Chapter Eleven

Asset Markets

Assets

An asset is a commodity that provides a flow of services over time.

E.g. a house, or a computer. A financial asset provides a flow of

money over time -- a security.

Selling An Asset

Q: When should an asset be sold? When its value is at a maximum? No. Why not?

Selling An Asset

Suppose the value of an asset changes with time according to

V t t t( ) =− + −1000 1000 10 2

Selling An Asset

V t t t( ) =− + −1000 1000 10 2

Maximum value occurs whenV t t'( ) = − =1000 20 0

That is, when t = 50.

Selling An Asset

0 10 20 30 40 50 60-1000

4000

9000

14000

19000

24000

Value

Years

Max. valueof $24,000is reachedat year 50.

Selling An Asset

The rate-of-return in year t is the income earned by the asset in year t as a fraction of its value in year t.

E.g. if an asset valued at $1,000 earns $100 then its rate-of-return is 10%.

Selling An Asset

Q: Suppose the interest rate is 10%. When should the asset be sold?

A: When the rate-of-return to holding the asset falls to 10%.

Then it is better to sell the asset and put the proceeds in the bank to earn a 10% rate-of-return from interest.

Selling An AssetThe rate-of-return of the asset at time t is

V tV t'( )( ).

In our example,.t10t10001000)t(V 2−+−=

V t t'( ) = − =1000 20 0V tV t

t

t t

'( )( )

.= −

− + −

1000 20

1000 1000 10 2so

Selling An Asset

The asset should be sold whenV tV t

t

t t

'( )( )

= −

− + −= ⋅1000 20

1000 1000 100 12

1000 - 20t = -100 +100t-t2

t2 - 120t +1100 = 0(t - 10)(t - 110) = 0So sell when t = 10

Selling An Asset

0 10 20 30 40 50 60-1000

4000

9000

14000

19000

24000

Value

Years

Max. valueof $24,000is reachedat year 50.Sell at 10 years

even though theasset’s value isonly $8,000.

slope= 0.1

Selling An Asset

What is the payoff at year 50 from selling at year 10 and then investing the $8,000 at 10% per year for the remaining 40 years?

Selling An Asset

What is the payoff at year 50 from selling at year 10 and then investing the $8,000 at 10% per year for the remaining 40 years?

$8, ( ) $362, $24,000 1 0 1 074 00040× + ⋅ = >

Selling An Asset

So the time at which an asset should besold is determined by

Rate-of-Return = r, the interest rate.

Arbitrage Arbitrage is trading for profit in

commodities which are not used for consumption.

E.g. buying and selling stocks, bonds, or stamps.

No uncertainty all profit opportunities will be found. What does this imply for prices over time?

Arbitrage

The price today of an asset is p0. Its price tomorrow will be p1. Should it be sold now?

The rate-of-return from holding the asset is

I.e.

Rp p

p= −1 0

0

( ) .1 0 1+ =R p p

Arbitrage

Sell the asset now for $p0, put the money in the bank to earn interest at rate r and tomorrow you have

( ) .1 0+ r p

Arbitrage

When is not selling best? When

I.e. if the rate-or-return to holding the asset the interest rate, then keep the asset.

And if then

so sell now for $p0.

( ) ( ) .1 10 0+ > +R p r p

R r>

R r<( ) ( )1 10 0+ < +R p r p

Arbitrage

If all asset markets are in equilibrium then for every asset.

Hence, for every asset, today’s price p0 and tomorrow’s price p1 satisfy

R r=

p r p1 01= +( ) .

Arbitrage

p r p1 01= +( )

I.e. tomorrow’s price is the future-value oftoday’s price. Equivalently,

pp

r01

1=

+.

I.e. today’s price is the present-valueof tomorrow’s price.

Arbitrage in Bonds

Bonds “pay interest”. Yet, when the interest rate paid by banks rises, the market prices of bonds fall. Why?

Arbitrage in Bonds A bond pays a fixed stream of payments

of $x per year, no matter the interest rate paid by banks.

At an initial equilibrium the rate-of-return to holding a bond must be R = r’, the initial bank interest rate.

If the bank interest rate rises to r” > r’ then r” > R and the bond should be sold.

Sales of bonds lower their market prices.

Taxation of Asset Returns

rb is the before-tax rate-of-return of a taxable asset.

re is the rate-of-return of a tax exempt asset.

t is the tax rate. The no-arbitrage rule is:

(1 - t)rb = re

I.e. after-tax rates-of-return are equal.

Financial Intermediaries

Banks, brokerages etc.– facilitate trades between people with

different levels of impatience– patient people (savers) lend funds to

impatient people (borrowers) in exchange for a rate-of-return on the loaned funds.

– both groups are (in principle) better off.