3.4 Capital Market Line and Sharpe Ratio
23
by noting that C
−1
= 1/σ
2
M and r = r M − r f . Inserting these simplifications we
obtain
ρ − r f
σ
=
r M − r f
σ M
(3.31)
which shows that the Sharpe ratio for our portfolio on the left-hand-side of the
equation equals that of the entire market, but we can adjust our desired return ρ to
our preference, albeit linked to a specific risk or volatility σ. Therefore, the Sharpe
ratio is sometimes called the price of risk; if we want a higher pay-off or return on
our investment, we have to accept a higher exposure to risk.
The line, that was the capital allocation line for a specific portfolio is called the
capital market line when we consider the whole market. It derives from a portfolio
that is representative of the market. Note that the (red) market allocation line in
Fig. 3.2 always lies above the (blue) efficient frontier, as given by investing in risky
assets only. This is no surprise, because we have an extra option; either investing
in the risk-free asset or borrowing at the risk-free rate and investing the borrowed
money in stocks. Optionality is good!
Note also that the intercept of the capital market with the vertical axis describes
a situation where all our available funds are invested in the risk-free asset. Conversely, the point where the capital market line touches the blue efficiency-frontier
line describes the situation where all our funds are invested in risky assets of the
market portfolio. The coordinates of this point at which the capital market line is
tangent to the efficient frontier can be easily calculated from the requirement that
all available funds are invested in risky assets, namely e
T w = 1. Using (3.28) and
solving for the return of the tangent portfolio ρ t , we find
ρ t = r f +
r
T ˆ
C
−1
r
e T ˆ
C −1 r
(3.32)
and inserting in (3.30), we find for the risk σ t of the tangent portfolio
σ
2
t =
r
T ˆ
C
−1
r
e T ˆ
C −1 r
2 .
(3.33)
In Fig. 3.2 the tangent point is indicated by a red asterisk. The section of the capital
market line between the intercept and tangent point describes a mixture of risk-free
and optimal market portfolio that matches our taste for risk. If we are risk-averse,
we use a mix that corresponds to a point on the line located towards the left, near
the intercept where the return ρ is close to the risk-free rate r f . If we are more risktolerant, we choose a mix on the line located closer to the tangent point. If we are
even more adventurous, we can borrow money at the risk-free rate (negative weight
w f ) and invest the borrowed money into the market portfolio. This is called leveraged
investment, which however, carries a high risk.
23
by noting that C
−1
= 1/σ
2
M and r = r M − r f . Inserting these simplifications we
obtain
ρ − r f
σ
=
r M − r f
σ M
(3.31)
which shows that the Sharpe ratio for our portfolio on the left-hand-side of the
equation equals that of the entire market, but we can adjust our desired return ρ to
our preference, albeit linked to a specific risk or volatility σ. Therefore, the Sharpe
ratio is sometimes called the price of risk; if we want a higher pay-off or return on
our investment, we have to accept a higher exposure to risk.
The line, that was the capital allocation line for a specific portfolio is called the
capital market line when we consider the whole market. It derives from a portfolio
that is representative of the market. Note that the (red) market allocation line in
Fig. 3.2 always lies above the (blue) efficient frontier, as given by investing in risky
assets only. This is no surprise, because we have an extra option; either investing
in the risk-free asset or borrowing at the risk-free rate and investing the borrowed
money in stocks. Optionality is good!
Note also that the intercept of the capital market with the vertical axis describes
a situation where all our available funds are invested in the risk-free asset. Conversely, the point where the capital market line touches the blue efficiency-frontier
line describes the situation where all our funds are invested in risky assets of the
market portfolio. The coordinates of this point at which the capital market line is
tangent to the efficient frontier can be easily calculated from the requirement that
all available funds are invested in risky assets, namely e
T w = 1. Using (3.28) and
solving for the return of the tangent portfolio ρ t , we find
ρ t = r f +
r
T ˆ
C
−1
r
e T ˆ
C −1 r
(3.32)
and inserting in (3.30), we find for the risk σ t of the tangent portfolio
σ
2
t =
r
T ˆ
C
−1
r
e T ˆ
C −1 r
2 .
(3.33)
In Fig. 3.2 the tangent point is indicated by a red asterisk. The section of the capital
market line between the intercept and tangent point describes a mixture of risk-free
and optimal market portfolio that matches our taste for risk. If we are risk-averse,
we use a mix that corresponds to a point on the line located towards the left, near
the intercept where the return ρ is close to the risk-free rate r f . If we are more risktolerant, we choose a mix on the line located closer to the tangent point. If we are
even more adventurous, we can borrow money at the risk-free rate (negative weight
w f ) and invest the borrowed money into the market portfolio. This is called leveraged
investment, which however, carries a high risk.