Econ110B_Chapter15

7/30/2019 Econ110B_Chapter15

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Financial Markets and Expectations


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Arbitrage and Stock Prices Choice: 1-year bonds versus holding a stock for one year

Stock: Buy a stock today, receive a divident next year andsell the stock. With $1, we can buy (1/$Qt) stocks today.Next year, we get expected divident $Det+1, and sell thestock. The expected price of the stock next year is $Qet+1

Today (t) One year from now (t+1)1-year Bond $1 $1 ( 1 + i1t)

Stock $1 ($1/$Qt) ($Det+1 + $Q

et+1)


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Arbitrage and Stock Prices (cont) Thus, if investors only care about expected return,

arbitrage says that the expected return on the two assetshave to be equal:

$1 (1 i1t) $1

$Qt ($Dt1

e$Qt1

e )

$Qt $Dt1e

$Qt1e

(1 i1t)

$Qt$Dt1

e

(1 i1t)

$Qt1

e

(1 i1t)


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Arbitrage and Stock Prices (cont)What determines $Qet+1 ?

Next year, investors will face again the choice betweenstocks and 1-year bonds same arbitrage relation willhold:

$Qt1e

$Dt2

e

(1 i1t1e )

$Qt2

e

(1 i1t1e )

Since $Qt$Dt1

e

(1 i1t)

$Qt1e

(1 i1t), thus

$Qt$Dt1

e

(1 i1t)

$Dt2

e

(1 i1t)(1 i1t1e )

$Qt2

e

(1 i1t)(1 i1t1e )


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Arbitrage and Stock Prices (cont) Keep replacing expected stock price

$Qt$Dt1

e

(1 i1t)

$Dt2

e

(1 i1t)(1 i1t1e

)

$Qt2

e

(1 i1t)(1 i1t1e

)

$Qt $Dt1

e

(1 i1t)

$Dt2e

(1 i1t)(1 i

1t

1

e )

$Dt3e

(1 i1t)(1 i

1t

1

e )(1 i1t

2

e )

$Qt3

e

(1 i1t)(1 i1t1e )(1 i1t2

e )


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Arbitrage and Stock Prices (cont) Keep replacing expected stock price

Take a look at the last term:

$Qet+n / (( 1 + i1t) ( 1 + ie1t+n-1)) as long as people do not expect

stock price to explode in the future, then as we keep replacing$Qet+n and n increases, this term will go to zero.

$Qt

$Dt1e

(1 i1t)

$Dt2e

(1 i1t)(1 i1t1e )

$Dt3e

(1 i1t)(1 i1t1e )(1 i1t2e )

K $Dtn

e

(1 i1t)(1 i1t1e )L (1 i1tn1

e )

$Qtne

(1 i1t)(1 i1t1e )L (1 i1tn1

e )


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Arbitrage and Stock Prices (cont) In that case

The stock price is equal to the present discounted value ofthe entire stream of expected dividends in the future

$Qt

$Dt1e

(1 i1t)

$Dt2e

(1 i1t)(1 i1t1e )

$Dt3e

(1 i1t)(1 i1t1e )(1 i1t2e )

K $Dtn

e

(1 i1t)(1 i1t1e )L (1 i1tn1

e )K


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Note 1: Rational Bubbles

Note 1:

Rational Bubbles people expect large increases in the stockprice in the future. The c0ndition that the expected stockprice in the future does not explode is not satisfied stockprice is not equal to the PV of expected future dividends

$Qt$Dt1

e

(1 i1t)

$Dt2e

(1 i1t)(1 i1t1e )

$Dt3

e

(1 i1t)(1 i1t1e )(1 i1t2

e )

K $Dtn

e

(1 i1t)(1 i1t1e )L (1 i1tn1

e )

$Qtne

(1 i1t)(1 i1t1e )L (1 i1tn1

e )


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Note 2: Accounting for riskAccounting for risk in order to hold stocks rather than bonds,we need a risk premium (or equity premium = ). Thus thediscount rate will be = i + , instead of only I

Thus, if risk premium is , people will hold stocks only if theexpected return on stocks exceeds expected rate of return onshort-term bonds by:

which will lead to:

$1 (1 i1t) $1$Qt

($Dt1e $Qt1

e )


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Note 2: Accounting for risk

$Qt$Dt1

e

(1 i1t)

$Dt2

e

(1

i1t

)(1

i1t1e

)

K

K $Dtn

e

(1 i1t)(1 i1t1e

)L (1 i1tn1e

) ...

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Econ110B_Chapter15