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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.
Assets
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
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 ) = −1000 + 1000t − 10t

2
Value

Selling An Asset

24000
19000
14000
9000
4000
-1000 0

10

20

30

40

50

60

Years
Selling An Asset
V ( t ) = −1000 + 1000t − 10t
Maximum value occurs when

V'( t ) = 1000 − 20t = 0

That is, when t = 50.

2
Value

Selling An Asset

24000

Max. value
of $24,000
is reached
at year 50.

19000
14000
9000
4000
-1000 0

10

20

30

40

50

60

Years
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 Asset
The rate-of-return of the asset at time t is

In our example,

V'( t )
.
V( t )

2

V (t ) = −1000 + 1000 t − 10 t .

so

V'( t ) = 1000 − 20t = 0
V'( t )
1000 − 20t
=
.
V ( t ) − 1000 + 1000t − 10t 2
Selling An Asset
The asset should be sold when

V'( t )
1000 − 20t
=
= 0⋅1
V ( t ) − 1000 + 1000t − 10t 2
That is, when t = 10.
Value

Selling An Asset

24000

Max. value
of $24,000
is reached
at year 50.

19000
14000

slope
= 0.1

9000
4000
-1000 0

10

20

30

40

50

60

Years
Value

Selling An Asset

24000

Max. value
of $24,000
is reached
at year 50.

19000
14000

slope
= 0.1

9000
4000
-1000 0

10

Sell at 10 years
even though the
asset’s value is
only $8,000.
20

30

40

50

60

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?
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?
40

$8,000 × (1 + 0 ⋅ 1)

= $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
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
p1 − p0
asset is
R=
I.e.

p0

(1 + R )p0 = p1 .
Arbitrage
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 .
Arbitrage
When is not selling best? When
(1 + R )p0 > (1 + r )p0 .
I.e. if the rate-or-return to holding the
asset R > r the interest rate, then
keep the asset.
And if R < r then
(1 + R )p0 < (1 + r )p0
so sell now for $p0.
Arbitrage
If all asset markets are in equilibrium
then R = r for every asset.
Hence, for every asset, today’s price
p0 and tomorrow’s price p1 satisfy

p1 = (1 + r )p0 .
Arbitrage
p1 = (1 + r )p0
I.e. tomorrow’s price is the future-value of
today’s price. Equivalently,

p1
p0 =
.
1+r

I.e. today’s price is the present-value
of 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 better off.

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Ch11

  • 2. 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.
  • 3. Assets 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.
  • 4. Selling An Asset Q: When should an asset be sold? When its value is at a maximum? No. Why not?
  • 5. Selling An Asset Suppose the value of an asset changes with time according to V ( t ) = −1000 + 1000t − 10t 2
  • 7. Selling An Asset V ( t ) = −1000 + 1000t − 10t Maximum value occurs when V'( t ) = 1000 − 20t = 0 That is, when t = 50. 2
  • 8. Value Selling An Asset 24000 Max. value of $24,000 is reached at year 50. 19000 14000 9000 4000 -1000 0 10 20 30 40 50 60 Years
  • 9. 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%.
  • 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.
  • 11. Selling An Asset The rate-of-return of the asset at time t is In our example, V'( t ) . V( t ) 2 V (t ) = −1000 + 1000 t − 10 t . so V'( t ) = 1000 − 20t = 0 V'( t ) 1000 − 20t = . V ( t ) − 1000 + 1000t − 10t 2
  • 12. Selling An Asset The asset should be sold when V'( t ) 1000 − 20t = = 0⋅1 V ( t ) − 1000 + 1000t − 10t 2 That is, when t = 10.
  • 13. Value Selling An Asset 24000 Max. value of $24,000 is reached at year 50. 19000 14000 slope = 0.1 9000 4000 -1000 0 10 20 30 40 50 60 Years
  • 14. Value Selling An Asset 24000 Max. value of $24,000 is reached at year 50. 19000 14000 slope = 0.1 9000 4000 -1000 0 10 Sell at 10 years even though the asset’s value is only $8,000. 20 30 40 50 60 Years
  • 15. 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?
  • 16. 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? 40 $8,000 × (1 + 0 ⋅ 1) = $362,074 > $24,000
  • 17. Selling An Asset So the time at which an asset should be sold is determined by Rate-of-Return = r, the interest rate.
  • 18. 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?
  • 19. 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 p1 − p0 asset is R= I.e. p0 (1 + R )p0 = p1 .
  • 20. Arbitrage 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 .
  • 21. Arbitrage When is not selling best? When (1 + R )p0 > (1 + r )p0 . I.e. if the rate-or-return to holding the asset R > r the interest rate, then keep the asset. And if R < r then (1 + R )p0 < (1 + r )p0 so sell now for $p0.
  • 22. Arbitrage If all asset markets are in equilibrium then R = r for every asset. Hence, for every asset, today’s price p0 and tomorrow’s price p1 satisfy p1 = (1 + r )p0 .
  • 23. Arbitrage p1 = (1 + r )p0 I.e. tomorrow’s price is the future-value of today’s price. Equivalently, p1 p0 = . 1+r I.e. today’s price is the present-value of tomorrow’s price.
  • 24. Arbitrage in Bonds Bonds “pay interest”. Yet, when the interest rate paid by banks rises, the market prices of bonds fall. Why?
  • 25. 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.
  • 26. 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.
  • 27. 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 better off.