The Portfolio Risk-Return Trade-Off
- Investors face a trade-off between risk and expected return.
Shifting funds from the risky portfolio to the risk-free asset is the simplest
way to reduce risk. Another method involves diversification of the risky
portfolio. This is the focus here.
- A risky investment portfolio (referred to here as the risky asset)
can be characterized by its reward-to-variability ratio. This ratio is the
slope of the capital allocation line (CAL), the line that goes from the risk-free
asset through the risky asset. All combinations of the risky and risk-free
assets lie on this line. Investors would prefer a steeper sloping CAL, because
that means higher expected returns for any level of risk. If the borrowing
rate is greater than the lending rate, the CAL will be "kinked" at the point
corresponding to investment of 100% of the complete portfolio in the risky
- An investor's preferred choice among the portfolios on the capital
allocation line will depend on risk aversion. Risk-averse investors will
weight their complete portfolios more heavily toward Treasury bills. Risk-tolerant
investors will hold higher proportions of their complete portfolios in the
- The capital market line is the capital allocation line that results
from using a passive investment strategy that treats a market index portfolio
such as the Standard & Poor's 500 as the risky asset. Passive strategies
are low-cost ways of obtaining well-diversified portfolios with performance
close to that of the market as a whole.
Optimal Portfolio Diversification
- The variance of a portfolio is a sum of the contributions of the
component-security variances plus terms involving the correlation among assets.
- Even if correlations are positive, the portfolio standard deviation
will be less than the weighted average of the component standard deviations,
as long as the assets are not perfectly positively correlated. Thus, portfolio
diversification is of value as long as assets are less than perfectly correlated.
- The contribution of an asset to portfolio variance depends on
its correlation with the other assets in the portfolio, as well as on its
own variance. An asset that is perfectly negatively correlated with a portfolio
can be used to reduce the portfolio variance to zero. Thus, it can serve
as a perfect hedge.
- The efficient frontier of risky assets is the graphical representation
of the set of portfolios that maximizes portfolio expected return for a
given level of portfolio standard deviation. Rational investors will choose
a portfolio on the efficient frontier.
- A portfolio manager identifies the efficient frontier by first
establishing estimates for the expected returns and standard deviations, and
the correlations among them. The input data are then fed into an optimization
program that produces the investment proportions, expected returns, and standard
deviations of the portfolios on the efficient frontier.
- In general, portfolio managers will identify different efficient
portfolios because of differences in the methods and quality of security analysis.
Managers compete on the quality of their security analysis relative to their
- If a risk-free asset is available and input data are identical,
all investors will choose the same portfolio on the efficient frontier, the
one that is tangent to the CAL. All investors with identical input data will
hold the identical risky portfolio, differing only in how much each allocates
to this optimal portfolio and to the risk-free asset. This result is characterized
as the separation principle of portfolio selection.
- The single-index representation of a single-factor security market
expresses the excess rate of return on a security as a function of the market
excess return: Ri = (xi + PiRm + ei) This equation also can be interpreted
as a regression of the security excess return on the market index excess return.
The regression line has intercept cLi and slope Pi, and is called the security
- In a single-index model, the variance of the rate of return on
a security or portfolio can be decomposed into systematic and firm-specific
risk. The systematic component of variance equals Beta times the variance
of the market excess return. The firm-specific component is the variance
of the residual term in the index model equation.
- The beta of a portfolio is the weighted average of the betas of
the component securities. A security with negative beta reduces the portfolio
beta, thereby reducing exposure to market risk.
- The unsystematic risk of a portfolio approaches zero as the portfolio
becomes more diversified.
Optimizing Fixed-Income Portfolios
- Duration measures the percentage price change of a bond for a
1% change in yield. Convexity measures the degree to which duration changes
as the yield to maturity changes.
- Duration is additive, so the duration of a portfolio of bonds
is the weighted sum of the duration of the individual bonds. Because duration
and convexity measure price risk, they can be helpful in bond portfolio management.
- Immunization strategies are characteristic of passive fixed-income
portfolio management. Such strategies attempt to render the individual or
firm immune from movements in interest rates. This may take the form of immunizing
net worth or, instead, immunizing the future accumulated value of a fixed-income
portfolio. Immunization of a fully funded plan is accomplished by matching
the durations of assets and liabilities.
- When the weighted duration of assets does not equal that of the
liabilities, derivatives (such as futures or swaps) can be employed to match
asset with liability duration.
- Duration changes as time passes and as yields change. Therefore
to maintain an immunized position, the portfolio must be periodically rebalanced.
- A more precise form of immunization is dedication or cash flow
matching. If a portfolio is perfectly matched in cash flow with projected
liabilities, rebalancing will be unnecessary.