Computing Returns
Return and Risk
Return =
An Example
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Percentage Return or Dollar
Return?
You bought IBM stock at $40 last month. The
price of IBM stock is $45 today. IBM paid $1
dividend yesterday. What is your holding
period return?
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Dollar return per share: $45-$40+$1=$6
Rate of return: $6 / $40 = 15%
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Percentage Return or Dollar
Return?
So the rate of return presents the complete
picture.
In this course, return means the rate of return.
0.01% is called a basis point.
Return is not unitless. We tend to annualize
returns.
Returns are bounded below at -100%.
Ending Price - Beginning Price + Dividends
Beginning Price
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It is more convenient to use the percentage
return, or the rate of return.
A $1,000 return is quite good for an initial
investment of $1,000, but not so impressive if
the initial investment is $1 million.
A 10% return implies that if you invest $1000
you would make $100 and if you invest $1
million you will make $100,000.
10% or 10?
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We like to express return in percentage
terms. 10% is the same as 0.1 but not the
same as 10.
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P = $1 / (10 - 5) = $0.20
P = $1 / (10% - 5%) = $20
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Averaging Returns
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Averaging Returns
Suppose you invested half of your money in X
and half of your money in Y.
The return of X is 100% and the return of Y is
-50%.
What is the average return?
Future Return is unknown
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Suppose you invested all of your money in X
The return of X is 100% last year and -50%
this year.
What is your average return over the two
years?
Return and Risk
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P − P + Dt
Rt +1 = t +1 t
Pt
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Under uncertainty, we measure “reward” by
using the expected (or average) return.
We measure “risk” by using the variance of
returns
Investments are risky.
Pt+1 is not known at time t; hence Rt+1 is
not known at time t.
How do we capture the randomness of
Rt+1?
Expected Return and
Variance
Sample Mean and Variance
N
µ X = E ( X ) = p1 X 1 + p2 X 2 + L + p N X N = ∑ pi X i
X=
i =1
N
Var ( X ) = σ = ∑ pi [ X i − E ( X )]
2
X
i =1
2
Vaˆr ( X ) =
1
N
N
∑X
i =1
i
2
1 N
∑ (X i − X )
N − 1 i =1
2
Annualize Return
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There are two ways you can annualize
returns.
Suppose R is the per period return and T is
the number of periods per year.
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APR = R × T
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EAR = ( 1 + R ) T - 1
Excess Returns
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The raw return includes compensation for
both the time value of money and the risk of
the security.
An excess return or risk premium is the
compensation for risk bearing alone.
Excess Return or Risk Premium = Ri − R f = Rie
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Variance is proportional to time
Annualized Variance = σ2 × T
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Standard deviation is proportional to the
square root of time
Annualized Standard Deviation = σ × T0.5
Expected Return of A Portfolio
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Adding more assets to a portfolio does
not make the calculation of expected
return more difficult.
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E(Rp) = w1E(R1) + w2E(R2) + … wnE(Rn)
This is the raw return less the risk-free rate.
Variance of A Portfolio
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Annualize Variance
We are often interested in the variance of a
portfolio. If there are two assets in the
portfolio,
Var(w1X +w2 Y) = w12Var(X) + w22Var(Y)
+ 2w1w2Cov(X,Y)
More on Covariance
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Var(w1X + w2Y + w3Z) = w12Var(X) + w22Var(Y)
+w32Var(Z) + 2w1w2Cov(X,Y)+ 2w1w3Cov(X,Z)+
2w2w3Cov(Y,Z)
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where Cov(X,Y)=Corr(X,Y)σ(X)σ(Y)
Now consider the three asset case.
The formula gets very complicated when
there are many assets in the portfolio.
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Matrix Notation
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R is a column vector of expected returns
W is a column vector of portfolio weights
∑ is the covariance matrix
Example:
0.01 ⎤
⎡ 0.1 ⎤
⎡0.4⎤
⎡0.04 0.01
R = ⎢⎢ 0.2 ⎥⎥ W = ⎢⎢0.3⎥⎥ Σ = ⎢⎢ 0.01 0.09 − 0.01⎥⎥
⎢⎣0.15⎥⎦
⎢⎣0.3⎥⎦
⎢⎣ 0.01 − 0.01 0.04 ⎥⎦
Portfolio Return and Risk
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Using matrix notation for portfolio return and
risk,
E (R p ) = w1 R1 + w2 R2 + ... + wn Rn = w′R
Var (R p ) = w12σ 12 + w22σ 22 + ... + wn2σ n2
+ 2w1w2σ 1, 2 + 2w1w3σ 1,3 + ...
= w' Σw
Mean
Standard Deviation
Excel Functions
Excel Functions
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Excel Functions
Covariance
Correlation
Data Analysis
Covariance
Covariance Matrix
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