Risk and Return
- A security's return is often measured by its holding-period return:
the change in price plus any income received, expressed as a percentage of
the original price. A better measure would take into account the timing of
dividends or other payments, and the rates at which they are reinvested.
- The total return on an investment has two components: the expected
return and the unexpected return. The unexpected return comes about because
of unanticipated events. The risk from investing stems from the possibility
of an unanticipated event.
- The total risk of a security refers to the extent to which realized
returns may deviate from the expected return. A common measure is standard
deviation, although for many investors the downside risk is more important
than the sheer dispersion of returns. For funds managers, the risk of underperforming
a benchmark may be the most relevant risk.
- If one assumes returns are nomally distributed, then variance
(or its square root, standard deviation) is a reasonable measure of risk,
since a normal distribution is symmetrical and fully described by its expected
value and variance. There is evidence that stock-price returns are more leptokurtic
(fat-tailed) than would be predicted by the standard normal distribution.
- Systematic risks (also called market risks) are unanticipated
events that affect almost all assets to some degree because the effects are
economywide. Unsystematic risks are unanticipated events that affect
single assets or small groups of assets. Unsystematic risks are also called
unique or asset-specific risks.
- Investors face a trade-off between risk and expected return.
Historical data confirm our intuition that assets with low degrees of risk
provide lower returns on average than do those of higher risk.
- 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.
- U.S. T-bills provide a perfectly risk-free asset in nominal terms
only. Nevertheless, the standard deviation of real rates on short-term T-bills
is small compared to that of assets such as long-term bonds and common stocks,
so for the purpose of our analysis, we consider T-bills the risk-free asset.
Besides T-bills, money market funds hold short-term safe obligations such
as commercial paper and CDs. These entail some default risk but relatively
little compared to most other risky assets. For convenience, we often refer
to money market funds as risk-free assets.
- 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.
Interest Rate Risk
- Even default-free bonds such as Treasury issues are subject to
interest rate risk. Longer term bonds generally are more sensitive to interest
rate shifts than short-term bonds. A measure of the average life of a bond
is Macaulay's duration, defined as the weighted average of the times until
each payment made by the security, with weights proportional to the present
value of the payment.
- Macauley's duration measures the time horizon when a bond's yield
will be realized. During that time, losses (gains) from price change will
be offset by gains (losses) from reinvestment of coupon interest.
- Modified Duration is a direct measure of the sensitivity of a
bond's price to a change in its yield. Modified Duration is equal to Macauley's
- Duration is only an approximation of the percentage price change
of a bond for a 1% change in yield. It assumes parallel changes in a flat
yield curve, and only works for small changes (such as 10 basis points) in
- The longer the maturity, the lower the yield, and the smaller
and less frequent the bond's coupon, the greater is the duration. The Macauley's
duration of a zero-coupon bond is equal to its maturity.
- Convexity measures the degree to which duration changes as the
yield to maturity changes.
- Positive convexity, which characterizes most straight (plain,
non-callable) bonds, refers to the fact that price sensitivity, as measured
by duration, declines as the yield increases, and rises as the yield decreases.
Positive convexity is regarded as a desirable feature of a bond, particularly
when yields are volatile. Callable bonds such as US mortgage-backed securities
have negative convexity over some yield range.
- 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. To maintain an immunized
position as time passes and interest rates change, the portfolio must be
- A more direct 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.
Quantifying Credit Risk
- For many years, academics and financial insitutions have sought
to predict losses from credit risk. The best-known methodogy is based on
Altman's Z-score, which seeks to predicts defaults using company financial
- The newer CreditMetrics approach estimates volatility from upgrades,
downgrades, and defaults. Historical data are used to attribute a likelihood
of possible credit events, including upgrades and downgrades, not just defaults.
- For example, CreditMetrics calculates the probability that a
bond’s current rating will shift to any other rating within a given time.
Each shift results in an estimated change in value (derived from historical
credit spread data or recovery rates in default). Each value out-come is
weighted by its likelihood to create a distribution of value across each
credit state, from which each asset’s expected value and volatility of value
- To compute the volatility of portfolio value from the volatility
of individual asset values requires estimates of correlation in credit quality
changes. Since these cannot be directly observed from historical data, one
approach is to infer these from historical asset correlation data derived
from equity price series. Several different approaches, including a simple
constant correlation, can be used.