Managerial economics guide 7

CAPM

T0: the consumer has 1 unit of wealth

During this period, he/she decided to using part of this unit to invest in financial assets, the rest is

invested in a government bond ⟨⟩

+ ∑ = 1

0

=1

T1:

The consumer is compensated in two ways: the time value of the government bond and the risk

premium from the financial assets

̃ ⟨⟩

= + ∑

0

=1

̃

⟨⟩⟨⟩ ↔ = + ∑ −

( )

⏟ ⏟

=1

Our objective is to maximize the utility of this return

(̃

max )

̃

↔ =0

(̃

′

↔ [ ∗ − = 0

) ( )]

(̃ (̃

′ ′

↔ [ ∗ = ∗ [

) ] )]

Knowing that: (̃ (̃ (̃

′ ′ ′

[ = [ ∗ − [ ∗ ( )

); ] ) ] )]

We can find that (̃ (̃

′ ′ (̃

′

∗ [ − [

)] ); ] [

); ]

)

( = = −

(̃ (̃

′ ′

[ [

)] )]

To specify this formula, we can use the quadratic utility function. Under the assumption of quadratic

utility, mean-variance analysis is optimal. For investors who prefer more to less, the quadratic utility

function may only represent their preferences over a restricted range of wealth. the absolute risk

aversion function demonstrates that the quadratic utility function exhibits increasing absolute risk

aversion. Thus, the quadratic function is consistent with investors who reduce the nominal amount

invested in risky assets as their wealth increases. By definition, a quadratic utility function must exhibit

increasing relative risk aversion. These characters correspond to our consumer’s preference

2

() = −

(̃ 2̃

′

→ = 1 −

)

2̃ ]

[1 − ;

)

( = −

2̃ ]

[1 −

[̃

] ]

[1; ;

)

( = − + 2

2̃ 2̃

] ]

[1 − [1 −

]

[1; = 0

In real life, interest rate from financial assets will always be less than 1, therefore

[̃ ]

;

)

→ ( = + 2 < >

2̃ ]

[1 −

̃ [̃ [̃

]

= ; = ]

When (for ex: index fund),

2[̃ ]

2̃ ]

→ [1 − =

)

( −

Plugging this to <III>, we will have )

( −

2[̃ ]

)

→ ( = + ∗ ;

2[̃ ]

[̃ ]

;

) )

→ ( = + ∗ − < >

[( ]

[̃ ]

[̃ ]

;

= is the coefficient which tells us how close the risk return rate tracks total asset return or

[̃ ]

how aggregate risk affects the return of asset. )

−

[( ]

Using the <IV> formula, we can graph the secure market line with intercept and slope

which is also known as market risk premium

Keeping in mind that the rate of return, can also be thought as the return divided by the price of

the asset, the formula can clearly be seen as a theory of asset pricing. So the SML is a graph presentation

of the CAPM model

A mutual fund is an investment vehicle made up of a pool of funds collected from many investors for the

purpose of investing in securities such as stocks, bonds, money market instruments and similar assets.

Mutual funds are operated by money managers, who invest the fund's capital and attempt to produce

capital gains and income for the fund's investors. A mutual fund's portfolio is structured and maintained

to match the investment objectives stated in its prospectus.

Source: Mutual Fund http://www.investopedia.com/terms/m/mutualfund.asp#ixzz4inCkBRkd