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
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.
