Risk and Return
𝑅𝑒𝑡𝑢𝑟𝑛=
(𝑊ℎ𝑎𝑡 𝑐𝑜𝑚𝑒𝑠 𝑡𝑜 𝑦𝑜𝑢𝑟 𝑝𝑜𝑐𝑘𝑒𝑡(𝑖𝑛𝑓𝑙𝑜𝑤)−𝑊ℎ𝑎𝑡 𝑔𝑜𝑒𝑠 𝑜𝑓𝑓 𝑦𝑜𝑢𝑟 𝑝𝑜𝑐𝑘𝑒𝑡(𝑜𝑢𝑡𝑓𝑙𝑜𝑤))/(𝑊ℎ𝑎𝑡 𝑔𝑜𝑒𝑠 𝑜𝑓𝑓
𝑦𝑜𝑢𝑟 𝑝𝑜𝑐𝑘𝑒𝑡(𝑜𝑢𝑡𝑓𝑙𝑜𝑤))
There are 2 types of returns :
Ex-Post Returns and Ex-Ante Returns
Ex-Post Returns :
Are past or historical returns
-
Total return on a bond :
total return = capital gain (loss) + income yieldtotal return =
-
𝑃1 − 𝑃0 𝐶𝐹1 𝐶𝐹1 + 𝑃1 − 𝑃0
+
=
𝑃0
𝑃0
𝑃0
Total return on a stock :
𝑅𝑒𝑡𝑢𝑟𝑛 =
(𝑃1−𝑃0)+𝐷1
𝑃0
=
𝑃1−𝑃0
𝐷1
+𝑃
𝑃0
0
Measuring Returns: Arithmetic Mean :
𝑆𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑟𝑒𝑡𝑢𝑟𝑛𝑠
𝐴𝑟𝑖𝑡ℎ𝑚𝑒𝑡𝑖𝑐 𝑀𝑒𝑎𝑛 =
=
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠
𝐴𝑟𝑖𝑡ℎ𝑚𝑒𝑡𝑖𝑐 𝑀𝑒𝑎𝑛 =
𝑛
𝑖=1 𝑟𝑖
𝑛
=
𝑛
𝑖=1 𝑟𝑖
𝑛
4+5+6+7+3+1−2−4
20
=
= 2.5%
8
8
Measuring Returns: Geometric Mean :
Geometric Mean is the compound growth rate over multiple period
1
𝑛
𝑛
𝐺𝑒𝑜𝑚𝑒𝑡𝑟𝑖𝑐 𝑀𝑒𝑎𝑛 =
1 + 𝑟𝑖
−1
1 + 𝑟3
1 + 𝑟4 … 𝑛 − 1
𝑖=1
𝐺𝑒𝑜𝑚𝑒𝑡𝑟𝑖𝑐 𝑀𝑒𝑎𝑛 =
1 + 𝑟1
1 + 𝑟2
1
Measuring Risk: Standard Deviation :
Below is the formula for standard deviation (ex-post condition):
𝜎𝐸𝑥𝑃𝑜𝑠𝑡 =
𝑛
2
𝑖=1 𝑟𝑖 − 𝑟
𝑛−1
Ex-Ante Returns :
Future returns, which means there is a certain probability that’ll let us use a certain return :
•
Below is the formula for Expected Return:
Measuring Risk: Standard Deviation :
𝜎
𝐸𝑥𝐴𝑛𝑡𝑒 =
𝑛
𝑖=1 𝑃𝑟𝑜𝑏 𝑖
EXPECTED RETURN PORTFOLIOS :
Note that portfolio weights must sum to 1
n
ER P   ( wi  ER i )
i 1
𝑟 𝑖 −𝐸𝑅 𝑖 2
RISK FOR PORTFOLIOS :
Covariance :
n
COV AB   Pr obi ( rA,i  ER A )( rB ,i  ER B )
i 1
Standard deviation :
 P  w   w   2w A wB COV AB
2
A
2
A
2
B
2
B
THE CORRELATION COEFFICIENT
The correlation coefficient will range between -1 and +1
𝝆𝑨𝑩 =
𝑪𝑶𝑽𝑨𝑩
𝝈𝑨 𝝈𝑩
 P  w A2  A2  wB2  B2  2w A wB  AB  A B
But if we have more than 2 portf
 P  w A2 A2  wB2 B2  wC2 C2  2w AwB  AB A B  2w AwC  AC  A C  2wBwC  BC  B C
Exo à part :
Assume that you have decided to invest in a combination of the BMF and the risk free asset
(Rf = 8%). What fraction of your wealth should be invested in the BMF so that your portfolio
earns a target return of 22%?
GO BACK TO CHAPTER 8-1, Slides about CASE 1, 2 and 3!!!
Chapter 8-2
Expected return :
E(R): Expected return (also denoted as 𝑅̅ )
pi: probability of state i
Ri: return of the asset in state i
n
E ( R )   pi Ri
i 1
Variance and Standard Deviation :
n
σ   pi ( Ri  E ( R )) 2
2
E(R) : Expected return
i 1
pi : probability of state i
Ri : return of the asset in state i
(Ri-E(R)) : deviation of actual return from the mean return
Asset portfolios :
A portfolio is a collection of assets
Portfolio’s expected return:
E(RP): expected return of the portfolio
E[ RP ]   w j  E[ R j ]
wj: the fraction of the portfolio wealth invested in asset j
E(Rj): expected return of asset j
We can also calculi it this way :
Expected return of each asset j can be calculated using the following formula:
𝑗
𝑗
𝑗
𝐸(𝑅 𝑗 ) = 𝑅̅ 𝑗 = 𝑃𝑟1 ∗ 𝑅1 + 𝑃𝑟2 ∗ 𝑅2 + 𝑃𝑟3 ∗ 𝑅3
Portfolio expected return
𝐸(𝑃) = 𝑃̅ = 𝜔𝑋 𝑅̅ 𝑋 + 𝜔𝑌 𝑅̅𝑌 + 𝜔𝑍 𝑅̅ 𝑍
N
j 1
Portfolio Variance :
For N Assets
N
N
   w    2   i , j  wi w j i j
2
P
2
j
2
j
j 1
i , j 1
̅ 𝑿 )(𝑹𝒀𝟏 − 𝑹
̅ 𝒀) +
𝑪𝒐𝒗(𝑿, 𝒀) = 𝝈𝑿𝒀 = 𝝈𝑿 𝝈𝒀 𝝆𝑿𝒀 = 𝑷𝒓𝟏 ∗ (𝑹𝑿𝟏 − 𝑹
̅ 𝑿 )(𝑹𝒀𝟐 − 𝑹
̅ 𝒀 ) + 𝑷𝒓𝟑 ∗ (𝑹𝑿𝟑 − 𝑹
̅ 𝑿 )(𝑹𝒀𝟑 − 𝑹
̅ 𝒀)
𝑷𝒓𝟐 ∗ (𝑹𝑿𝟐 − 𝑹
n
COV AB   Pr obi ( rA,i  ER A )( rB ,i  ER B )
i 1
• If ρ = +1, they are perfectly correlated and move in the same direction.
XY
• If ρ = -1, they are perfectly negatively correlated, they move in opposite directions.
XY
• If ρ = 0, then on average they don't move together or in opposite directions
XY
WHEN : ρX,Y=-1 then : Zero variance portfolio
Which means
𝜎𝑌
⁄(𝜎 +𝜎 ) =
𝑋
𝑌
9%/(7.5% + 9%)=0.545
𝜔𝑋 =
Beta : Risk of an asset regarding to the market
The average beta across all securities when weighted by the proportion of each security’s market
value to that of the market portfolio is 1.
i 
Cov( Ri , RM )
 ( RM )
2
The risk premium = expected return – risk-free rate
Capital Asset Pricing Model (CAPM)
E(RA) = Rf + A(E(RM) – Rf)
= Rf + B x risk prem
Portfolio Beta:
Pi is like w, it’s for wages
(E(RM) – Rf) : risk premium