Portfolio Theory
Was developed by Harry Markowitz in 1952 and later improved by other economists,
e.g., Sharpe and Linter
John Burr Williams’ 1938 book, ‘The Theory of Investment Value’, proposed that the
value of a stock should equal the current value of future dividends, i.e., the Dividend
Discount Model. This thinking resonated well with the thinking of the time because
investors were mostly interested in acquiring good stocks at the best price.
During Williams’ time, information circulated slowly, and the markets were weakly
regulated.
The above environment increased the perception that investing was a gamble for the
wealthy.
Markowitz came across Williams’ book while searching for a doctoral thesis topic. He
discovered a gap in the book, i.e., no consideration for risk in a particular investment, and
this inspired him to write an article, ‘Portfolio Selection’, which was published in the
March 1952 Journal of Finance.
Markowitz’s work languished in dusty libraries for a decade, mostly because it contained
too many graphs and numerical doodles
Markowitz devised a model of reducing risk through investing in a diversified portfolio.
The ideal form of diversification is to engage in activities that behave in opposite
directions
There are two conditions necessary for total elimination of portfolio risk, and these are: 2
investments with a perfect negative relationship/correlation and proper weightings of the
investments.
We are going to demonstrate how portfolio risk can be reduced through diversification.
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Example
Consider Investment C and D and their probability distributions
Scenario Probability of Return on Investment Return on Investment
scenario C D
Boom 30% 20% -10%
Normal 50% 0% 0%
Recession 20% -20% 45%
An equal amount of capital was invested in each asset.
Find the risk of the portfolio.
Points to remember
The risk of an investment refers to the probability of its returns deviating from what is
expected; hence, it is a standard deviation.
For portfolio standard deviation to be calculated, the following metrics must be
determined first: Weights of the investment in each asset, variance of each asset in the
portfolio, and the portfolio covariance
Workings
1. Expected rate of return for the portfolio, ER p
To get the portfolio, ER p, we first find the expected rate of return for each asset and
multiply it by the weight invested in it, in our case 50%, we sum up the weighted
expected returns.
For an individual asset, the formula we use to calculate the expected rate of return is:
ER C = ∑ Ex
Where, ER C is the expected rate of return from Investment C.
E represents the actual return on the asset
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x represents the scenario’s probability of occurrence
Investment C: ERC = (0.3×20 ¿+ ( 0.5 ×0 ) + ( 0.2 ×−20 )=¿ 2%
Investment D: ER D= (0.3×−10 ¿+(0.5 ×0)+(0.2× 45)=6 %
Portfolio Expected rate of return, ER p= ( 2× 0.5 ) + ( 6 ×0.5 )=4 %
2. Variances of the assets
Var, σ 2= ∑ ¿¿ ¿ p
Where p stands for the probability of occurrence of the scenario
When applying the formula in tabular fashion
For Investment C:
Return Expected Difference Difference Probability Diff weighted by
Return squared probability
(3) (5)
(1) (2) (4) (6)
(3¿2
(1)−¿(2) (4)×(5)
20 2 18 324 0.3 97.2
0 2 2 4 0.5 2
−20 2 −22 484 0.2 96.8
Variance, σ 2= 196
Standard deviation for Investment C, σ P = 1960.5 =14%
For investment D, we deliberately decide to apply the formula differently:
2 2
σ =¿0.3 + (0−6 ¿ ×0.5+ ¿=399
Standard deviation for Investment D, σ D=3990.5 =19.97%
3. Covariance between the asset
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Covariance measures how the assets move together. A negative value shows that
the assets move in opposite directions, i.e., an inverse relationship, and is
desirable in diversification. It means gains made in one asset will offset losses
suffered in the other asset; hence, the expected portfolio returns will not fluctuate
much, and in fact, they will be constant if the inverse relationship is a perfect one.
A positive value means the assets move in the same direction, i.e. positive
relationship. It means that in good times, both assets realize profits, but in bad
times, the whole portfolio suffers losses. This then means that the portfolio’s
returns fluctuate much, thereby posing much uncertainty to risk-averse investors.
A covariance of zero means that there is no co-movement between the assets.
COV CD =∑ ¿ ¿]
Where, RC is the return realized from Investment C
ERc is the expected return from Investment C.
R D is the return realized from Investment D.
ER D is the return expected from Investment D.
piis the probability of any pair of values occurring.
There are several ways of applying the formula; here, we elect to apply it in tabular form:
RC ER C RD ER D (RC −ERC ) ( Product Probability Product
R D−ER D ¿ weighted
by
probability
20% 2% −10 % 0.06 +18 % −16 % ∓ 288 0.3 −86.4
0% 2% 0 0.06 −2 % −6 % +12 0.5 +6
−20 % 2% + 45 % 0.06 −22 % +39 % −858 0.2 −171.6
−252
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A covariance of -252 only informs us of a negative relationship, but not its strength. To
see the strength of the relationship, we should transform the COV into a standard
measurement called the correlation coefficient, r.
The correlation coefficient’s scale of measurement is −1 to +1 inclusive, with −1
showing a perfect negative relationship, which eliminates risk, and a +1 showing a
perfect positive relationship, which increases the risk to its maximum.
COV CD
Correlation coefficient for Investments C &D, r CD =
σ C ×σ D
, simply put, the COV for the 2
assets is expressed as a fraction of the product of the standard deviations of the 2 assets
making up the portfolio.
COV CD −252
r CD = = = −0.901352028 ≈−0.9013
σ C ×σ D 14 ×19.97
Comment:
−0.9013 is close to −¿1, so it shows a strong inverse relationship between the performance of
Investments C and D. The portfolio is an ideal one because it is reducing risk through
diversification.
4. Portfolio variance and standard deviation
Portfolio variance = W 2c σ 2c +W 2D σ 2D +2 COV CD W C W D
Where, W C is the proportion of capital invested in Investment C
W D is the proportion of capital invested in Investment D
2
σ C is Investment C’s variance and,
2
σ D is Investment D’s variance
∴ Portfolio variance = 0.52 ¿+0.52 ¿+2(-252)(0.5(0.5)
= 22.700225
≈ 22.70
The variance is a squared value; therefore, it has to be unsquared to get the standard deviation.
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Portfolio standard deviation, σ p= √ 22.70 = 4.76%
Comment:
The portfolio risk, as shown by its standard deviation, is 4.76% which is below the standard
deviation of either asset making up the portfolio (14% for Investment C and 19.97% for
Investment D). We conclude that diversifying in assets with a negative relationship reduces
portfolio risk.
5. The optimal portfolio
An optimal portfolio is simply the one that offers the lowest level of risk by
investing more in a less risky asset, i.e., one with the lower standard deviation,
and in our case, it is Investment C. to calculate the proportion of capital which
should be invested in Investment C to get an optimal portfolio we use the formula:
2
σ D −2COV CD
α C= 2 2
σ C + σ D −2 COV AB
399−2 (−252)
=
(196+399)−2(−252)
= 82.1656051%
≈ 82%
Comment:
The lowest portfolio risk can be obtained by investing 82% in Investment C and 18% in
Investment D.
Evaluate the effectiveness of the portfolio theory in a real-world environment.
Lecture: Chitsinde E- 0773426836
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