# Math362 Fundamentals of Mathematical Finance Solution

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```					 MATH362 – Fundamentals of Mathematical Finance

Solution to Test One

Fall, 2007                                                               Course Instructor: Prof. Y.K. Kwok

1. (a) Solve for v such that Ωv = 1, then normalize the components such that the sum of
components equal 1.

2 1           v1             1               v1       0.4
=           giving           =         .
1 3           v2             1               v2       0.2

The weight vector of the minimum variance portfolio is then

w g = (2/3   1/3)T .

0.03
(b) µh = w T r = (0.5
h              0.5)                = 0.04.
0.05
The expected gain over one year = \$100 × 0.04 = \$4, so you expect to have \$104 at the
end of one year.
2     1           2 1    0.5
(c) cov(rg , rh ) = w T Ωw h =
g w
3     3           1 3    0.5
2   1       1.5          5
=                        =      .
3   3        2           3
(d) For any portfolio with weight vector w P , we have

1T Ω−1 Ωw P
w
cov(rg , rP ) =             ,       a = 1T Ω−1 1,
a

and since 1T w P = 1, giving cov(rg , rP ) = 1/a = 0. Hence, we cannot ﬁnd another
portfolio P such that its return rP is uncorrelated with rg .
(e) For a given level of risk, an eﬃcient fund maximizes the expected rate of return. On the
other hand, a minimum variance portfolio minimizes the portfolio variance for a given level
of expected rate of return. According to the mean-variance portfolio model, the minimum
variance portfolio lies on the left boundary of the feasible region, which is represented
by a hyperbola. Only those minimum variance portfolios which lie on the upper part of
the hyperbola are eﬃcient funds. Hence, a minimum variance portfolio is not always an
eﬃcient fund.

1
P
efficient funds are on the upper portion

P

2. (a) Consider a portfolio P which lies inside the feasible region of risky assets only. Let F (0, r)
denote the riskfree point. Since r < µg , F lies below the global minimum variance point
(the most left point of the frontier of the feasible region). Any combination of portfolio
P and the riskfree asset (no short selling of portfolio P ) can be represented by points on
the half-line joining P and F (see the upper dotted half-lined).
P

g
llin sset
rt se     a
P    sho skfree
of  ri
x

F x
s ho r
t
of p selling
ortf
olio
P
P

If short selling of portfolio P is allowed, then the various combinations of short position
of portfolio P and long position of the riskfree asset correspond to portfolios that lie on
the lower dotted half-line. As we vary the choice of portfolio P within the feasible region,
the totality of these lines form a solid inﬁnite triangular wedge, with the upper boundary
being the upper tangent line from the riskfree point to the frontier of the feasible region.
P

(   g   , g)

P

When short selling of the riskfree asset is not allowed, the upper half-line terminates at P
and cannot go beyond P . In particular, we cannot extend the upper tangent line beyond
the tangency point M . On the other hand, the lower half line can be extended to inﬁnity.
The new feasible region is depicted in the ﬁgure below.

2
P

M

P

(b) One-fund Theorem: Any eﬃcient portfolio can be expressed as a combination of the
riskfree asset and the tangency portfolio t.
Let α and (1 − α) be the weight invested on the tangency portfolio t and the riskfree asset.
Equating the expected return:

µP = αµt + (1 − α)r

giving
µp − r
α=          .
µt − r

3. (a) The ﬁrst order conditions are

µ    w      1
τµ − Ωw ∗ + λ1 = 0
w T 1 = 1.

(b) We obtain w ∗ = τ Ω−1µ + λΩ−11. Applying the constraint condition:

1 = 1T w ∗ = τ1T Ω−1µ + λ1T Ω−11
1          1

so that

1 − τ1T Ω−1µ
1
λ=       T −1
.
1 Ω 1

Finally, we obtain

1T Ω−1µ −1
w∗ = τ   Ω−1µ −            Ω 1 + wg .
1 T Ω−11

ri − r
4. (a) Sharpe ratio =          , where r i is the expected rate of return of the risky asset, r is the
σi
riskfree rate and σi is the standard deviation of the rate of return.

3
(b) According to CAPM:

σiM             ρiM σi σM
ri − r =     2 (r M − r) =     2     (r M − r)
σM                 σM

where ρiM is the correlation coeﬃcient between ri and rM . We have

ri − r       rM − r
= ρiM
σi          σM

so that the Sharpe ratio of the risky asset is ρiM times that of the market portfolio,
−1 ≤ ρiM ≤ 1. For an eﬃcient portfolio e on the capital market line, we have ρeM = 1 so
that
re − r   rM − r
=        .
σe       σM

cov(ri , rm )
(c) βi =        2        so that βi < 0 ⇔ cov(ri , rM ) < 0.
σM

4

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