# Understanding Asset Allocation: In Search of the Upside

- The Measurement of Risk
- Asset Allocation and Retirement
- Ranging the Possibilities: Monte Carlo Simulations
- Summary

It is
striking
how
little
most
people
understood
about
risk
as recently
as three
decades
ago.
Risk,
of course,
is that
piece
of information
all
investors
need
to know—and
should
desire
to keep
as low
as possible
in relation
to the
returns
they
expect
to see
on their
investments.
Fortunately,
developments
in modern
portfolio
theory
provide
a framework
for
addressing
the
ways
risk
can
affect
expected
returns.^{1} The
developments
have
been
nothing
short
of dramatic.

## The Measurement of Risk

We now have the Sharpe ratio at our disposal, a well-known formula useful for
evaluating alternative investments and determining when to add additional assets
to a portfolio.^{2}

The *Sharpe ratio* summarizes two measures—mean return and
variance—within a single measure. *Mean return* can be considered
the average return an investment or investment class is expected to deliver over
time, while *variance* can be considered the average range of asset
performance around the mean return. To calculate the Sharpe ratio, subtract the
*risk-free rate* returns (that is, Treasury bill [T-bill] returns) from
the asset returns in question and divide that result by the standard deviation
of the return of the asset class in question less that of the risk-free rate. In
this manner, risk is pinpointed. One way to think of this is to consider a
person who borrows money to invest. After doing so, that person’s net gain
is the difference between the return of the investment and the funds borrowed;
the greater the difference (on the positive side), the greater the reward.
Similarly, the higher the Sharpe ratio, the lower the risk in relation to the
reward. The Sharpe ratio is calculated using the mean and standard deviation of
an excess return. That is the net of the asset class return and the risk free
rate (that is, three months’ T-bill yields). A related measure is obtained
when the ratio is calculated based on the mean and return of a single
investment. This ratio is also known as the *information ratio*.

Then, there’s the *capital asset pricing model* (CAPM), which
similarly looks at the relationship between an investment’s risk and its
expected market return—or, more specifically, the ways investment risk
should affect its expected return.^{3} One major insight of the CAPM is
that not all risks should affect asset prices. As would be the case if two
assets moved in the same direction, the volatility of the portfolio consisting
of the two assets would remain the same as the individual assets. In contrast,
if the two assets move in the opposite direction, the volatil-ity of a portfolio
consisting of the two assets would be much lower than that of each of the assets
by themselves. The latter represents an example of a risk that can be
diversified away by combining it with other assets in a portfolio, which should
not be considered a risk. Hence, when considering adding asset to a portfolio,
one needs to take into account whether the asset moves with the portfolio and
whether the addition of the asset will reduce or increase the volatility of the
portfolio. If the asset does not add to the volatility of the portfolio, it
should not be priced for risk, or more plainly, investors would not demand an
additional return or premium over and above the current expected return. The
only risk that should be priced is the risk that cannot be diversified away, the
*residual risk* or *systematic risk*. The CAPM is firm on this
point. What should matter to the investor, therefore, is the incremental impact
on the overall portfolio volatility—not the individual investment
volatility. With this in mind, an investor can effectively apply the Sharpe
ratio: When adding an asset to a portfolio improves the Sharpe ratio, the asset
adds to the return of a portfolio over and above the increased volatility of the
new overall portfolio.

Investing suddenly seems very simple. Indeed, in the days of the Sharpe ratio
and the CAPM, the *market portfolio*—a portfolio that has bought
the market (given that the overall market is in equilibrium)—has become
the efficient portfolio. An *efficient portfolio* is a portfolio that
contains returns that have been maximized in relation to the risk level that
individual investors desire. In a market that is in equilibrium, where the
number of winners and losers must balance out, adding one additional asset class
or stock does not increase the portfolio’s risk return ratio. This means
the portfolio containing risky assets with the highest Sharpe ratio must be the
market portfolio.