Mid-Term and Final Examination Formula Sheet with Accompanying Notes PDF

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FNCE30001 InvestmentsFormulae SheetRisk and ReturnAnnual Percentage Rate (APR/BEY)𧰀()*=𧰀+,-=𧰀)./%×𠐀)./%3/Effective Annual Rate (EAR)𧰀,(*=( 1 +𧰀)./%)5− 1𧰀,(*=( 1 +𧰀()*𠐀)5− 1Where; both equations are equal, 𧰀)./%equals/"#$5.Annualised ReturnAnnualised Return=( 1 +Total Return)!5− 1Interest Ac...

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FNCE30001 Investments

Formulae Sheet

Risk and Return

Return on Asset

R =#

Revenue − Cost Cost

R =#

Revenue − Cost Initial#Margin

Return on Short Sale

Holding Period Return

𝑟! =

𝑃! − 𝑃" + 𝐷! 𝑃"

Margin Percent

Margin#Percent = # Margin#Percent = # Margin#Percent = #

Assets − #Liabilities Value#of#Stock#Purchased

Net#Equity#In#Account Value#of#Stock#Purchased

Value#of#Stock − Loan Value#of#Stock#Purchased

Loan- To- Value

LTV =

Loan Value#of#Stock#Purchased

LTV = 1 − Margin#Percent

Arithmetic Return

1 1 𝑟I= (𝑟! + 𝑟# + ⋯ + 𝑟$ ) = N 𝑟% 𝑛 𝑛# $

%&!

Geometric Return ! 𝑟' = O( 1 + 𝑟!)(1 + 𝑟#) … (1 + 𝑟$ ) − 1

𝑟' = [( 1 + 𝑟!)(1 + 𝑟# ) … (1 + 𝑟$)] $ − 1 !

Annual Percentage Rate (APR/BEY (APR/BEY))

𝑟()* = 𝑟+,- = 𝑟)./%01 × 𝑁)./%012 3.4/

Effective Annual Rate (EAR)

𝑟,(* = (1 + 𝑟)./%01 )5 − 1 𝑟,(* = (1 +

𝑟()* 5 ) −1 𝑁

Where; both equations are equal, 𝑟)./%01 equals

/"#$ 5

.

Annualised Return

Annualised#Return = (1 + Total#Return)5 − 1 !

Interest Accrued

𝑟 Interest#Accured = Loan#Amount# × V(1 + )5 − 1W 𝑁 Loan Payable Inclusive of Interest Accrued

𝑟 Loan#Payable#with#Interest#Accured = Loan#Amount# × V(1 + )5 W 𝑁 Real Return

𝑟*,(6 =

1 + 𝑟57895(6 −1 1+𝑖

Expected/Scenario Return :

Where; s#=#particular#scenario.

𝐸[ ⏞ 𝑟( ] = N 𝑝( 𝑠) × 𝑟( ( 𝑠) 2&!

Historic Variance of One Risky Asset

1 N (𝑟; − 𝑟I)# 𝜎 = 𝑇−1 <

#

;&!

Scenario Variance of One Risky Asset :

𝜎 = N 𝑝(𝑠)(𝑟% (𝑠) − 𝐸[𝑟 ⏞ % ])# #

2&!

Expected Return on a Portfolio with One Risky and One Risk- free Asset

𝐸[⏞𝑟) ] = 𝑤% 𝐸[𝑟⏞% ] + (1 − 𝑤% )𝑟= Risk (Standard Deviation) on a Portfolio with One Risky and One Risk- free Asset

𝜎) = 𝑤% 𝜎% Price of an Asset >

𝑃% = N

;&!

e ;f 𝐸b𝐶𝐹 (1 + 𝐸[ 𝑟⏞% ]);

Price of an Asset with a Dividend in Perpetuity

𝑃% =

𝐸 bg𝐷f 𝐸 [𝑟h %]

Modern Portfolio Theory and Index Models

Mean Mean-- Variance Utility

𝑈 = 𝐸 [ 𝑟⏞ 0] −

1 𝐴𝜎 # 2

Mean Mean-- Variance Maximising Utility

1 Max. U = 𝑤) 𝐸 [⏞𝑟) ] +(1 − 𝑤) )𝑟= − 𝐴𝑤)# 𝜎)# 2 Markowitz’s Utility Maximising Weight in Risky Portfolio

𝑤)∗ =

𝑟 @ f − 𝑟= 𝐸 b⏞ 𝐴𝜎)#

Markowitz’s Utility Maximising Risk Premium

𝐸 [𝑟⏞ )] − 𝑟= = 𝐴𝜎)# This assumes the weight in the risky/market portfolio is 1, i.e. 100%. Markowitz’s Level of Risk Risk-- Aversion

𝐴=

𝐸 [𝑟⏞8 ] − 𝑟= 𝜎8#

Again, this assumes the weight in the market portfolio is 1, i.e. 100%, as the representative investor assumes the full market portfolio. Sharpe Ratio or Reward- to- Variability

𝑆) =

⏞ ) ] − 𝑟= 𝐸[𝑟 𝜎⏞/#

Expected Return on a Portfolio with ‘N’ Securities 5

𝐸[𝑟⏞ )] = N 𝑤5 𝐸[𝑟⏞) ] $&!

Risk (Variance) on a Portfolio with Two Risky Securities

𝜎)# = 𝑤(# 𝜎(# + 𝑤+# 𝜎+# + 2𝑤( 𝑤+ 𝜎(+, or,

𝜎)# = 𝑤(# 𝜎(# + 𝑤+# 𝜎+# + 2𝑤( 𝑤+ 𝜎( 𝜎+ 𝜌(+

Where; 𝜎(+#=#Covariance#between#A#&#B, 𝜌(+ #=#Correlation#between#A#&#B. Historic Covariance

𝜎(+

1 = Ns 𝑟(,; − (𝑟Its 𝑟+,; − +𝑟It 𝑇−1 <

;&!

Scenario Covariance :

𝜎(+ = N 𝑝(𝑠 )s𝑟(,2 − 𝐸[⏞𝑟( ]ts𝑟+,2 − 𝐸[𝑟 ⏞ +]t 2&!

Covariance between Two Risky Securities

𝜎%,C = 𝛽%,C #𝜎C

𝜎(+ = 𝜌(+ 𝜎( 𝜎+ Correlation

Corrs𝑟⏞( , ⏞𝑟 + t =

Cov s𝑟⏞( , ⏞ 𝑟 +t

St. Devs𝑟 ⏞( t#St. Devs𝑟 ⏞+ t

𝜌(+ =

𝜎(+ 𝜎( 𝜎+

Net Asset Value (NAV)

NAV =

Mkt. Value#of#Assets − Liabilities # Shares#Outstanding

Index Model D

𝑟%,; − 𝑟=,; = 𝛼% +N 𝛽%,Cs𝑟C,; − 𝑟=,; t + 𝑒%,; C&!

Realised Beta of a Security from an Index Model

𝛽%,C = 𝛽%,C =

𝐶𝑜𝑣(𝑟}% , 𝑟}C ) 𝑉𝑎𝑟(𝑟}C )

𝐶𝑜𝑣s 𝑟%} − 𝑟= , 𝑟}C − 𝑟= t 𝑉𝑎𝑟(𝑟}C ) 𝛽%,C =

𝜎%,C 𝜎C#

𝑦# − 𝑦! 𝛽%,C = 𝑥# − 𝑥! Index Market Model

𝑟%,; − 𝑟=,; = 𝛼% + 𝛽s𝑟 t + 𝑒%,; % 8,; − 𝑟=,; Realised Risk of an Individual Security from an Index Model # 𝜎#/& + 𝜎.#% 𝜎/#% = 𝛽%,8

𝜎/#%

# # 𝜎/,8 𝛽%,8 = 𝑅#

# 𝜎#/&=#Covariance/systematic#risk and 𝜎.#%=#firmWhere; total risk is a factor of: 𝛽%,8 specific/residual#risk.

Equation of the Security Characteristic Line

𝐸b𝑟 ⏞% − ⏞ 𝑟8 − 𝑟=f = 𝛼†% + 𝛽‡%s𝑟8 − 𝑟= t 𝑟= … RR-squared squared

𝑅 # = 𝜌# =

Explained#Variance Total#Variance

𝑅 # = 𝜌# =

# # 𝛽%,8 𝜎/,8 # 𝛽%,8 𝜎#/& + 𝜎.#%

Treynor Treynor-- Black Portfolio Optimisation i. Compute alpha of the individual securities.

𝛼% = 𝐸[ 𝑟⏞%] − ˆ𝑟= + 𝛽% s𝐸 [⏞ 𝑟 8 ] − 𝑟= t‰ ii. Compute residual variances (or firm-specific risk) of the individual securities.

𝜎.#% = 𝜎#.% 1. Compute initial position in the active portfolio.

𝑤%0 =

𝛼% 𝜎.#%

Known as the information ratio or appraisal ratio. 2. Rescale the pseudo-weights to create the true weight of each asset in the active portfolio.

𝑤% = #

𝑤%" 5 ∑%&! 𝑤%"

3. Compute the alpha and beta of the entire portfolio. 5

𝛼( = N 𝑤% 𝛼% %&! 5

𝛽( = N 𝑤% 𝛽% %&!

4. Compute the residual variance.

𝜎.#"

5

= N 𝑤%# 𝜎.#% %&!

5. Compute the initial weight of the active portfolio in the overall risky portfolio.

𝑤(0 =

𝛼( ⁄𝜎.#"

⏞ 8 ] − 𝑟= t Œ( 𝜎8# ) s𝐸[𝑟

Numerator is the information ratio. Denominator is the market risk premium divided by the market variance. 6. Adjust the initial weight allocated in the active portfolio.

𝑤(∗ =

𝑤(0 1 + 𝑤(0 (1 − 𝛽()

7. Calculate the weight in the passive, benchmark/market fund. ∗ 𝑤8 = 1 − 𝑤(∗

Return, Risk and Sharpe Ratio of a Treynor Treynor-- Black Optimised Portfolio 1. Return.

𝐸 [ 𝑟⏞ 0] − 𝑟= = 𝑤∗8 s𝐸 [𝑟⏞ 8] − 𝑟= t + 𝑤(∗ •𝛼( + 𝛽( s𝐸 [𝑟⏞ 8] − 𝑟=tŽ 𝐸[ ⏞ 𝑟0] − 𝑟= = 𝑤(∗ 𝛼( + ( 𝑤8∗ + 𝑤(∗𝛽()s 𝐸[ ⏞ 𝑟8 ] − 𝑟= t

2. Risk. # 𝜎0# = (𝑤8∗ + 𝑤(∗ 𝛽( )# 𝜎8 + (𝑤(∗ 𝜎." )#

3. Sharpe Ratio.

𝛼( # + ( )# 𝑆0 = • 𝑆8 𝜎 ."

𝑆0 = •

𝐸[ 𝑟⏞0 ] − 𝑟= 𝜎0

Asset Pricing

Single, Market Factor Capital Asset Pricing Model (CAPM)

𝐸[⏞𝑟%] − 𝑟= = 𝛽%,8s𝐸 [𝑟⏞8 ] − 𝑟=t 𝐸[𝑟⏞ %] − 𝑟= = 𝐶𝑜𝑣s𝑟}% − 𝑟= , 𝑟}C − 𝑟= t

(𝐸[ 𝑟⏞8 ] − 𝑟= ) 𝑉𝑎𝑟(𝑟h 8 − 𝑟= )

Treynor Measure

𝑇) =

𝐸 [⏞𝑟) ] − 𝑟= 𝛽)

Where; 𝑇) the return per unit of systematic risk. Combining Beta 5

𝛽) = N 𝑤% 𝛽% %&!

Equation of Security Characteristic Line

𝐸 ˆ𝑟h 𝑟8 − 𝑟= ‰ = 𝛼 † % + 𝛽‡% (𝑟8 − 𝑟= ) =… % − 𝑟h

Where; 𝛼†% is the intercept, 𝛽‡% is the slope.

Merton’s Intertemporal CAPM (ICAPM)

𝐸[⏞ 𝑟% ] − 𝑟= = 𝛽%,8s𝐸[ ⏞ 𝑟8 ] − 𝑟= t + 𝛽%,...