FORMULAS IA
Part 1
5% probability that portfolio will fall in value by more than VaR over one year period.
𝑉𝑎𝑅$%%&$' = 𝑊+ 1.64𝜎$%%&$' − 𝜇$%%&$'
𝑉𝑎𝑅$%%&$' = 𝑊+ 12 ×1.64𝜎56%+7'8 − 12 ×𝜇56%+7'8
GUISE: 𝑥:.;: = 0.3125𝑥:.:; + 0.4375𝑥:.:A + 0.2500𝑥:.;: 𝑥B = 𝑊: ×𝑒𝑥𝑝 𝐻𝜇 + 𝑧B 𝜎 𝐻
𝑟H = 𝑟I + 𝛽H 𝑟5 − 𝑟I
𝜎K = 𝑦𝜎M
𝐶𝑜𝑣 𝑟H , 𝑟5
𝛽H =
𝑉𝑎𝑟 𝑟5
U
𝜌ST = VW 𝜎ST = 𝜌ST × 𝜎S × 𝜎T
UV UW
𝑟K = 𝑤S 𝑟S + 𝑤T 𝑟T
Y
𝜎ST = 𝑤SY 𝜎SY + 𝑤TY 𝜎TY + 2𝑤S 𝑤T 𝜎ST
1
𝑈 𝐶 = 𝐸 𝑟K − 𝐴 ×𝑉𝑎𝑟 𝑟K
2
Matrix notation
𝑤; 1
𝑤= ⋮ 𝜄= ⋮ weight of risk-free asset 𝑤`a = 1 − b
Hc; 𝑤H = 1 − 𝑤 d𝜄
𝑤^ 1
𝐸 𝑟; − 𝑟I
Return of combination portfolio: 𝑟K = 𝑟I + 𝑤 d 𝑟Me where 𝑤 d = 𝑤; … 𝑤^ and 𝑟Me = ⋮
𝐸 𝑟^ − 𝑟I
𝐸 𝑟; − 𝑟I
d e ⋮ e
Expected return: 𝐸 𝑟K = 𝑟I + 𝑤 𝜇 where 𝜇 =
𝐸 𝑟^ − 𝑟I
𝜎;Y ⋯ 𝜎;b 𝑤;
Variance: 𝑉𝑎𝑟 𝑟K = 𝑤 d
𝑤 = 𝑤; … 𝑤^ ⋮ ⋱ ⋮ ⋮
𝜎b; ⋯ 𝜎bY 𝑤^
;
Find optimal complete portfolio weights (maximize U): 𝑈 𝐶 = 𝐸 𝑟K − 𝐴 ×𝑉𝑎𝑟 𝑟K
Y
;
= 𝑟I + 𝑤 d 𝜇 e − 𝐴 ×𝑤 d 𝑤
Y
e
𝜇 − 𝐴Σ𝑤 = 0
𝐴Σ𝑤 = 𝜇 e
𝑤 ∗ = 𝐴k; Σ k; 𝜇 e
, Optimal portfolio with exogenous risk
𝑟K = 𝑟I + 𝑤 d 𝑟Me + 𝑞 × 𝑟m
𝐸 𝑟K = 𝑟I + 𝑤 d 𝜇 e + 𝑞𝜇m
𝑉𝑎𝑟 𝑟K = 𝑤 d Σ𝑤 + 𝑞 Y 𝜎mY + 2𝑤 d 𝜎`n 𝑞
𝑤 ∗ = 𝐴k; Σ k; 𝜇 e − 𝑞Σ k; 𝜎`n
Part 2
Equity
𝑚𝑎𝑟𝑘𝑒𝑡 𝑝𝑟𝑖𝑐𝑒
𝑇𝑜𝑏𝑖𝑛𝑠 𝑞 =
𝑟𝑒𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑎𝑠𝑠𝑒𝑡𝑠
T S• € T •• k •‚
Expected stock return: 𝐸 𝑟 = 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑌𝑖𝑒𝑙𝑑 + 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 𝐺𝑎𝑖𝑛 𝑌𝑖𝑒𝑙𝑑 =
•ƒ
T S• € T ••
Intrinsic value: 𝑉: =
;€^
1. DDM:
S• S„ S… € •…
a. Finite horizon: 𝑉: = + +⋯+
;€^ ;€^ „ ;€^ …
‡ S
b. Infinite horizon: 𝑉6 = +c; † †
;€^
c. Dividend growth:
i. Constant dividend growth model: 𝑔 = 𝑅𝑂𝐸 × 𝑏
1. 𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠+€; = 𝐵𝑜𝑜𝑘𝑣𝑎𝑙𝑢𝑒 +€; ×𝑅𝑂𝐸
2. 𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠+€; = 𝐵𝑜𝑜𝑘𝑣𝑎𝑙𝑢𝑒+ + 𝐵𝑜𝑜𝑘𝑣𝑎𝑙𝑢𝑒+ ×𝑏 ×𝑅𝑂𝐸 ×𝑅𝑂𝐸
a. 𝑅𝑒𝑡𝑎𝑖𝑛𝑒𝑑 = 𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠 ×𝑏
b. 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑝𝑎𝑦𝑒𝑑 𝑜𝑢𝑡 = 𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠 × 1 − 𝑏
Ž ;€ Œ• † S• ;€ Œ„
ii. Multistage growth models: 𝑃: = 𝐷: +c; ;€^ † + ^k Œ ;€^ •
„
S• ;€Œ
1. Determine price at end with 𝑃: = , add and discount
^kŒ
2. P/E ratios and growth opportunities:
𝑃𝑟𝑖𝑐𝑒 = 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝐴𝑠𝑠𝑒𝑡𝑠 𝑖𝑛 𝑃𝑙𝑎𝑐𝑒 + 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝐺𝑟𝑜𝑤𝑡ℎ 𝑂𝑝𝑝𝑜𝑟𝑡𝑢𝑛𝑖𝑡𝑖𝑒𝑠
•ƒ ; ••‘’
a. = 1+ T
T• ^ ^
T•
i. If PVGO=0, 𝑃: =
^
S• T•
ii. 𝑃𝑉𝐺𝑂 = 𝑃𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 − 𝑁𝑜𝑔𝑟𝑜𝑤𝑡ℎ 𝑣𝑎𝑙𝑢𝑒 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 = −
^kΠ^
•ƒ ;k”
b. =
T• ^k•’T×”
• ;k”
c. =
T ^kŒ
Empirical Evidence CAPM
1. Estimating betas
a. Theory: 𝐸 𝑟H = 𝑟I + 𝛽– 𝐸 𝑟— − 𝑟I
b. Data: 𝑹𝒊𝒕 − 𝑹𝒇𝒕 = 𝜶𝒊 + 𝜷𝒊 𝑹𝒎𝒕 − 𝑹𝒇𝒕 + 𝜺𝒊𝒕 estimate using time series regressions
2. Regressing returns on betas (average returns on constant β, for every month on time-varying β)
a. Theory: 𝐸 𝑅H+ − 𝑅I+ = 𝜆+ 𝛽+ cross-sectional return-beta relationship
b. Data: 𝑹𝒊𝒕 − 𝑹𝒇𝒕 = 𝝀𝟎𝒕 + 𝝀𝟏𝒕 𝜷𝒕 + 𝜺𝒊𝒕 estimating SML
Part 1
5% probability that portfolio will fall in value by more than VaR over one year period.
𝑉𝑎𝑅$%%&$' = 𝑊+ 1.64𝜎$%%&$' − 𝜇$%%&$'
𝑉𝑎𝑅$%%&$' = 𝑊+ 12 ×1.64𝜎56%+7'8 − 12 ×𝜇56%+7'8
GUISE: 𝑥:.;: = 0.3125𝑥:.:; + 0.4375𝑥:.:A + 0.2500𝑥:.;: 𝑥B = 𝑊: ×𝑒𝑥𝑝 𝐻𝜇 + 𝑧B 𝜎 𝐻
𝑟H = 𝑟I + 𝛽H 𝑟5 − 𝑟I
𝜎K = 𝑦𝜎M
𝐶𝑜𝑣 𝑟H , 𝑟5
𝛽H =
𝑉𝑎𝑟 𝑟5
U
𝜌ST = VW 𝜎ST = 𝜌ST × 𝜎S × 𝜎T
UV UW
𝑟K = 𝑤S 𝑟S + 𝑤T 𝑟T
Y
𝜎ST = 𝑤SY 𝜎SY + 𝑤TY 𝜎TY + 2𝑤S 𝑤T 𝜎ST
1
𝑈 𝐶 = 𝐸 𝑟K − 𝐴 ×𝑉𝑎𝑟 𝑟K
2
Matrix notation
𝑤; 1
𝑤= ⋮ 𝜄= ⋮ weight of risk-free asset 𝑤`a = 1 − b
Hc; 𝑤H = 1 − 𝑤 d𝜄
𝑤^ 1
𝐸 𝑟; − 𝑟I
Return of combination portfolio: 𝑟K = 𝑟I + 𝑤 d 𝑟Me where 𝑤 d = 𝑤; … 𝑤^ and 𝑟Me = ⋮
𝐸 𝑟^ − 𝑟I
𝐸 𝑟; − 𝑟I
d e ⋮ e
Expected return: 𝐸 𝑟K = 𝑟I + 𝑤 𝜇 where 𝜇 =
𝐸 𝑟^ − 𝑟I
𝜎;Y ⋯ 𝜎;b 𝑤;
Variance: 𝑉𝑎𝑟 𝑟K = 𝑤 d
𝑤 = 𝑤; … 𝑤^ ⋮ ⋱ ⋮ ⋮
𝜎b; ⋯ 𝜎bY 𝑤^
;
Find optimal complete portfolio weights (maximize U): 𝑈 𝐶 = 𝐸 𝑟K − 𝐴 ×𝑉𝑎𝑟 𝑟K
Y
;
= 𝑟I + 𝑤 d 𝜇 e − 𝐴 ×𝑤 d 𝑤
Y
e
𝜇 − 𝐴Σ𝑤 = 0
𝐴Σ𝑤 = 𝜇 e
𝑤 ∗ = 𝐴k; Σ k; 𝜇 e
, Optimal portfolio with exogenous risk
𝑟K = 𝑟I + 𝑤 d 𝑟Me + 𝑞 × 𝑟m
𝐸 𝑟K = 𝑟I + 𝑤 d 𝜇 e + 𝑞𝜇m
𝑉𝑎𝑟 𝑟K = 𝑤 d Σ𝑤 + 𝑞 Y 𝜎mY + 2𝑤 d 𝜎`n 𝑞
𝑤 ∗ = 𝐴k; Σ k; 𝜇 e − 𝑞Σ k; 𝜎`n
Part 2
Equity
𝑚𝑎𝑟𝑘𝑒𝑡 𝑝𝑟𝑖𝑐𝑒
𝑇𝑜𝑏𝑖𝑛𝑠 𝑞 =
𝑟𝑒𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑎𝑠𝑠𝑒𝑡𝑠
T S• € T •• k •‚
Expected stock return: 𝐸 𝑟 = 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑌𝑖𝑒𝑙𝑑 + 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 𝐺𝑎𝑖𝑛 𝑌𝑖𝑒𝑙𝑑 =
•ƒ
T S• € T ••
Intrinsic value: 𝑉: =
;€^
1. DDM:
S• S„ S… € •…
a. Finite horizon: 𝑉: = + +⋯+
;€^ ;€^ „ ;€^ …
‡ S
b. Infinite horizon: 𝑉6 = +c; † †
;€^
c. Dividend growth:
i. Constant dividend growth model: 𝑔 = 𝑅𝑂𝐸 × 𝑏
1. 𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠+€; = 𝐵𝑜𝑜𝑘𝑣𝑎𝑙𝑢𝑒 +€; ×𝑅𝑂𝐸
2. 𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠+€; = 𝐵𝑜𝑜𝑘𝑣𝑎𝑙𝑢𝑒+ + 𝐵𝑜𝑜𝑘𝑣𝑎𝑙𝑢𝑒+ ×𝑏 ×𝑅𝑂𝐸 ×𝑅𝑂𝐸
a. 𝑅𝑒𝑡𝑎𝑖𝑛𝑒𝑑 = 𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠 ×𝑏
b. 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑝𝑎𝑦𝑒𝑑 𝑜𝑢𝑡 = 𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠 × 1 − 𝑏
Ž ;€ Œ• † S• ;€ Œ„
ii. Multistage growth models: 𝑃: = 𝐷: +c; ;€^ † + ^k Œ ;€^ •
„
S• ;€Œ
1. Determine price at end with 𝑃: = , add and discount
^kŒ
2. P/E ratios and growth opportunities:
𝑃𝑟𝑖𝑐𝑒 = 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝐴𝑠𝑠𝑒𝑡𝑠 𝑖𝑛 𝑃𝑙𝑎𝑐𝑒 + 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝐺𝑟𝑜𝑤𝑡ℎ 𝑂𝑝𝑝𝑜𝑟𝑡𝑢𝑛𝑖𝑡𝑖𝑒𝑠
•ƒ ; ••‘’
a. = 1+ T
T• ^ ^
T•
i. If PVGO=0, 𝑃: =
^
S• T•
ii. 𝑃𝑉𝐺𝑂 = 𝑃𝑟𝑖𝑐𝑒 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 − 𝑁𝑜𝑔𝑟𝑜𝑤𝑡ℎ 𝑣𝑎𝑙𝑢𝑒 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒 = −
^kΠ^
•ƒ ;k”
b. =
T• ^k•’T×”
• ;k”
c. =
T ^kŒ
Empirical Evidence CAPM
1. Estimating betas
a. Theory: 𝐸 𝑟H = 𝑟I + 𝛽– 𝐸 𝑟— − 𝑟I
b. Data: 𝑹𝒊𝒕 − 𝑹𝒇𝒕 = 𝜶𝒊 + 𝜷𝒊 𝑹𝒎𝒕 − 𝑹𝒇𝒕 + 𝜺𝒊𝒕 estimate using time series regressions
2. Regressing returns on betas (average returns on constant β, for every month on time-varying β)
a. Theory: 𝐸 𝑅H+ − 𝑅I+ = 𝜆+ 𝛽+ cross-sectional return-beta relationship
b. Data: 𝑹𝒊𝒕 − 𝑹𝒇𝒕 = 𝝀𝟎𝒕 + 𝝀𝟏𝒕 𝜷𝒕 + 𝜺𝒊𝒕 estimating SML
