University of Michigan Literature, Science and the Arts (LSA)

Department of Math Actuarial & Financial

Winter 2014

Math 423 Sec. 2 & 4

Mathematics of Finance

Capinski-Zastawniak (Second Edition)

B. Roger Natarajan, PhD, FSA, MAAA

Chapter 3 C

Agenda

Market Portfolio - (, )

Capital Market Line (CML)

= R + ( R)

Capital Asset Pricing Model (CAPM)

= + ( )

Security Market Line

2 5/22/2014

Feasible Portfolios (Example 3.31)

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0.15

(2, 2)

(3, 3)

0.10

0.20

0.15 0.20 0.25 0.30

(1, 1)

Market Portfolio - Maximize

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( , )

(, )

(0, )

Feasible Set (Risky Only)

3.3.4 Market Portfolio - (, ) plane Start with a Risky Portfolio V represented by V( , ). Now let us allow Risk-Free Security (0, R) in our portfolio. R < All Extended Portfolios that include V and Risk-Free Security are on

the two rectilinear half-lines meeting at the point (0, R). The slope of the straight line connecting V( , ) and (0, R) is

given by

. It represents the extra return for a unit change in .

We want to Maximize

=

w m

R

w C w subject to w. u

= 1

F(w, ) = w m

R

w C w (w. u

1)

w C wm(w m

R) wC

w C w

w C w u = 0

m ()

wC

2 = u

m

() wC

3 = u (1)

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3.3.4 Market Portfolio - (, ) plane

(1) x w

m w

() wCw

3= u w

=

1

{- ( R) } =

=

(1) X m ()

2 wC = u = R u

()2

wC = (m R u) x C1 ()

2w = (m R u)C1 (2)

(2) X u

(R)

2 w u

= (m R u)C1 u

(R)2

= (m R u)C1 u

Use this in (2)

w= (m R u)C1

(m R u)C1 u = = Market Portfolio

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Theorem 3.33 & Capital Market Line If det |C | 0 and the risk-free rate R is lower than the expected return of the Minimum Variance Portfolio, then the Market Portfolio M exists and its weights are given by

w= (m R u)C1

(m R u)C1 u

Proof: We already derived the expression for w.

We need to prove (m R u)C1 u

0 ; Given > R

= w m

=uC1

uC1 u m

> R

u C1 m

> R u C1 u

m C1 u

= u C1 m

> R u C1 u

Denominator is +ve ---------------------------------------------------------------------------------------------

Capital Market Line = The Line connecting the Market Portfolio ( , ) and (0, R) in the (, ) plane

( R) = (R)

( 0) = R +

( R)

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Capital Asset Pricing Model - Basics The models so far are based on Risk expressed as .

Let = 1/n. Assume 2 L for each i = 1, 2, ., n

2 =

2=

2 1

2 n L +

1

2 (2 - n) c assuming c = c for all i and j

2

1

2 n L + ( 1 -

1

) c

2 c as n

It is the covariance that is more important to measure the risk of a portfolio (consisting of many securities) than the variances.

Diversifiable or Specific Risk: This can be reduced to Zero by expanding the portfolio.

Systematic, Material, or UnDiversifiable Risk: Cannot be avoided since the

securities are linked to the market. 5/22/2014

8

, 1

i i

i

j j

j

n

ww c

2 2

, 1

n

j j

j j

i

i i

n

ii iw ww c

Capital Asset Pricing Model

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(0, )

(, ) ( , )

Capital Asset Pricing Model - Construction Let (, ) be the Market Portfolio with weights

Consider any other portfolio ( , )

Now consider all portfolios constructed from M and V

Assertion: The CML must be tangent to the above Hyperbola. If the hyperbola were to intersect the CML, it would contradict the fact that the slope of CML is Maximal.

Let P be the portfolio with weights and (1- ) of V and M

= 22+ (1 )2 2 + 2(1 ) Cov (KV, KM)

= + (1- )

Slope of the tangent of Hyperbola at M = { ()

/

()

} at =0

)

= 222(1)2 + (2 4) Cov (KV,KM)

2 22+ (1 )2 2 + 2(1 ) Cov (KV,KM)

)

= Cov (KV,KM) 2

at = 0

()

=

5/22/2014 10

Capital Asset Pricing Model - Construction Slope of the tangent of Hyperbola at M =

Cov (KV,KM)

2

Slope of CML =

Cov (KV,KM)

2

=

( ) 2 = ( ) {Cov (KV, KM)

2 }

2 = ( ) Cov (KV, KM) +

2

= + Cov (KV,KM)

2 ( )

= + ( ) where = Cov (KV,KM)

2

( ) = Risk Premium

= Beta factor = Linkage to the Market

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Capital Market Line Security Market Line

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(, )

( , )

=1

=0

(, )

Capital Market Line

Security Market Line

Case 2 Self Funding - Pension

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1. Pension fund based on a regular savings invested in a Bank Account at 5% per annum.

2. When you retire after 40 years, you want to receive a Pension equal to 50% of your final salary and payable for 20 years.

3. Your earnings (salaries) are assumed to grow at 2% annually and you want the Pension payments to grow at the same rate.

S = Salary now (t=0); g = growth rate of Salary = 2%;

r= Return on Investment = 5%

= Contribution = . S . (1 + ) t=1, 2, 3, ., 40

= % of Current Salary to be saved in order to self-fund retirement goals

= 0.5 . (1+)40(1+)20[(1+)20(1+)20 ]

[(1+)40(1+)40 ] (2)

= 10.05%

Case 3 Self Funding - Pension

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Expand the investment horizon to include Risky Securities.

1 = 8% 2 = 10% 3 = 14% R = 5%

1 = 0.15 2 = 0.22 3 = 0.26

12= 0.5 23= 0.3 13= 0.3

w= (m R u)C1

(m R u)C1 u ( 0.40, 0.20, 0.80 )

We dont want to do any Short Selling.

We solve for Market Portfolio that includes only 1 and 3.

w = ( 0.5646, 0, 0.4354 ) and

= 10.61% = 11.93%

Let P be the portfolio that includes (1 y) % in Risk-Free and y % in Market Portfolio without Short Selling

Case 3 Self Funding - Pension

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Let us assume that we want to take only 10% Risk ()

= 10% = y = y (11.93%) y = .8381

Therefore 83.81% in and 16.19% in Risk-Free

= (83.81%x 10.61%) +(16.19%x5%) = 9.70%

() = 2.2% [ using 9.70% for r in equation (2) )

--------------------------------------------------------------------------------------

Given , solve for y; calculate and use this for r in (2) to get

-------------------------------------------------------------------------------

Risk = 0% = 5% = 10%

% invest in Market y =0.00% y =41.91% y =83.81%

% invest Risk-Free 100.00% 58.09% 16.19%

5.00% 7.35% 9.70%

% Sal to Pension = 10.05% = 4.76% =2.20%

Feasible Set (Proposition 3.18)

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0

0

1(1, 1)

2(2, 2)

(0, 0)

All possible portfolios that can be constructed from 1 and 2 The one with Minimum Variance is MVP.

s0 = 22c12

12+222c12

0 = 122 +212 (1+2)c12

12+222c12

2 =

122

2 c122

12 + 222c12

2 A2 (0)2 = 0

2

A2= 12+ 222c12

(12)2

Minimum Variance Portfolio (n-Securities) The portfolio with the smallest variance among ALL feasible portfolios will be called the Minimum Variance Portfolio(MVP).

w = uC1

uC1u =wm

2 =wCw

= (0)

(0) w = [1, 2, ., ]; u = [1, 1, 1, , 1]

= [1, 2, ., ] C =

c11 c12 c1c21 c22 c2c1 c2 c

c= Cov(, )

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Minimum Variance Line (n-Securities) The portfolio with the smallest variance among all feasible portfolios but with Expected Return will be called the MVP().

w() = a + b

a = 111mC1 + 21

1uC1

b = 121mC1 + 22

1uC1

M = mC1m

uC1m

mC1u

uC1u

1 = 11

1 121

211 22

1

= w()m

2() =w()Cw()

All the w() for various forms the Minimum Variance Line

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Theorem 3.30 Two-Fund Theorem

Assume that |C | 0 and m and u are linearly independent vectors.

Let w1 and 2 be the weights of any two PORTFOLIOS 1 and 2, on the MVL with different expected returns 1 and 2. Then EACH

portfolio V on the MVL can be obtained as a linear combination of these two portfolios.

w = w1+ (1 ) 2

The MVL can be viewed as if it were a two-asset line constructed from the two portfolios 1 and 2 (regarded for this purpose as if they were individual risky assets).

We can therefore conclude that the MVL is a Hyperbola of the kind

2 - A2 (0)2 = 0

2

Where MVP (of all feasible set of n-securities) is obtained by

w = uC1

uC1u

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Market Portfolio & Capital Market Line

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(, )

(0, )

Feasible Set (Risky Only)

Minimum Variance Line

( , )

1(1 , 1)

2(2 , 2)

Feasible Set (Risky + Risk-Free)