CAPM

Question 1

Return of an asset (probabilities)

Answer 1

r_a = p1r1 + p2r2 + p3*r3

Question 2

Variance of an asset (probabilities)

Answer 2

Var_a = p1(r_a1 - r_a)^2 + p2(r_a2 - r_a)^2 + p3*(r_a3 - r_a)^2

Question 3

Covariance (probabilities)

Answer 3

cov = p1(r_a1 - ra)(r_b1 - r_b) + p2(r_a2 - ra)(r_b2 - r_b) + …

Question 4

Return of a portfolio

Answer 4

r_p = xa * ra + xb * rb

Question 5

Variance of portfolio (basic)

Answer 5

Var_p = xa^2 * Var_a^2 + xb^2 * Var_b^2 + 2xaxb*cov_ab

Question 6

Variance of portfolio (3 assets)

Answer 6

Var_p = xa^2 * Var_a^2 + xb^2 * Var_b^2 + xc^2 * Var_c^2 + 2xaxbcov_ab + 2xaxccov_ac + 2xbxc*cov_bc

Question 7

Variance of portfolio (Cov_ap)

Answer 7

Var_p = xacov_ap + xbcov_bp

Question 8

Variance of portfolio (with beta)

Answer 8

Var_p = (beta*sd_m)^2 + Var(e)^2

Question 9

Covariance ab

Answer 9

Cov_ab = Cor_ab * sd_a * sd_b

Question 10

Covariance ap

Answer 10

Cov_ap = xavar_a + xbcov_ab

Question 11

Covariance ap (3 assets)

Answer 11

Cov_ap = xavar_a + xbcov_ab + xc*cov_ac

Question 12

Correlation ab

Answer 12

Cor_ab = Cov_ab/(sd_a * sd_b)

Question 13

Beta_a (Var_m)

Answer 13

Beta_a = Cov_am/Var_m

Question 14

Beta equality

Answer 14

1 = xabeta_a + xbbeta_b

Question 15

Beta_a (premiums)

Answer 15

Beta_a = (ra-rf)/(rm-rf)

Question 16

Beta_p

Answer 16

Beta_p = xa * beta_a + xb * beta_b

Question 17

R^2

Answer 17

Sqrt(R2) = Cor_am = (Beta_a * Sd_M) / Sd_A

Question 18

Total, systematic, non-systematic variance

Answer 18

Var_a = Beta_a^2 * Var_m + Var(e_a)

Question 19

Proportion of total risk that can be diversified away

Answer 19

Sd(e_a)/Sd_a = 1 - (Beta_a*Sd_m)/Sd_a = (Sm - Sa) / Sm

Question 20

MVP

Answer 20

Xa (MVP) = (Var_b - Cov_ab) / (Var_a + Var_b - 2*Cov_ab)

Question 21

Risk contribution

Answer 21

Risk cont = Beta_a * Xa

Question 22

Holding Period Return

Answer 22

HPR = (P1 - P0 + d)/P0 = P1/P0 - 1

Question 23

Total return

Answer 23

R = 1 + r

Question 24

Real return

Answer 24

r_r = (1+r)/(1+i) - 1

Question 25

Risk premium

Answer 25

Risk prem. = ra - rf

Question 26

Market risk premium

Answer 26

Market risk prem. = rm - rf

Question 27

Arithmetic average

Answer 27

ra = average

Question 28

Geometric average x2

Answer 28

``` (1+r_g) = ((1+r1)*(1+r2)*...*(1+rn))^(1/n) r_g = (Wn/W0)^(1/n) - 1 ```

Question 29

Effective Annual Rate

Answer 29

EAR = (1+APR/k)^k - 1

Question 30

Continuously compounded return

Answer 30

R_t = 1 + r = e^(rcc*t)

Question 31

Common stock total return

Answer 31

Total return = Capital gains + Dividen yield

Question 32

Price of a share of a company

Answer 32

Div. next year / (r-g)

Question 33

Annuity

Answer 33

PVann = C*(1-(1+r)^-N)/r

Question 34

Growing annuity

Answer 34

PVgann = C/(r-g) * (1-((1+g)/(1+r))^N)

Question 35

Perpetuity

Answer 35

PVperp = C/r

Question 36

Growing perpetuity

Answer 36

PVgperp = C/(r-g)

Question 37

Discounting

Answer 37

PV = C / (1+r)^N

Question 38

Mortgage calculations

Answer 38

Annuity factor = (1-(1+r)^-N)/r | Annual payment = Loaned sum / Annuity factor

Question 39

CAPM formula

Answer 39

r_a = rf + beta_a * (rm - rf)

Question 40

Treynor black model formula (combining portfolio with an underpriced asset)

Answer 40

r_a = alpha_a + rf + beta_a * (rm - rf)

Question 41

3-factor model formula

Answer 41

ra-rf = Beta_market * r_market + Beta_size*r_size + Beta_btm*r_btm

Question 42

Arbitrage pricing theory

Answer 42

ra = E(ra) + Beta_1*r_factor1 + beta_2*r_factor2 + ... + noise_a

Question 43

CML formula

Answer 43

ra = rf + Sd_a * (rm-rf)/Sd_m

Question 44

SML formula

Answer 44

ra = rf + beta_a * (rm-rf)/Beta_m

Question 45

Skewness

Answer 45

Skewness = Mu^3 / Sd^3

Question 46

Kurtosis and excess kurtosis

Answer 46

Kurtosis = Mu^4 / Sd^4 | Ex. kurtosis = Mu^4 / Sd^4 - 3

Question 47

Sharpe ratio

Answer 47

Sp = (rp - rf)/Sd_p | [Slope of the CAL line]

Question 48

Modigliani Modigliani | full

Answer 48

``` 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 ```

Question 49

Treynor index

Answer 49

Tp = (rp-rf)/beta_p | [Slope of the SML line]

Question 50

T^2

Answer 50

T^2 = Tp - Tm = Alpha_p/Beta _p | [Difference between the Tp and Tm lines]

Question 51

Jensnes measure (alpha)

Answer 51

alpha_p = rp - (rf + beta_p*(rm - rf))

Question 52

Appraisal/information ratio (AR)

Answer 52

ARp = alpha_p/Sd(e) | [Gain over non-systematic risk]

Question 53

S^2 of optimal portfolio

Answer 53

S^2 = Sm^2 + (Alpha_a/Sd_e)^2 | [Highest alpha per tracking error]

Question 54

WACC 1,2

Answer 54

``` r_assets = r_debt *(D/V) + r_equity (E/V) r_assets = rf + beta_assets*(rm-rf) ```

Question 55

r_equity 1,2,3

Answer 55

``` 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) ```

Question 56

Beta_assets 1,2

Answer 56

``` Beta_assets = Beta_debt * D/V + Beta_equity * E/V Beta_assets = (E/V) * Beta_equity ```

Question 57

Beta_equity

Answer 57

Beta_equity = Beta_assets * (1+D/E)

Question 58

Total Beta

Answer 58

Total beta = Beta_a/Cor_am = Sd_a/Sd_m

Question 59

Weights in market portfolio

Answer 59

Xa = (Sd_b^2*(ra-rf) - Cov_ab*(rb-rf)) / (Sd_a^2*(rb-rf) + Sd_b^2*(ra-rf) - 2*Cov_ab*(ra+rb-2rf))