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S&P500 with IBM. Beta preserves this ordering, too, because both covariances are divided by the
same variance (the variance of rS&P500 ). It is only the correlation that would have misleadingly
˜
indicated that Sony would have been the better diversi¬er. This, then, is our main insight:



Important: The beta of any security i with respect to our portfolio (called βi,Y )
is a measure of the security™s risk contribution to the portfolio, because it properly
takes scale into account. The lower the beta of security i with respect to portfolio Y,
the better security i works at diversifying portfolio Y™s overall risk.



The covariance would have worked equally well, but it is more di¬cult to intuitively interpret. The other two measures.
Correlation is not suitable as a quantitative measure, because it fails to recognize investment
scales.
The remainder of this section just provides some additional intuition: Beta has a nice slope Beta has a nice slope
interpretation, too.
interpretation. If you graph the rate of return on your overall portfolio Y (e.g., the S&P500) on
the x axis, and the rate of return on the new security i (e.g., Sony) on the y axis, then beta
represents the slope of a line that helps you predict how the rate of return of security i will
turn out if you know how the rate of return on your portfolio turned out.
Figure 15.3 shows three securities with very di¬erent betas with respect to your portfolio. In Beta can be visually
judged by graphing the
graph (c), security POS has a positive beta with respect to your portfolio. You would therefore
rate of return on a
expect security POS to not help you much in your quest to diversi¬y: when your portfolio Y security against the rate
does better, so does U. In the ¬gure, U has a beta with respect to your portfolio Y of 3. of return on the overall
portfolio. It is the slope
This indicates that if Y were to ear an additional 5% rate of return, the rate of return on the
of the line.
security POS would be expected to change by an additional 15%. More importantly, if your
portfolio Y hits hard times, asset POS would be hit even harder! You would be better o¬
purchasing only a very, very small amount of such a security; otherwise, you could quickly end
up with a portfolio that looks more like POS than like Y. In contrast, in graph (a), you see that if
you purchased security NEG would expect this NEG to help you considerably in diversi¬cation:
when your Y does worse, NEG does better! With a beta of ’3, the security NEG tends to go
up by an additional 15% when the rest of your portfolio Y goes down by an additional 5%.
Therefore, NEG provides excellent “insurance” against downturns in Y. Finally, in graph (b),
the security ZR has a zero beta, which is the case when ZR ™s rates of return are independent
of P™s rates of return. You already know that securities with no correlation can help you quite
nicely in helping diversify your portfolio risk.

The intercept in Figure 15.3 is sometimes called the “alpha,” and it can measure how much
Side Note:
expected rate of return the security is likely to o¬er, holding its extra risk constant. For example, if stocks i
and j have lines as follows

E (˜i ) ≈ +15% + 1.5 · E (˜P ) ,
r r

(15.19)
E (˜j ) ≈ ’10% + 1.5 · E (˜P ) ,
r r

E (˜k ) ≈ + βk,P · E (˜P ) ,
r ±k r
the ± of stock i is 15%, while the ± of stock j is “10%. It appears that stock i o¬ers a holder of portfolio P a lot
of positive return compared to j, holding its exposure to portfolio P constant. Naturally, you would like stocks
with high alphas and low betas”but such opportunities are di¬cult to ¬nd, especially if you already hold a
reasonably good (well-diversi¬ed) portfolio.
¬le=diversi¬cation-g.tex: LP
366 Chapter 15. The Principle of Diversi¬cation.




Figure 15.3. Three Di¬erent Beta Measures of Risk for Security i.


*




Historical Rate of Return on Security n
**
**
* *
*
* ** *
* ** ***




5
* *
* ***
*
* **
*
**
*
* *
* * **
* * ** * *
* *
* ** * * *
**
**
(a) *




0
* * **
* ** * * * * *
** *
**
*** ** * *
* *
* **
** *
* * *
** * *
** *




’5
*
** *
* *

’1 0 1 2 3

Historical Rate of Return on Your Portfolio P




*
Historical Rate of Return on Security z

6




*

*
*
4




* * *
* *
* ** **
**
** * *
*
** ** *
2




*
* ** **
*
**
** ** *
* *
* **
*
**
(b) * *
*
*
* ** *
* ** *
0




*
** * ***
*
* ** * * *
** * *
*
* ** *
* * * * ** ** *
* *
* *
’2




*
* *
*
* *
’4




*
*
*

’1 0 1 2 3

Historical Rate of Return on Your Portfolio P




*
*
Historical Rate of Return on Security p




** *
*
*
**
*
** ** *
*
5




* *
*
** * ***
* * * **
* *
*
* **
*
* *
** * *
* * * ** * ** *
*
(c) * * ** *
0




*
*
**
** *
* * **
*
* * ** ** *
* * ** *
* * *
**
* *
**
*
*
* **
’5




** *
*
* * *
*
*

’1 0 1 2 3

Historical Rate of Return on Your Portfolio P


The historical betas of these three securities, n, z, and p, with respect to your portfolio P are based on returns whose
true betas with respect to your portfolio P are ’3, 0, and +3 in the three graphs, respectively. Historical data is
indicative, but not perfect in telling you the true beta.
¬le=diversi¬cation-g.tex: RP
367
Section 15·3. Diversi¬cation: The Formal Way.

Solve Now!
Q 15.1 Assume that every security has a mean of 12%, and a standard deviation of 30%. Further,
assume that each security has no covariation with any security. What are the risk and reward
of a portfolio of N stocks that invests equal amount in each security?


Q 15.2 Assume that every security has a mean of 12%, and a standard deviation of 30%. Further,
assume that each security has 0.0025 covariation with any security. What are the risk and reward
of a portfolio of N stocks that invests equal amount in each security? If you ¬nd this di¬cult,
solve this for 2 stocks. Can you guess what the risk is for a portfolio of in¬nitely many such
stocks?


Q 15.3 Compute the variance of a two-stock portfolio if the two securities are not perfectly posi-
tively, but perfectly negatively, correlated.


Q 15.4 You own a $1,000 portfolio P, whose expected rate of return has a mean of 10% and a
standard deviation of 20%. You are considering buying a security Q that has a mean of 15% and
a standard deviation of 50%. The correlation between the rates of return on P and Q is 20%.

(a) What is the covariance between the rate of return of P and Q?

(b) What is the beta between the rate of return on P and Q?

(c) Consider purchasing $1,000 in Q. What would the portfolio risk be?

A new security, named N has appeared. It has a mean of 150% and a standard deviation of
500%, and the same 20% correlation with P. (Sidenote: Such a security could be created by a
fund that borrows in order to purchase more than 100% in Q.)

(d) What is the covariance between the rate of return of P and N?

(e) What is the beta between the rate of return on P and N?

(f) Consider purchasing $1,000 in N. What would the portfolio risk be?

(g) Q and N have equal correlation with portfolio P. Does it follow that they would both be
equal risk-contributors, if added to the portfolio?




Anecdote: Value-At-Risk (VAR)
The latest in risk measurement techniques among banks and other ¬nancial institutions”and a great step
forward if executed correctly”is VaR (Value-at-Risk). It often replaces older risk-scoring systems, in which (for
example) all commercial loans received one score, government loans another, etc. The goal of Value at Risk
is to compute the risk (standard deviation) when all investment (loans) are evaluated in a portfolio framework.
Value at Risk can come to very di¬erent conclusions than these older systems, especially if payo¬s to loans are
very negatively or very positively correlated.
¬le=diversi¬cation-g.tex: LP
368 Chapter 15. The Principle of Diversi¬cation.

15·4. Does Diversi¬cation Work in the Real World?

You now understand the theory. But can you make it work in the real world?


15·4.A. Diversi¬cation Among The Dow-Jones 30 Stocks

To see whether portfolio diversi¬cation works in the real world, you should look at some speci¬c
Diversi¬cation depends
on the speci¬c portfolio. securities. Clearly, the degree to which diversi¬cation works must depend on the weights of
the speci¬c securities you look at within the context of your portfolio.
You can get an intuitive feel for the e¬ectiveness of diversi¬cation in the U.S. stock markets if
The risk of investing in a
Dow-Jones 30 Stock. you look at the 30 stocks constituting the Dow-Jones 30. The Dow-Jones company has chosen
30 stocks in its Dow-Jones 30 Index (see also Section 12·3.C, Page 297) to avoid industry and
¬rm-type concentration. However, the Dow-Jones contains only very large ¬rms, both in market
capitalization and sales.
Table 15.1 shows the risks (standard deviations of the rate of returns) for the 30 Dow-Jones
The average stock in the
Dow-Jones has a risk of stocks, measured either from January 1997 to October 2002 or from January 1994 to October
about 30% to 40%; the
2002. Their risks ranged from about 18% to 50% (annualized); the typical stock™s risk averaged
overall portfolio has a
about 30% from 1994 to 2002, and 35% from 1997 to 2002. Just investing in one randomly
risk just about half.
chosen stock is fairly risky. However, when you compute the risk of the index itself, the index™s
risk was only 16.5% from 1994 to 20002 and 18.7% from 1997 to 2002. Naturally, this is still
pretty risky. But the Dow-Jones index risk is only about half the risk of its average stock.
Are the returns of these 30 stocks positively correlated? Note that 18% is only about half of
Inferring correlations.
the 35% risk that would obtain if all the securities™ returns were perfectly positively correlated.
So, diversi¬cation works: the correlation among the Dow-Jones 30 stock returns is not close to
+1. However, the correlation among the Dow-30 stocks is also not zero. Let us do some back-
of-the-envelope calculations. If the Dow-Jones 30 Index were an equal-weighted portfolio (it is
not!) of uncorrelated securities (it is not!), you would have expected it to have a risk of about
1/30 · 35%≈ 6.4% per year. Instead, the risk of the Dow-Jones portfolio is 2.8 times as high at
18% per year. In sum, this evidence suggests that these 30 stocks move together, i.e. that they
tend to have mutual positive correlations. This reduces the e¬ectiveness of diversi¬cation
among them. This is actually a broader e¬ect. When the U.S. stock market does well, most
stocks do well at the same time (and vice-versa).



Important: The Dow-Jones 30 Market Index Portfolio has a risk of about 15%
to 20% per year. (Broader U.S. stock market indexes, like the S&P500 Index, tend
to have slightly lower risks.)
The 30 component stocks in the Dow-Jones 30 Index are mutually positively cor-
related, which limits the e¬ectiveness of diversi¬cation”but not so much as to
render diversi¬cation useless.



The square root in the portfolio standard deviation formula suggests that most of the diversi-
Make sure not to buy
just tech stocks, or just ¬cation typically comes from the ¬rst 10 to 50 stocks. Therefore, it is more important to be
growth stocks, or ...
suitably diversi¬ed across di¬erent types of stocks (to avoid mutual positive covariances) than
it is to add every single possible stock to a portfolio. If you holds the Dow-Jones 30 stocks if you
want to further diversify using U.S. stocks, you should consider adding small or high-growth
¬rm stocks to your portfolio (or, better, invest in a mutual fund that itself holds on many small
¬rms).
¬le=diversi¬cation-g.tex: RP
369
Section 15·4. Does Diversi¬cation Work in the Real World?.




Table 15.1. Risk and Reward for Dow Jones Constituents, Based on Monthly Rates of Returns,
Then Annualized.

about 10 years about 5 years
1994/01-2002/10 1997/01-2002/10
Asset Mean StdDev Mean StdDev
alcoa 18.4% 36.0% 15.1% 41.3%
american express 21.4% 27.9% 17.3% 31.4%
’2.9%
boeing 8.9% 30.1% 34.6%
citigroup 27.7% 33.6% 24.9% 37.4%
caterpillar 14.0% 31.2% 9.6% 34.0%
du pont 12.2% 26.2% 4.4% 29.0%
disney 6.7% 28.8% 0.3% 32.2%
’5.5%
eastman kodak 4.5% 30.0% 34.6%
general electric 17.1% 25.1% 12.8% 28.4%
general motors 5.2% 32.6% 5.1% 36.7%
home depot 18.5% 31.3% 22.8% 35.1%
honeywell 11.2% 37.9% 6.0% 44.5%
hewlett packard 14.8% 42.9% 4.0% 48.2%
ibm 25.8% 36.0% 20.2% 39.6%
intel 28.7% 46.7% 15.8% 53.1%
international paper 7.5% 31.6% 6.0% 36.0%
johnson and johnson 23.2% 23.9% 19.5% 26.4%
jp morgan 15.0% 36.8% 5.9% 42.5%
coca-cola 13.1% 26.5% 3.7% 30.7%
mcdonalds 6.2% 25.6% 0.6% 28.8%
3m 14.8% 23.2% 12.8% 26.0%
philip morris 13.5% 29.9% 6.9% 33.0%
merck 19.2% 28.6% 12.1% 32.0%
microsoft 35.8% 42.6% 28.4% 50.0%
proctor and gamble 16.7% 24.7% 13.7% 28.4%
sbc communications 9.4% 28.9% 8.1% 34.4%
’4.4% ’5.7%
att 36.5% 40.3%
united technologies 21.9% 29.2% 18.1% 34.1%
wal-mart 20.6% 28.8% 31.5% 31.1%
exxon 10.0% 16.8% 7.1% 18.3%
Average 15.3% 31.0% 10.6% 35.0%
Typical (Median) 14.8% 29.9% 8.1% 34.1%

dow jones 30 index 10.5% 16.5% 6.3% 18.7%

For comparison, the S&P500 index had a mean of 8.6% (risk of 16.1%) from 1994“2002, and a mean of 4.8% (risk of
18.5%) from 1997”2002.

In Chapter 13·6.A (Page 324), I claimed that historical standard deviations of rates of return
Side Note:
tend to be relatively stable. This table shows that, although the riskiness of ¬rms does change over time (it is
di¬erent over 5 years and 10 years), it does change only slowly, even for the individual Dow-30 stocks. (There
would be even more stability if you considered asset class portfolios instead of just stocks.) This stability gives
us con¬dence in using historical risk measures (e.g., standard deviations) as estimates of future risk.
¬le=diversi¬cation-g.tex: LP
370 Chapter 15. The Principle of Diversi¬cation.

So, what matters more in determining portfolio variance: covariances or variances? The num-
A short digression:
covariance matters more ber of covariance terms in the portfolio risk formula 15.12 (Page 362) increases roughly with the
than variance!
square of the number of securities”by N 2 ’ N to be exact. The number of variances increases
lineary”by N. For example, for 100 securities, there are 100 variance terms and 9,900 covari-
ance terms (4, 450 if you do not want to double-count the same pairwise covariance). It should
come as no surprise that as the number of securities becomes large, the risk of a portfolio is
determined more by the covariance terms than by the variance terms.
The Dow-Jones Index is not alone in bene¬ting from the e¬ects of diversi¬cation. Academic
A short digression:
covariance matters more research has shown that if you look at the average stock on the N.Y.S.E., about 75% of its risk
than variance!
can be diversi¬ed away (i.e., disappears!), while the remaining 25% of its risk cannot be diversi-
¬ed away. Put di¬erently, undiversi¬able co-movements among all stocks in the stock market
are responsible for about one-quarter of the typical stocks™ return variance; three-quarters are
idiosyncratic day-to-day ¬‚uctuations, which average themselves away if you hold a highly di-
versi¬ed stock market index like portfolio.


15·4.B. Mutual Funds

Diversi¬cation clearly reduces risk, but it can also be expensive to accomplish. How can you
There are too many
stocks for you to buy purchase 500 securities with a $50,000 portfolio? The transaction costs of purchasing $100
them all.
in each security would be prohibitive. With about 10,000 publicly traded equities in the U.S.
stock market, even purchasing just $1,000 in every stock traded would require $10 million,
well beyond the ¬nancial capabilities of most retail investors.
Mutual funds, already mentioned in Section 12·3.B, come to the rescue. As already described
Mutual funds are
investment vehicles to in Chapter 12, a mutual fund is like a ¬rm that consists of nothing but holdings in other assets,
accomplish
usually ¬nancial assets. In a sense, a mutual fund is a large portfolio that can be purchased as
diversi¬cation.
a bundle. However, there is one small catch”isn™t there always? On the one hand, investing
in a mutual fund rather than in its individual underlying assets can reduce your transaction
costs, ranging from the time necessary to research stocks and initiate transactions, to the direct
trading costs (the commission and bid-ask spread). But, on the other hand, mutual funds often
charge a variety of fees and may force you to realize taxable gains in a year when they would
rather not.


15·4.C. Alternative Assets

A common error committed by investors is that they focus only on the diversi¬cation among
Other Assets Are Equally
or More Important. stocks traded on the major U.S. stock exchanges. But there are many other ¬nancial and non-
¬nancial instruments that can aid investors in diversifying their risk. Because these instruments
are often less correlated with an investor™s portfolio than domestic U.S. stocks, these assets can
be especially valuable in reducing the portfolio risk. Among possible investment assets are:

• Savings accounts.
• Bonds.
• Commodities (such as gold).
• Other futures (such as agricultural commodities).
• Art.
• Real estate.

Anecdote: Portfolios of Finance Professors
Many ¬nance professors invest their own money into passively managed, low-cost mutual funds, often Index
Funds, which buy-and-hold a wide cross-section of assets and avoid active trading. They also require minimal
investment selection abilities by their managers, and usually incur minimal trading costs. Vanguard funds are
particularly popular, because Vanguard is not only the largest mutual fund provider”though neck-in-neck with
Fidelity”but it also does not even seek to earn a pro¬t. It is a “mutual” mutual fund, owned by the investors
in the funds themselves.
¬le=diversi¬cation-g.tex: RP
371
Section 15·4. Does Diversi¬cation Work in the Real World?.

• Mortgage and corporate bonds.
• Labor income.
• International stocks.
• Hedge funds.
• Venture and private equity funds.
• Vulture and bankruptcy funds.

In addition, if you are a smart investor, you would not only consider the diversi¬cation within
your stock portfolio, but across your entire wealth. Your wealth would include your house, your
education, your job, etc. Many of these alternative investments could also have low covariation
with your overall wealth.

Side Note: Some of the above mentioned asset categories may be better held in modest amounts. Further-
more, some of these areas do not resemble the highly liquid, fair, and e¬cient ¬nancial markets that U.S. stock
investors are used to. Instead, some are rife with outright scams. Therefore, it might be wise to hold such
assets through sophisticated and dedicated professional investors (mutual funds), who have the appropriate
expertise.

Solve Now!
Q 15.5 In Table 15.1, Exxon had the lowest standard deviation among the Dow-30 stocks. Why
not just purchase Exxon by itself?


Q 15.6 How much does diversifying over all 500 stocks in the S&P500 help in terms of risk
reduction relative to investing in the 30 stocks of the Dow Jones-30?


Q 15.7 Why do mutual funds exist?


Q 15.8 Should you just own U.S. stocks?


Q 15.9 Does the true value-weighted market portfolio just contain stocks?
¬le=diversi¬cation-g.tex: LP
372 Chapter 15. The Principle of Diversi¬cation.

15·5. Diversi¬cation Over Time

Many investors think of diversi¬cation across securities within a portfolio, but do not realize
Explaining the Figure
Illustrating Time that diversi¬cation can also work over time”although the sidenote below explains why aca-
Diversi¬cation.
demics are divided on this issue. Figure 15.4 illustrates this point by showing the risk and
reward if you had invested in the S&P 500 from 1990“2002 for x consecutive trading days. The
left graph shows your average daily rate of return; the right side your total compounded rate
of return. For example, the right ¬gure shows that if you had held an S&P500 portfolio during
a random 25 day period (about a month), you would have earned a little less than 1%, but with
a risk (standard deviation) of about 5%. Graph (a) quotes this in average daily terms: over 25
days, a 1% mean and 5% risk was an average reward of about 0.04%/day with a risk of about
0.2%/day.
Note from graph (b) how the risk-reward trade-o¬ changes with time. Over an investment
Time Diversi¬cation at
Work! horizon of a full year (255 trading days), you would have expected to earn about 10% with
a risk of about 15%. The risk would have been 1.5 times the reward. In contrast, over an
investment horizon of one day, you would have expected to earn about 0.04% with a risk of
about 1.1%. Your risk would have been about 30 times your reward! If you stare at graph (b),
you should notice that the reward goes up a little more than linearly (the compounding e¬ect!),
while the risk goes up like a parabola, i.e., a square root function. Indeed, this is the case, and
the rest of this section shows why.
Recall that a portfolio that earns rt=1 in period 1, rt=2 in period 2, and so on until period T
˜ ˜
Over periods shorter
than a few years, the r
(˜t=T ), will earn an overall rate of return of
ordinary portfolio
return is roughly the
sum of time period rt=0,t=T = (1 + rt=0,1 ) · (1 + rt=1,2 ) · . . . · (1 + rt=T ’1,T ) ’1
˜ ˜ ˜ ˜
returns. (15.20)
≈ rt=0,1 + rt=1,2 + . . . + rt=T ’1,T + many multiplicative r terms .
˜ ˜ ˜ ˜

The multiplicative terms re¬‚ect the power of compounding”which clearly matters over many
years”but perhaps less so over periods of just a few months or years. After all, even monthly
returns may typically be only on the order of 1%, so the multiplicative term would be on the
order of 1% · 1% = 0.01%.
Now, if you forget about the small multiplicative terms, you already know that if you expect to
Stock returns have to be
about independent earn 1% per period, then over x periods you expect to earn x%:
across time-periods to
avoid great
(15.21)
money-making E (˜t=0,t=T ) = T · E (˜t ) .
r r
opportunities. This
determines the riskiness
For the variance, let us assume that the return variance in each time period is about the same
of portfolios over
and can just be called Sdv(˜t ). Further assume that the covariance terms among returns in
r
multiple periods.
di¬erent times (e.g., rt=0,1 and rt=1,2 ) are about zero. After all, if they were not, this would
˜ ˜
mean that you could predict future returns with past returns. Su¬ce it to say that this sort of
prediction is not an easy task”if it were, you would quickly become rich! Put this all together
and use the variance formula 15.22:

Var(˜t=0,T ) = Var rt=0,1 + rt=1,2 + . . . + rt=T ’1,T
r ˜ ˜ ˜

(15.22)
+ many covariance terms, all about zero

≈ T · Var(˜t )
r ,

It follows that

(15.23)
Sdv(˜t=0,T ) ≈ T · Sdv(˜t ) .
r r

But this is just the relationship in the graph: The risk increases with the square root of time!
¬le=diversi¬cation-g.tex: RP
373
Section 15·5. Diversi¬cation Over Time.




Figure 15.4. Average Risk and Return over Time in the S&P500, 1990“2002
1.0
Rate of Return, Sdv and Mean, in Percent

0.8
0.6
0.4
0.2




x
xx x
x x
x
x
x
0.0




0 50 100 150 200 250

Holding Period in Days
(a)
20
Rate of Return, Sdv and Mean, in Percent

15




+
10




+
5




+

+
+
++ +
+
0




0 50 100 150 200 250

Holding Period in Days
(b)


The x-axis is the investment horizon, i.e., the number of consecutive investment days. The two top series (lines and
dots) are the standard deviation, the two bottom series are the means. Lines are the theoretical values computed
with the formulas below; circles and pluses are from simulated actual investment strategies.
¬le=diversi¬cation-g.tex: LP
374 Chapter 15. The Principle of Diversi¬cation.

Formula 15.23 is commonly used to “annualize” portfolio risk. For example, if an investment
Annualization
strategy has a monthly risk of 5% (i.e., the standard deviation of its rate of return), and the
question is what kind of risk such an investment strategy √ would have per annum, you can
compute the implied annual standard deviation to be about 12 · 5% ≈ √ 17%. Conversely, if the
¬ve-year variance is 39%, then the annualized variance is 17%, because 5 · 17% ≈ 39%.



Important: A quick and dirty (and common) method to annualize portfolio risk
(the standard deviation of the rate of return) is to multiply the single-period rate
of return by the square root of the number of periods. For example, the annual

portfolio risk is about 12 or 3.5 times as high the monthly risk.
The most common method, on Wall Street and in academia, is to compute risk and
reward from monthly rates of return, but to report their annualized values.



The most commonly used measure of portfolio performance is the Sharpe-ratio, named after
The Sharpe-ratio is a
measure of risk-reward Nobel Prize Winner William Sharpe. It is the expected rate of return of a portfolio above and
performance.
beyond the risk-free rate of return (rF ), divided by the standard deviation of the rate of the
return,
E (˜P ’ rF ) E (˜P ) ’ rF
r r (15.24)
Sharpe Ratio = = .
Sdv(˜P ’ rF ) Sdv( rP )
r ˜
Be aware that the Sharpe-ratio depends on the time interval that is used to measure returns. In
the example with 1% monthly mean and 5% monthly standard deviation, if the risk-free rate were
6% per year, the Sharpe-Ratio of the portfolio would be about (1%-0.5%)/5% = 0.1 if measured over
a one month time interval; (12%-6%)/17% = 0.35 if measured over a year; and (60%-30%)/39% = 0.77
if measured over ¬ve years. Although the Sharpe ratio makes it clear how mean and standard
deviation change with di¬erent time horizons, it is just a measure”it does not mean that you
are better o¬ if you hold stocks over longer time horizons. More importantly, you should be
warned: although the Sharpe ratio is intuitively appealing and although it is in widespread use,
it has the near-fatal ¬‚aw that it can easily be manipulated. You will ¬nd this out for yourself
in Question 15.16.

The overall division of assets between stocks and bonds is often called asset allocation. Many
Side Note:
practitioners suggest that you should put more of your money into risky stocks when you are young. Over the
very long run”and young people have naturally longer investment horizons”the expected rate of return vs.
risk relationship looks more favorable than it does over shorter investment horizons. After all, the mean goes
up with time, while the risk goes up only with the square-root of time. This means that the average rate of
return is less risky when you are young than when you are old.
But academics are divided on this advice. Some point out that the portfolio rate of return is the product of the
individual returns:
(15.25)
rt=0,T = (1 + r0,1 ) · (1 + r1,2 ) · ... · (1 + rT ’1,T ) ’1 .
˜ ˜ ˜ ˜
It should not matter whether you choose risky stocks over safer bonds at time t = 0 (when you are young) or at
time t = T (when you are old). That is, even though it is true that the risk of the average rate of return declines
over longer horizon, you should not be interested in the average rate of return, but in the total rate of return.
This argument suggests that your time horizon should not matter to your asset allocation.
Even more sophisticated arguments take into account that you can adjust better (e.g., by working harder) when
you are young if you experience a bad portfolio return; or that the stock market rate of return may be mean
reverting. That is, the market rate of return may be negatively correlated with itself over very long time periods,
in which case the long-run risk could be a little lower than the short-run risk.

Solve Now!
Q 15.10 What is a reasonable assumption for stock return correlations across di¬erent time
periods?


Q 15.11 Table 14.4 on Page 346 shows that the S&P500 has an annual standard deviation of
about 20% per year. As of November 2002, the S&P500 stood at a level of about 900. What
would you expect the daily standard deviation of the S&P500 index to be? Assume that there are
about 255 trading days in a year.
¬le=diversi¬cation-g.tex: RP
375
Section 15·5. Diversi¬cation Over Time.

Q 15.12 Assume you have a portfolio that seems to have a monthly standard deviation of 5%.
What would you expect its annual standard deviation to be?


Q 15.13 If the risk-free rate over the 1997-2002 sample period was about 3% per annum, what
was the monthly and what was the annual Sharpe-ratio of the Dow-Jones 30 index?


Q 15.14 Assume you have a portfolio that seems to have had a daily standard deviation of 1%.
What would you expect its annual standard deviation to be? Assume there are 255 trading days
in a year.


Q 15.15 The S&P500 was quoted in the ¬rst two weeks of June 2003 as

06/02/2003 967.00 06/06/2003 987.7606/12/2003 998.51
06/03/2003 971.56 06/09/2003 975.9306/13/2003 988.61
06/04/2003 986.24 06/10/2003 984.84
06/05/2003 990.14 06/11/2003 997.48

Compute the mean and standard deviation of annual returns of a portfolio that would have
mimicked the S&P500. Based on these returns, what would you expect to be the risk and reward
if you held the S&P500 for one year? Assuming a risk-free rate of return of 3%/annum, what
would be the proper estimate for a Sharpe-ratio of daily, monthly, and annual returns? Is this
consistent with the statistics in Table 14.4? Can you speculate why?


Q 15.16 Consider an investment strategy that has returned the following four rates of return:
+5%, +10%, +5%, +20%. These are quoted above the risk-free rate (or equivalently assume the
risk-free rate is 0%.)

(a) What was its Sharpe-ratio?

(b) Throw away 5% of the rate of return in the ¬nal period only. That is, if you had $200 and
you had ended up with $240 (20%), you would now be throwing away $10. So you would
end up with $230 for a remaining rate of return of 15% only. (How easy is it to throw away
money?) What is the Sharpe-ratio of this revised strategy?

(c) Which is the better investment strategy?
¬le=diversi¬cation-g.tex: LP
376 Chapter 15. The Principle of Diversi¬cation.

15·6. Summary

The chapter covered the following major points:

• Diversi¬cation”investment in di¬erent assets”reduces the overall risk (standard devia-
tion).

• Diversi¬cation works better when assets are uncorrelated.

• The beta of a new asset (with respect to the existing portfolio) is a good measure of the
marginal contribution of the new asset to the risk of the overall portfolio.

• Mutual funds invest in many assets, and thereby reduce retail investors™ cost of diversi¬-
cation. However, they charge fees for this service.

• Diversi¬cation works over time, too. The portfolio reward (expected rate of return) grows
roughly linearly over time, but the portfolio risk (standard deviation) grows roughly with
the square-root of time. That is, if the expected rate of return is 1% per month, then
the T month expected rate of return is approximately T · 1%. If the standard deviation
of the rate of rate of return is 10% per month, then the T month standard deviation is

approximately T · 10%. The latter formula is often used to “annualize” risk.

• The Sharpe-ratio is the most common measure of investment strategy performance”
although an awful one. One relatively minor problem is that it depends on the investment
horizon on which it is quoted. A Sharpe-ratio based on annual returns is typically about

T higher than a Sharpe-ratio based on monthly returns.
¬le=diversi¬cation-g.tex: RP
377
Section 15·6. Summary.

Solutions and Exercises





1. The mean is 12%, the standard deviation is 30%/ N.
2. The mean is 12%. The variance now still has N variance terms, but N·(N’1) terms that are each 1/N·1/N·0.001.
Thus, the variance is now


Var(˜P ) = N · (1/N 2 · (30%)2 ) + N·(N ’ 1) · 1/N · 1/N · (0.0025) . (15.26)
r

Therefore, for 2 stocks, the variance is 1/2 · .09 + 2 · 1 · (1/4 · 0.0025) = 0.045 + 1/2 · 0.0025 = 0.04625.
Sdv = 21.5%. This is a little higher than the 21.2% from the previous question. For many stocks, N ’ ∞ is


= N · (1/N 2 · (30%)2 ) + N·(N ’ 1) · 1/N · 1/N · (0.0025)
Var(˜P )
r

≈ N · (1/N 2 · (30%)2 ) + 0.0025
(15.27)
1/N · (.09)
= + .
0.0025

lim Sdv(˜P ) = =
r .
0.0025 5%
N’∞


3.
Var( 1/2 · r1 + 1/2 · r2 )
Var(˜P ) =
r ˜ ˜

= 1/22 · Var( r1 ) + 1/22 · Var( r2 ) + 2 · 1/2 · 1/2 · Cov( r1 , r2 )
˜ ˜ ˜˜

(15.28)
1/4 · 0.01 + 1/4 · 0.01 + 2 · 1/2 · 1/2 · (’0.01)
=

1/2 · 0.01 ’ 1/2 · 0.01 = 0.00
=

Sdv(˜P ) = Var(˜P ) = 0%
r r .
Therefore, with perfect negative correlation and equal investment weights, diversi¬cation works perfect!
4.
(a) The covariance is the correlation multiplied by the two standard deviations. For the Q portfolio,


Cov(˜Q , rP ) = Correlation(˜Q , rP ) · Sdv(˜Q ) · Sdv(˜P )
r˜ r˜ r r
(15.29)
= 20% · 20% · 50% = 0.02 .

(b)
Cov(˜Q , rP )
r˜ 0.02 (15.30)
βQ,P = = = 0.5 .
Var(˜P )
r 0.04

(c) Therefore, the new portfolio with weights of 0.5 each is


= 0.5 · rP + 0.5 · rQ
r .
˜ ˜ ˜

Var(˜) = wP · Var(˜P ) + wQ · Var(˜Q ) + 2 · wQ · wP Cov(˜Q , rP )
r r r r˜

0.52 · (20%)2 + 0.52 · (50%)2 + (15.31)
= 2 · 0.5 · 0.5 · 0.02

= 0.01 + 0.0625 + 0.01 = .
0.0825

Sdv(˜) =
r .
28.7%
¬le=diversi¬cation-g.tex: LP
378 Chapter 15. The Principle of Diversi¬cation.

(d) For the N portfolio,


Cov(˜N , rP ) = Correlation(˜N , rP ) · Sdv(˜N ) · Sdv(˜P )
r˜ r˜ r r
(15.32)
= 20% · 20% · 500% = 0.2 .

(e)
Cov(˜N , rP )
r˜ 0.2 (15.33)
βrN ,˜P = = =5.
˜r
Var(˜P )
r 0.04

(f) Therefore, the new portfolio with weights of 0.5 each is


= 0.5 · rP + 0.5 · rN
r .
˜ ˜ ˜

Var(˜) = wP · Var(˜P ) + wQ · Var(˜Q ) + 2 · wQ · wP Cov(˜Q , rP )
r r r r˜

0.52 · (20%)2 + 0.52 · (500%)2 + (15.34)
= 2 · 0.5 · 0.5 · 0.2

= 0.01 + 6.25 + 0.1 = .
6.36

Sdv(˜) =
r .
252.2%


(g) Even though N has an equal correlation with portfolio P, its diversi¬cation aid is overwhelmed by its
scale: N is just a lot riskier than Q. Thus, the portfolio with N is de¬nitely more risky than the portfolio
with Q. This lesser diversi¬cation e¬ect is re¬‚ected in the (higher) covariance and the (higher) beta with
our portfolio P, but not in the (equal) correlation.


5. This is not a question you might necessarily know how to answer. However, it should get you to think. Exxon
is indeed a low-risk stock. However, it is also a low mean stock. You would have done a little better purchasing
the diversi¬ed Dow-30 index (remember: indexes do not count dividends!). However, it is reasonably likely
that in the future, the diversi¬ed Dow-Jones 30 index will have lower risk than Exxon. More than likely, Exxon
was just lucky over the sample period. For example, oil prices were relatively stable.
6. There is almost no di¬erence in risk between these two indexes. To reduce risk further, you should instead
invest in asset classes that are di¬erent from large corporate equities.
7. They allow investors with limited amounts of money to own large numbers of securities, without incurring
large transaction costs.
8. No. You should hold all sorts of other assets that do not correlate too highly with your existing portfolio
(wealth).
9. No. There are many alternative asset classes, such as commodities, art, and real estate, etc., which are part
of the market portfolio.


10. That they are zero. If they were not zero, it would mean that you could use past returns to predict future
returns. Presumably, this would allow you to earn extra returns.
11. The standard deviation is likely to be


Sdv annual (˜) ≈ T · Sdv daily (˜)
r r

(15.35)
≈ 255 · Sdv daily (˜)
r
20%

’ Sdv daily (˜) ≈
r .
1.25%

This corresponds to a typical daily movement of about 11 points. In statistical terms, if the S&P500 is about
normally distributed, about 2/3 of all days, it should move up or down no more than 11 points. In about 9/10
of all days, it should move up or down no more than 22 points.

12 · 5% ≈ 17.3%.
12.
13. For the annual Sharpe-ratio, you can use the numbers in the Table 15.1:

6.3% ’ 3.0%
(15.36)
Annual Sharpe Ratio = ≈ 0.18 .
18.7%
For the monthly Sharpe-ratio, you need to de-annualize the mean and√standard deviation. The excess mean
is 3.3%/12 ≈ 0.275%. The monthly standard deviation is about 18.7%/ 12 ≈ 5.4%. Therefore, the monthly

0.275%
(15.37)
Monthly Sharpe Ratio = ≈ 0.05 .
5.4%
¬le=diversi¬cation-g.tex: RP
379
Section 15·6. Summary.

255 · 1% ≈ 16.0%.
14.
15. Over 10 real days, you would have earned a compound rate of return of 988.61/967 ’ 1 = 2.2%. With 36.5
such 10-day periods over the year, the compound annual return would have been 124%. As to the daily rates
of return, they were 0.47%, 1.51%, 0.40%, “0.24%, “1.20%, 0.92%, 1.28%, 1.03%, “1.00%. The simple arithmetic
average rate of return was an even higher 0.35%/day (3.5% over the 10 days). Clearly, this was a great ten
days, not likely to repeat. So, you should not trust these 10 day means. The standard deviation is 0.97%/day.

This indicates an annualized standard deviation of about 365 · 0.97% ≈ 18.5%. (Over 10 days, the estimated
standard deviation is 3.07%/(10 days).) As is fairly common, the annualized standard deviation is fairly
reasonable. Finally a risk-free rate of return of 3% per annum is less than 0.01%./day. The Sharpe-ratio would
therefore be approximately (0.35% ’ 0.01%)/0.97% ≈ 0.35. Quoted in annual terms, the Sharpe-ratio would
√ √
be approximately (0.35% ’ 0.01%)·365/(0.97%· 365) = 365 · 0.35 ≈ 6.7. Quoted in monthly terms, it would

be 30 · 0.35 ≈ 1.9.
16.
(a) The average rate of return is 10%. The variance is 37.5%%. The standard deviation is 6.12%. Therefore,
its Sharpe-ratio is 1.63.
(b) The new strategy has rates of return of +5%, +10%, +5%, +15%. It is very easy to accomplish this”give
me the money. The average rate of return has declined to 8.75%. The standard deviation has declined
to 4.146%. Therefore, the Sharpe ratio is 2.11.
(c) Obviously, the second strategy of throwing money away is terrible. The fact that the Sharpe-ratio comes
out higher tells us that the Sharpe-ratio is an awful measure of portfolio performance. And, yes, the
Sharpe-ratio is indeed the most common fund performance measure in practical use. Sharpe ratio
manipulation is particularly pro¬table for portfolio managers whose returns until October or November
were positive. In this case, managers who want to maximize Sharpe ratios should try to avoid really
high positive rates of returns. If it “happens,” they can always bring down the return, e.g., by paying
themselves more money, or by buying illiquid securities and then marking them down to less than their
values.




(All answers should be treated as suspect. They have only been sketched, and not been checked.)
¬le=diversi¬cation-g.tex: LP
380 Chapter 15. The Principle of Diversi¬cation.
CHAPTER 16
The Ef¬cient Frontier”Optimally Diversi¬ed
Portfolios

How much of each security?
last ¬le change: Feb 5, 2006 (18:24h)

last major edit: Aug 2004




This chapter appears in the Survey text only.


You already know the following:

1. Diversi¬cation reduces risk.

2. The covariance between investment asset returns determines the e¬ectiveness of diversi-
¬cation. Beta is an equally good measure.

3. Because risk typically decreases approximately by the square root of the number of se-
curities in the portfolio, it is especially important to have at least a handful and better a
dozen of very di¬erent types of assets.

4. Publicly traded stocks are mutually positively correlated, but not perfectly so. This leaves
diversi¬cation a useful tool, but not a perfect one.

But you do not yet know the optimal portfolio weights that you should assign to individual
assets in their portfolios. For example, should you purchase equal amounts in every security?
Should you purchase relatively more of securities with higher expected rates of return, lower
variances, or lower covariances? This chapter answers these questions.




381
¬le=optimalp¬o-g.tex: LP
382 Chapter 16. The E¬cient Frontier”Optimally Diversi¬ed Portfolios.

16·1. The Mean-Variance Ef¬cient Frontier

You know that diversi¬cation reduces risk. Therefore, you know that investors like diversi-
¬cation”but this does not tell you how much your investors should purchase in each security.
It may be better to purchase 25% in A and 75% in B, rather than 50% in each. So how do
you determine generally good investment weights”and the best investment weights for you?
This question is the primary subject of this chapter, and the optimal portfolio is the force that
ultimately shapes the CAPM formula.


16·1.A. The Mean-Variance E¬cient Frontier With Two Risky Securities

You can now tackle the problem of ¬nding the best portfolio, starting with only two possible
investment securities.


Graphing Portfolios in Mean-Standard Deviation Space
With two securities, you have just one decision to make: how much to invest in the ¬rst security.
For two securities,
compute the expected (The remainder is automatically allocated to the second security). For example, assume that
rate of return and risk
each security has a risk (standard deviation) of 10%. The ¬rst security (call it A) has an expected
computations for
rate of return of 10%, and the second security (call it B) has an expected rate of return of 5%.
possible portfolios.
Should you ever choose to purchase the second security, which is so obviously inferior to the
¬rst? You already know that the second security helps you diversify risk”so the answer may
be yes”but let us work out the requisite trade-o¬s.
Recall the formulas for the portfolio risk and reward. The formula for the portfolio reward, or
Recap the formulas you
need. expected rate of return, is

E (˜P ) = wA · 10% + (1 ’ wA ) · 5%
r
(16.1)
E (˜P ) = wA · E (˜A ) + (1 ’ wA ) · E (˜B ) .
r r r

You have also worked out the formula for the portfolio risk

2
Sdv(˜P ) = wA · 0.01 + (1 ’ wA )2 · 0.01 + 2 · wA · (1 ’ wA ) · Cov(˜A , rB )
r r˜
(16.2)
2
Sdv(˜P ) = · Var(˜A ) + (1 ’ wA · Var(˜B ) + 2 · wA · (1 ’ wA ) · Cov(˜A , rB ) ,
)2
r wA r r r˜

where wA is any fractional weight that you may choose to invest in the ¬rst security. These
two formulas allow you to work with any combination portfolio between the two securities.
Begin by assuming a particular covariance, say, Cov(˜A , rB ) of ’0.0075. (Given the standard

You can compute the
possible portfolio deviations, this implies a correlation of ’75%.) What reward can you achieve and at what
choices.
risk? You have only one decision variable, the weight wA , but you can choose any weight you
desire. Table 16.1 shows various risk/reward “performance pairs” (i.e., the portfolio mean and
standard deviation). For example, the portfolio with wA = 2/3 (and thus wB = 1/3) has an
expected rate of return of 8.30% and a standard deviation of 4.7%.
Figure 16.1 plots the data of Table 16.1 into a coordinate system, in which the overall portfolio
You can also draw each
portfolio in a coordinate standard deviation is on the x-Axis, and the overall portfolio expected rate of return is on the
system of risk vs. reward.
y-Axis. Each coordinate is a particular risk and reward characteristic. When you compute more
Each portfolio is one
risk-reward trade-o¬s (as in Table 16.1) that you can invest in, you ¬nd a curve that is called a
point in x-y space.
hyperbola. The points on the hyperbola are the risk-reward combinations that you can achieve
by varying wA . For example, verify visually that your boldfaced portfolio (wA = 2/3, wB = 1/3)
is on the hyperbola: it should be at x = 4.7% and y = 8.3%. I have made it easier by drawing
dashed lines to the X and Y axes for this portfolio. In this example, the smallest risk always
obtains when the two securities are equally weighted. Above and below wA = 1/2, the risk
steadily increases.
¬le=optimalp¬o-g.tex: RP
383
Section 16·1. The Mean-Variance E¬cient Frontier.




Table 16.1. Portfolio Performance Under a Negative 75% Correlation.

E (˜P ) Sdv (˜P ) E (˜P ) Sdv (˜P )
wA wB r r wA wB r r
1.0 0.0 10.0% 10.00% 0.5 0.5 7.5% 3.54%
0.9 0.1 9.5% 8.28% 0.4 0.6 7.0% 4.00%
0.8 0.2 9.0% 6.63% 0.3 0.7 6.5% 5.15%
0.7 0.3 8.5% 5.15% 0.2 0.8 6.0% 6.63%
2/3 1/3 8.3% 4.71% 0.1 0.9 5.5% 8.28%
0.6 0.4 8.0% 4.00% 0.0 1.0 5.0% 10.00%
Portfolios Involving Shorting
’0.1 ’0.1
1.1 4.5% 11.77% 1.1 10.5% 11.77%
’1 +2 +2 ’1
0.0% 28.3% 15.0% 28.3%



The mean and standard deviation of the portfolio wA = 2/3 (and thus wB = (1 ’ wA ) = 1/3) are in bold. They were
computed as follows: The portfolio has a mean of 2/3·10% + 1/3·5% ≈ 8.3%, a variance of (2/3)2 ·0.01 + (1/3)2 ·0.01 +

2·(2/3)·(1/3)·(’0.0075) ≈ 0.00022, and thus a risk of 0.00022 ≈ 4.71%.




Figure 16.1. The Risk-Reward Trade-o¬ With A Correlation of ’75%.
0.12




wA = 1
(There are no (wB = 0)
st
portfolios with
e be
re th s
risk’reward chara’
ed a ortfolio ire
A
tier"
0.10




requ
rp
and
Expected Rate of Return




cteristics here.)
pfios B to
Fron
Blue ’reward line
t io
cien (Red ng of pf
risk .
fi io A
e "Ef
ti
shor ore of pf
= Th m
buy
wA = 4/5
(Here would be
wA = 2/3
0.08




pfios with inferior
Minimum Variance Pfio
risk’reward
wA = 0.5
wA = 1/2 characteristics.)
(wB = 0.5)
wA = 1/3
0.06




wA = 1/5

B


wA = 0
0.04




(wB = 1)

0.00 0.02 0.04 0.06 0.08 0.10 0.12

Standard Deviation of Rate of Return

The portfolios that invest fully in either A or B are marked with arrows and name. Other portfolio combinations
are those on the curves. The fat blue curve is the mean-variance e¬cient frontier for portfolios that do not require
shorting”portfolios that maximize reward given a desired level of risk. The fat red curve is also part of the e¬cient
frontier, but consists of portfolios that require shorting B in order to purchase more of A. The lower part of the
hyperbola is not mean-variance e¬cient”you would never purchase these. Incidentally, if you were allowed to throw
away money, you could obtain any portfolio vertically below the e¬cient frontier”of course, you would never want
to do this, anyway.
¬le=optimalp¬o-g.tex: LP
384 Chapter 16. The E¬cient Frontier”Optimally Diversi¬ed Portfolios.

You know even more about where portfolios lie on the curves. Portfolios with similar weights
Similar portfolios are
close to one another. must be close to one another in the graph: after all, both the expected rate of return and the
standard deviations of a portfolio with 60% A and 40% B are very similar to the expected rate
of return and the standard deviations of a portfolio with 61% A and 39% B. Therefore, the two
portfolio points would have to be close. Thus, you can visually judge where a given portfolio
combination must lie. For example, a portfolio with 95% weight on the A security and 5% weight
on the B security would have to lie 95% of the way between the two on any of the curves, i.e.,
rather close to the wA = 1 portfolio.
All portfolios that have positive weights in both securities lie between the two portfolios marked
Where are portfolios
with both long and short by arrows (wA = 1, wB = 0 and wA = 0, wB = 1). But Table 16.1 and Figure 16.1 also show
positions?
portfolios that involve shorting (Section 12·2.B). For example, say you have $200, but you want
to invest $220 in security A. This means that you have to short $20 in security B, in order to
obtain the money to have the full $220 to invest in security A. Your investment weights are
now wA = +110% and wB = ’10%. Please convince yourself that this portfolio”and any other
portfolio that involves shorting security B”lie above the upper arrows in Figure 16.1. You can
read the mean and standard deviation in Table 16.1, and then mark it in Figure 16.1.


Achievable Combinations, Unachievable Combinations, and The E¬cient Frontier
Now look again at Figure 16.1. There are some unobtainable regions”for example, combina-
Unobtainable and
Inferior Risk-Reward tions that would give you a portfolio with a 4% risk and a 10% reward. Similarly, you cannot
Choices.
purchase a portfolio with a 3% risk and a 7% reward. There are also some portfolios that are out-
right “ugly.” You would never want to purchase more in security B than in security A”or you
end up on the lower arm of the hyperbola. In general, you would also never want any portfolio
in the red area (even if you could obtain it, though this is not possible in our example).
There is really only one set of portfolios that you would choose if you are sane”those on the
The highest achievable
expected rates of return upper boundary of the achievable set of portfolios. Only the portfolios on the upper arm of
for each choice of risk is
the hyperbola give you the highest reward for a given amount of risk. Which of these you
called the
would choose is a matter of your taste, however”if you are more risk-tolerant, you would
“Mean-Variance Ef¬cient
Frontier.”
purchase a portfolio farther to the right and top than if you are more risk-averse. The set of
points/portfolios yielding the highest possible expected rate of return for a given amount of
risk is called the mean-variance e¬cient frontier, or, in brief, the e¬cient frontier. This term
will be used so often that we shall abbreviate “mean-variance e¬cient” as MVE.
Actually, mean-variance e¬cient frontier is a doubly bad name. First, the traditional graph
Some confusasive
namings. draws the standard deviation on the “risk” or x axis. This graph should be called the “mean-
standard deviation” graph, but it is commonly called the “mean-variance” graph. Fortunately,
the two look fairly similar visually. And it does not matter whether you call it the mean-variance
frontier or the mean-standard deviation frontier”they are really the same, in that a portfolio
with lower variance also has lower standard deviation. Second, the e¬cient frontier here has
nothing to do with the concept of e¬cient markets that we discussed in Section 6·1.C and
which we will revisit in Chapter 19. Because the term e¬cient is so often used in Economics, it
is often confusing.
¬le=optimalp¬o-g.tex: RP
385
Section 16·1. The Mean-Variance E¬cient Frontier.

You can solve for the functional form of all weighted combinations of two portfolios by

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