CAPM Flashcards
Return of an asset (probabilities)
r_a = p1r1 + p2r2 + p3*r3
Variance of an asset (probabilities)
Var_a = p1(r_a1 - r_a)^2 + p2(r_a2 - r_a)^2 + p3*(r_a3 - r_a)^2
Covariance (probabilities)
cov = p1(r_a1 - ra)(r_b1 - r_b) + p2(r_a2 - ra)(r_b2 - r_b) + …
Return of a portfolio
r_p = xa * ra + xb * rb
Variance of portfolio (basic)
Var_p = xa^2 * Var_a^2 + xb^2 * Var_b^2 + 2xaxb*cov_ab
Variance of portfolio (3 assets)
Var_p = xa^2 * Var_a^2 + xb^2 * Var_b^2 + xc^2 * Var_c^2 + 2xaxbcov_ab + 2xaxccov_ac + 2xbxc*cov_bc
Variance of portfolio (Cov_ap)
Var_p = xacov_ap + xbcov_bp
Variance of portfolio (with beta)
Var_p = (beta*sd_m)^2 + Var(e)^2
Covariance ab
Cov_ab = Cor_ab * sd_a * sd_b
Covariance ap
Cov_ap = xavar_a + xbcov_ab
Covariance ap (3 assets)
Cov_ap = xavar_a + xbcov_ab + xc*cov_ac
Correlation ab
Cor_ab = Cov_ab/(sd_a * sd_b)
Beta_a (Var_m)
Beta_a = Cov_am/Var_m
Beta equality
1 = xabeta_a + xbbeta_b
Beta_a (premiums)
Beta_a = (ra-rf)/(rm-rf)
Beta_p
Beta_p = xa * beta_a + xb * beta_b
R^2
Sqrt(R2) = Cor_am = (Beta_a * Sd_M) / Sd_A
Total, systematic, non-systematic variance
Var_a = Beta_a^2 * Var_m + Var(e_a)
Proportion of total risk that can be diversified away
Sd(e_a)/Sd_a = 1 - (Beta_a*Sd_m)/Sd_a = (Sm - Sa) / Sm
MVP
Xa (MVP) = (Var_b - Cov_ab) / (Var_a + Var_b - 2*Cov_ab)
Risk contribution
Risk cont = Beta_a * Xa
Holding Period Return
HPR = (P1 - P0 + d)/P0 = P1/P0 - 1
Total return
R = 1 + r
Real return
r_r = (1+r)/(1+i) - 1
Risk premium
Risk prem. = ra - rf
Market risk premium
Market risk prem. = rm - rf
Arithmetic average
ra = average
Geometric average x2
(1+r_g) = ((1+r1)*(1+r2)*...*(1+rn))^(1/n) r_g = (Wn/W0)^(1/n) - 1
Effective Annual Rate
EAR = (1+APR/k)^k - 1
Continuously compounded return
R_t = 1 + r = e^(rcc*t)
Common stock total return
Total return = Capital gains + Dividen yield
Price of a share of a company
Div. next year / (r-g)
Annuity
PVann = C*(1-(1+r)^-N)/r
Growing annuity
PVgann = C/(r-g) * (1-((1+g)/(1+r))^N)
Perpetuity
PVperp = C/r
Growing perpetuity
PVgperp = C/(r-g)
Discounting
PV = C / (1+r)^N
Mortgage calculations
Annuity factor = (1-(1+r)^-N)/r
Annual payment = Loaned sum / Annuity factor
CAPM formula
r_a = rf + beta_a * (rm - rf)
Treynor black model formula (combining portfolio with an underpriced asset)
r_a = alpha_a + rf + beta_a * (rm - rf)
3-factor model formula
ra-rf = Beta_market * r_market + Beta_sizer_size + Beta_btmr_btm
Arbitrage pricing theory
ra = E(ra) + Beta_1r_factor1 + beta_2r_factor2 + … + noise_a
CML formula
ra = rf + Sd_a * (rm-rf)/Sd_m
SML formula
ra = rf + beta_a * (rm-rf)/Beta_m
Skewness
Skewness = Mu^3 / Sd^3
Kurtosis and excess kurtosis
Kurtosis = Mu^4 / Sd^4
Ex. kurtosis = Mu^4 / Sd^4 - 3
Sharpe ratio
Sp = (rp - rf)/Sd_p
[Slope of the CAL line]
Modigliani Modigliani
full
M^2 = ( (ra - rf) / Sd_a ) * Sd_m - (rm-rf) or M^2 = rp* - rm sd_p* = y*sd_p = sd_m rp* = y*rp + (1-y)*rf
Treynor index
Tp = (rp-rf)/beta_p
[Slope of the SML line]
T^2
T^2 = Tp - Tm = Alpha_p/Beta _p
[Difference between the Tp and Tm lines]
Jensnes measure (alpha)
alpha_p = rp - (rf + beta_p*(rm - rf))
Appraisal/information ratio (AR)
ARp = alpha_p/Sd(e)
[Gain over non-systematic risk]
S^2 of optimal portfolio
S^2 = Sm^2 + (Alpha_a/Sd_e)^2
[Highest alpha per tracking error]
WACC 1,2
r_assets = r_debt *(D/V) + r_equity (E/V) r_assets = rf + beta_assets*(rm-rf)
r_equity 1,2,3
r_equity = rf + Beta_equity*(rm-rf) r_equity = r_assets + D/E*(r_assets - r_debt) r_equity = r_debt + beta_equity*(rm - r_debt)
Beta_assets 1,2
Beta_assets = Beta_debt * D/V + Beta_equity * E/V Beta_assets = (E/V) * Beta_equity
Beta_equity
Beta_equity = Beta_assets * (1+D/E)
Total Beta
Total beta = Beta_a/Cor_am = Sd_a/Sd_m
Weights in market portfolio
Xa = (Sd_b^2(ra-rf) - Cov_ab(rb-rf)) / (Sd_a^2(rb-rf) + Sd_b^2(ra-rf) - 2Cov_ab(ra+rb-2rf))