and factor models
o According to CAPM, everyone is a passive market investor- no channel for
info de real economy to enter capital market- therefore prices may deviate
enough from their fundamental values for an analyst to achieve enough
excess returns
- Single index model (SIM/empirical model)
o Jeses alpha: h regress excess returns
against market excess returns
o according to CAPM, alpha(i)=0, if not, asset i is mispriced and alpha is a
measure of mispricing
- deviate from market weights to exploit alpha- select P* ≠ M icrease Er w/out
increasing volatility) highest Sharpe ratio (M where all assets are correctly priced)
- combine an active position in the mispriced asset w(A) w a passive position (1-w(A)):
- but incur some unsystematic risk and hence unpriced risk (residual risk) buying
the benefit of mispricing at the cost of taking on unsystematic risk at price
alpha(A)/sd2 buying excess return by taking on risk when selecting optimal risky
portfolio
- optimal weight:
- positive active weight (buying mispriced A) when alpha >0 and sell when alpha <0
- the bigger the magnitude of w(0), the more extra return per (residual) risk unit we
get from active asset (relative to return per risk unit we get in M- so increase weight
in active asset)
- ispriced asset allows us to costruct a ew efficiet frotier steeper
- asset information ratio (IR/appraisal ratio) = alpha/SD(epsilon)
o sharpe ratio(P)^2 = sharpe ratio(M)^2 + IR^2
- multiple mispriced: form an active portfolio (AP) form optimal risky portfolio of
market and active portfolio
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o each mispriced asset is given AP weight proportional to alpha to
unsystematic risk ratio (alpha/SD(E)^2)
o IR(AP)^2 = sum((IR of each asset)^2)
o Sharpe ratio(P)^2 = S(M)^2 + IR(AP)^2 = S(M)^2 + sum(IR^2)
o CAPM assumptions (rational utility optimisation, trade all A)
i. Theoretical prediction of CAPM may be wrong- A E(r) may nnot be det
by only its ß (det by cov(ir,mr)
Factor models
2. M- risk factor
asset expected return:
(variation of asset return from
CAPM)
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Document Summary

Single index model (sim/empirical model: je(cid:374)se(cid:374)(cid:859)s alpha: h against market excess returns regress excess returns, according to capm, alpha(i)=0, if not, asset i is mispriced and alpha is a measure of mispricing. (cid:373)ispriced asset allows us to co(cid:374)struct a (cid:858)(cid:374)ew efficie(cid:374)t fro(cid:374)tier(cid:859) (cid:894)steeper(cid:895) Asset information ratio (ir/appraisal ratio) = alpha/sd(epsilon: sharpe ratio(p)^2 = sharpe ratio(m)^2 + ir^2. Multiple mispriced: form an active portfolio (ap) form optimal risky portfolio of market and active portfolio: each mispriced asset is given ap weight proportional to alpha to unsystematic risk ratio (alpha/sd(e)^2) Factor models: m- risk factor, - loading on that factor asset expected return: Difference between 2 sharpe ratios = difference in slope of respective cals/prices of risk. Will inflate when risk of m increases: positive sq(m) will be larger and negative sq(m) will be smaller (more ve, quod need to scale the evaluated portfolio p by sd(m)/sd(p) when calculating sq(m)

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