范里安中微课件 (1)

Sell at 10 years even though the asset’s value is only $8,000.
20 30 40 50
10
60
Years
Selling An Asset
u 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
The rate-of-return of the asset at time t is
In our example,
V'( t ) . V( t )
V ( t ) 1000 1000 t 10 t .
V '( t ) 1000 20 t 0 V'( t ) 1000 20t . V ( t ) 1000 1000t 10t 2
Financial Intermediaries
u 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 better off.
Selling An Asset
u What
$8,000 ( 1 0 1)
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? 40
Arbitrage
u When
is not selling best? When ( 1 R )p 0 ( 1 r )p 0 . I.e. if the rate-or-return to holding the asset R r the interest rate, then keep the asset. u And if R r then ( 1 R )p 0 ( 1 r )p 0 so sell now for $p0.
is trading for profit in commodities which are not used for consumption. u E.g. buying and selling stocks, bonds, or stamps. u No uncertainty all profit opportunities will be found. What does this imply for prices over time?
p1 ( 1 r )p0 .
Arbitrage
p1 ( 1 r ) p 0
I.e. tomorrow’s price is the future-value of today’s price. Equivalently, p1 p0 . I.e. today’s price is the present-value of tomorrow’s price.
Asset Markets
Assets
u An
asset is a commodity that provides a flow of services over time. u E.g. a house, or a computer. u A financial asset provides a flow of money over time -- a security.
2
so
Selling An Asset
The asset should be sold when
V'( t ) 1000 20t 01 V ( t ) 1000 1000t 10t 2
That is, when t = 10.
Value
24000 19000 14000 9000 4000 -1000 0
uA
bond pays a fixed stream of payments of $x per year, no matter the interest rate paid by banks. u At an initial equilibrium the rate-of-return to holding a bond must be R = r’, the initial bank interest rate. u If the bank interest rate rises to r” > r’ then r” > R and the bond should be sold. u Sales of bonds lower their market prices.
Selling An Asset
u Q:
Suppose the interest rate is 10%. When should the asset be sold? u A: When the rate-of-return to holding the asset falls to 10%. u Then it is better to sell the asset and put the proceeds in the bank to earn a 10% rate-of-return from interest.
R
I.e.
p0
( 1 R )p0 p1 .
Arbitrage
u Sell
the asset now for $p0, put the money in the bank to earn interest at rate r and tomorrow you have
(1 r )p0 .
Taxation of Asset Returns
u rb
is the before-tax rate-of-return of a taxable asset. u re is the rate-of-return of a tax exempt asset. u t is the tax rate. u The no-arbitrage rule is: (1 - t)rb = re u I.e. after-tax rates-of-return are equal.
Arbitrage
u The
price today of an asset is p0. Its price tomorrow will be p1. Should it be sold now? u The rate-of-return from holding the asset is p1 p0
Selling An Asset
u Q:
When should an asset be sold? u When its value is at a maximum? u No. Why not?
Selling An Asset
u Suppose
the value of an asset changes with time according to
10
20
30
40
50
60
Years
Selling An Asset
u 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. u E.g. if an asset valued at $1,000 earns $100 then its rate-of-return is 10%.
V ( t ) 1000 1000 t 10 t
2
Value
24000 19000 14000 9000 4000 -1000 0
Selling An Asset
10
20
30
40
50
60
Years
Selling An Asset
V ( t ) 1000 1000 t 10 t
$362,074 $24,000
Selling An Asset
So the time at which an asset should be sold is determined by Rate-of-Return = r, the interest rate.
Arbitrage
u Arbitrage
Maximum value occurs when
2
V '( t ) 1000 20 t 0
That is, when t = 50.
Value
24000 19000 14000 9000 4000 -10000
Selling An Asset
Max. value of $24,000 is reached at year 50.
1rLeabharlann Arbitrage in Bonds
u Bonds
“pay interest”. Yet, when the interest rate paid by banks rises, the market prices of bonds fall. Why?
Arbitrage in Bonds
Arbitrage
u If
all asset markets are in equilibrium then R r for every asset. u Hence, for every asset, today’s price p0 and tomorrow’s price p1 satisfy
Assets
u Typically
asset values are uncertain. Incorporating uncertainty is difficult at this stage so we will instead study assets assuming that we can see the future with perfect certainty.
Selling An Asset
slope = 0.1 Max. value of $24,000 is reached at year 50.
10
20
30
40
50
60
Years
Value
24000 19000 14000 9000 4000 -1000 0
Selling An Asset
slope = 0.1 Max. value of $24,000 is reached at year 50.