Risk
and Return
Ferdinand L. Timbang
CPA, REB, MSCF, DBA
Risk
• Risk is the exposure to uncertainty or danger resulting to changes in
the expected return in a given investment.
• The chance that some unfavorable event will occur (Brigham, 4th
edition).
Ranking of Risk - Securities
1. Treasury bills/bonds
2. Bonds – corporate bonds
3. Mutual funds
4. Stocks
The Risk- Return Trade-Off
• To attract investors to take more risk, an investment should provide higher
expected returns.
• The higher the risk, the higher the return.
Investment
return
Good investments:
The
return
is
sufficient
to
compensate for the
perceived risk
Risk-return
trade-off
Bad investments: The
return is insufficient to
compensate for the
perceived risk
Investment risk
Probability
A listing of all possible outcomes, and the probability of each occurrence.
For instance:
State of
Economy
Recession
Normal
Prosperity
Probability
distributions (pi)
0.20
0.60
0.20
The expected return is the return after considering the probabilities of
occurrence, the state of economy, individual’s expected outcomes. Needless to
say, the risk has already been factored in this type of return.
Probability
XYZ Corporation is planning to invest in Stock A. The firm expects that the
possible returns are dependent on the state of the economy. Determine the
expected return on their planned investment.
State of Economy
Recession
Normal
Prosperity
Expected return (ř)
Possible
returns (ri)
10%
15%
20%
Probability
distributions (pi)
0.20
0.60
0.20
= (10% x 0.20) + (15% x 0.60) + (20% x 0.20)
= 0.15 or 15%
Assets Risk
The asset’s risk can be analyzed in two ways:
1. Stand-alone basis. It is where the asset is considered by itself.
Example: T-bill vs. Stock
2. Portfolio basis. It is where the assets is held as one of a number of
assets bin a portfolio.
Measuring Risk – Stand Alone
 = Standard deviation
  Variance  
N
2
ˆ
   (r - r) Pi
i=1
2
Measuring Risk – Stand alone
For FLI
State of Economy Return ( ri ) Probability (pi)
ri x pi
Recession
-8.00%
0.15
-1.20
Normal
15.00%
0.70
10.50
Prosperity
35.00%
0.15
5.25
ř = 14.55
=
=
139.16
11.80%
( ri - ř )2
508.50
0.21
418.20
(ri - ř )2(pi)
76.28
0.15
62.73
2=139.16
Measuring Risk Using Historical Data
Year
2016
Return
42.0%
(ri - ř )2
(0.42 – 0.20)2
2017
2018
2019
12.0%
-30.0%
56.0%
(0.12 – 0.20)2
(-0.30 – 0.20)2
(0.56 – 0.20)2
Ave. 20.0%
σ = square root of variance/(n-1)
= (0.4344/3)
= 38.05%
0.0484
0.0064
0.2500
0.1296
Variance = 0.4344
Conclusion - Measuring Risk
•
Standard deviation (σi) measures total, or stand-alone, risk.
•
The larger σi is, the lower the probability that actual returns will be
closer to expected returns.
•
Larger σi is associated with a wider probability distribution of returns.
Covariance
It is a measure of the degree to which two variables move together relative to their individual
mean values over time.
A positive covariance means that the rates of return for two investments tend to move in the
same direction relative to their individual means during the same time period.
A negative covariance indicates that the rates of return for two investments tend to move in
different directions relative to their means during specified time intervals overtime.
For two assets, i and j, we define the covariance rates of return as:
Covij = E [Ri – E(Ri)] [Rj – E(Rj)] when applied to the monthly rates of return during the
n
year, it becomes
1
∑ [Ri – E(Ri)] [Rj – E(Rj)]
n – 1 t=1
If the correlation coefficient is given, covariance can be computed as Cov = rijδiδj
Covariance
January
February
March
April
May
June
July
August
September
October
November
December
Mean
Wilshire 5000
Stock Index (Ri)
(1.51)
0.96
3.77
3.33
(1.55)
(3.47)
1.12
3.46
2.03
(4.73)
(0.76)
(6.23)
(0.30)
A
Wilshire 5000
Lehman Brothers
Stock Index
Treasury Bonds (Rj)
(Ri - Ri)
(0.16)
(1.21)
1.66
1.26
(0.05)
4.07
0.52
3.63
(0.90)
(1.25)
(0.04)
(3.17)
1.66
1.42
1.57
3.76
0.54
2.33
0.79
(4.43)
3.07
(0.46)
0.08
(5.93)
0.73
B
Lehman Brothers
Treasury Bonds
(Rj - Rj)
(0.89)
0.93
(0.78)
(0.21)
(1.63)
(0.77)
0.93
0.84
(0.19)
0.06
2.34
(0.65)
AxB
1.08
1.17
(3.17)
(0.76)
2.04
2.44
1.32
3.16
(0.44)
(0.27)
(1.08)
3.85
9.34
Covariance
The covariance between the rates of return for Wilshire and Lehman is
Covi,j =
=
1
(12 – 1)
0.85
(9.34)
The relationship was not very strong because there are months that the two
assets moved counter to each other.
The 0.85 might indicate a weak positive relationship if the two individual
indexes were volatile, but would reflect a strong positive relationship if the
two indexes were stable.
Covariance
Correlation Coefficient
To standardize the covariance by the product of the individual standard deviations
yields the correlation coefficient rij. It can only vary from -1 to +1.
A +1 indicates a perfect positive linear relationship between Ri and Rj. It means
that the return for the two assets move together in a completely linear manner.
A -1 indicates a perfect negative relationship between the two return indexes.
To compute for the correlation coefficient, we have
rij = Covij
δiδj
where rij = the correlation coefficient of returns
δi = the standard deviation of Rit
δj = the standard deviation of Rjt
Correlation Coefficient
January
February
March
April
May
June
July
August
September
October
November
December
Mean
Variance
Std. Deviation
Wilshire 5000
Stock Index (Ri)
(1.51)
0.96
3.77
3.33
(1.55)
(3.47)
1.12
3.46
2.03
(4.73)
(0.76)
(6.23)
(0.30)
11.00
3.32
Lehman Brothers
Treasury Bonds (Rj)
(0.16)
1.66
(0.05)
0.52
(0.90)
(0.04)
1.66
1.57
0.54
0.79
3.07
0.08
0.73
1.19
1.09
rij = Covij
δiδj
=
0.85
(3.32)(1.09)
=
0.235
Portfolio Risk
p = w2A2A+w2B2B+2wAwB CovA,B
Or
p = w2A2A+w2B2B+2wAwB ● (ρ) ABAB
Portfolio risk is associated with the total risks of the portfolio consisting of
systematic and unsystematic risks.
Portfolio Risk
Weight is 50% for Wilshire and 50% for Lehman
p = w2W2W+w2L2L+2wWwL ● (ρ) WLWL
= (0.50)(11.0) + (0.50)(1.19) + 2 (0.50)(0.50) (0.235)(3.32)(1.09)
= 2.55
p =
w2W2W+w2L2L+2wWwL CovW,L
= (0.50)(11.0) + (0.50) (1.19) + 2 (0.50)(0.50) (0.85)
= 2.55
Equal Risk and Return with Changing Correlations
Example:
E(R1) = 0.20
E(R2) = 0.20
E(δ1) = 0.10
E(δ2) = 0.10
Consider the following five correlation coefficients and the covariances they yield.
a. r1,2 = 1.00
Cov1,2 = (1.0)(0.10)(0.10) = 0.01
b. r1,2 = 0.50
Cov1,2 = (0.50)(0.10)(0.10) = 0.05
c. r1,2 = 0.00
Cov1,2 = (0.00)(0.10)(0.10) = 0.00
d. r1,2 = -0.50
Cov1,2 = (-0.50)(0.10)(0.10) = -0.005
e. r1,2 = -1.00
Cov1,2 = (-1.00)(0.10)(0.10) = -0.01
Computing the standard deviation for each using the portfolio standard
deviation, we have:
a. δport(a) = (0.5)2 (0.10)2 + (0.5)2 (0.10)2 + (2)(0.5)(0.5)(0.01)
= 0.10
Equal Risk and Return with Changing Correlations
Computing the standard deviation for each using the portfolio standard
deviation, we have:
a. δport(a) = (0.5)2 (0.10)2 + (0.5)2 (0.10)2 + (2)(0.5)(0.5)(0.01)
= 0.10
b. δport(b) = (0.5)2 (0.10)2 + (0.5)2 (0.10)2 + (2)(0.5)(0.5)(0.05)
= 0.0866
c. δport(c) = (0.5)2 (0.10)2 + (0.5)2 (0.10)2 + (2)(0.5)(0.5)(0.00)
= 0.0707
d. δport(d) = (0.5)2 (0.10)2 + (0.5)2 (0.10)2 + (2)(0.5)(0.5)(-0.005)
= 0.05
e. δport(e) = (0.5)2 (0.10)2 + (0.5)2 (0.10)2 + (2)(0.5)(0.5)(-0.01)
= 0.0
Equal Risk and Return with Changing Correlations
Combining Stocks with Different Returns and Risk
Asset
1
2
Case
A
B
C
D
E
E(Ri)
0.10
0.20
Correlation
Coefficient (r1,2)
+1.00
+0.50
0.00
-0.50
-1.00
Wi
0.50
0.50
Variance (δ2) Std. Dev(δ)
0.0049
0.07
0.0100
0.10
Covariance
(r1,2,δ1δ2)
0.0070
0.0035
0.0000
-0.0035
-0.0070
Er = 0.10(0.50 + 0.20(0.50)
Combining Stocks with Different Returns and Risk
Computing the standard deviation for each using the portfolio standard
deviation, we have:
Notice that the
2
2
2
2
a. δport(a) = (0.5) (0.07) + (0.5) (0.10) + (2)(0.5)(0.5)(0.0070)
perfect negative
= 0.085
correlation did
not result to
2
2
2
2
b. δport(b) = (0.5) (0.07) + (0.5) (0.10) + (2)(0.5)(0.5)(0.0035)
zero
standard
= 0.07399
deviation. It is
because
the
2
2
2
2
c. δport(c) = (0.5) (0.07) + (0.5) (0.10) + (2)(0.5)(0.5)(0.00)
different
= 0.0610
examples have
equal weights,
2
2
2
2
d. δport(d) = (0.5) (0.07) + (0.5) (0.10) + (2)(0.5)(0.5)(-0.0035)
but the asset
= 0.0444
standard
deviations are
2
2
2
2
e. δport(e) = (0.5) (0.07) + (0.5) (0.10) + (2)(0.5)(0.5)(-0.0070)
not equal.
= 0.0015
Constant Correlation With Changing Weights
Assume that the correlation coefficient is 0.
Case
F
G
H
I
J
K
L
A1
0.10
0.10
0.10
0.10
0.10
0.10
0.10
A2
0.20
0.20
0.20
0.20
0.20
0.20
0.20
W1
0.00
0.20
0.40
0.50
0.60
0.80
1.00
W2 E(Ri)
1.00 0.20
0.80 0.18
0.60 0.16
0.50 0.15
0.40 0.14
0.20 0.12
0.00 0.10
Asset
1
2
Variance Std.
(δ2)
Dev(δ)
0.0049
0.07
0.0100
0.10
Constant Correlation With Changing Weights
Computing the standard deviation for each using the portfolio standard
deviation, we have:
δport(f) = (0.0)2 (0.07)2 + (1.0)2 (0.10)2 + (2)(0.0)(0.1)(0.00)
= 0.10
δport(g) = (0.20)2 (0.07)2 + (0.80)2 (0.10)2 + (2)(0.20)(0.80)(0.00)
= 0.0812
δport(h) = (0.40)2 (0.07)2 + (0.60)2 (0.10)2 + (2)(0.40)(0.60)(0.00)
= 0.0662
δport(j) = (0.60)2 (0.07)2 + (0.40)2 (0.10)2 + (2)(0.60)(0.40)(0.00)
= 0.0580
δport(k) = (0.80)2 (0.07)2 + (0.20)2 (0.10)2 + (2)(0.80)(0.20)(0.00)
= 0.0595
Constant Correlation With Changing Weights
Summary where the correlation coefficient is 0.
Case
F
G
H
I
J
K
L
A1
0.10
0.10
0.10
0.10
0.10
0.10
0.10
A2
0.20
0.20
0.20
0.20
0.20
0.20
0.20
W1
0.00
0.20
0.40
0.50
0.60
0.80
1.00
W2 E(Ri)
1.00 0.20
0.80 0.18
0.60 0.16
0.50 0.15
0.40 0.14
0.20 0.12
0.00 0.10
E(δport)
0.1000
0.0812
0.0662
0.0610
0.0580
0.0595
0.0700
Three-Asset Portfolio
E(ri)
E(δi)
Wi
Stocks (S)
0.12
0.20
0.60
Bonds (B)
0.08
0.10
0.30
Cash Equivalent (C)
0.04
0.03
0.10
Asset Classes
The correlation coefficients are
RS,B = 0.25
RS,C = -0.08
RB,C = 0.15
E(Rport) = (0.60)(0.12) + (0.30)(0.08) + (0.10)(0.04) = 0.10
δ2port = [w2S2S + w2B2B + w2C2C] + [2wSwB ● ρSBSB] + [2wSwC ●ρSCSC] + [2wBwC ●ρBCBC]
= [(0.60)2(0.20)2 + (0.30)2(0.10)2 + (0.10)2(0.03)2 ] + [2(0.60)(0.30)(0.25)(0.20)(0.10)] +
= [2(0.60)(0.10)(-0.08)(0.20)(0.03)] + [2(0.30)(0.10)(0.15)(0.10)(0.03)]
= 0.015309 + 0.0018 - 0.0000576 + 0.000027
= 0.170784
δport = √0.170784 = 0.1306 0r 13.06%