Lecture 2: Valuation, Atomic Prices, Complete
and Incomplete Markets
Economics of Finance
School of Economics, UNSW1
Valuation
Definition: Valuation is the process of determining the present
value of a security or productive investment.
Example: How much is a tree worth today (at time 0)?
48 apples Bad weather
63 apples Good weather
t = 0 t = 1
Present Value of a tree: PV = 0.285 · 63 + 0.665 · 48 = 49.875
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In matrix notation
The vector of atomic prices: (1×states)
p =
(
0.285 0.665
)
Good Weather Bad Weather
The vector of quantities: (states×1)
q =
(
63
48
)
Good Weather
Bad Weather
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Present value
The Present Value:
PV = p · q = (0.285 0.665)(63
48
)
= 49.875
Check the dimensions: (1× states)︸ ︷︷ ︸
p
· (states× 1)︸ ︷︷ ︸
q
= (1× 1)︸ ︷︷ ︸
PV
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Python matrix operations
The following Python commands will do the job:
p = np.array([0.285,0.665])
q = np.array([63,48])
pv=np.matmul(p,q.transpose())
print(pv)
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Net Present Value
The net present value of a set of claims is based on future
payments and any required payment in the present. If the Tree
is purchased for 49.875 apples
NPV =
(
1.00 0.285 0.665
)×
 −49.87563
48

NPV = −49.875 + 0.285× 63 + 0.665× 48 = 0
The net present value of a fairly priced investment is zero.
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Net Present Value (cont’d)
Assume you discover how to plant 60 apples in a way that will
produce 100 apples if the weather is good and 50 apples if the
weather is bad. Compute the net present value:
NPV = −60 + 0.285× 100 + 0.665× 50 = 1.75
Should you do it? YES. Why?
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Riskless Securities
Definition: A riskless security pays the same amount at a
given time, no matter what state of the world occurs.
• A riskless security is equivalent to a bundle of equal
amounts of atomic claims for a time period.
• In our example, a riskless security pays a fixed amount (say
X apples) at time period 1, whether the weather has been
good or bad.
• Equivalently, it is a bundle of X good weather apples (GA)
and X bad weather apples (BA).
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Riskless Securities
X apples Bad weather
X apples Good weather
t = 0 t = 1
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Riskless Securities
Question: What is the present value of a riskless security that
pays 20 apples in time 1?
The vector of atomic prices: (1×states)
p =
(
0.285 0.665
)
Good Weather Bad Weather
The vector of quantities: (states×1)
q =
(
20
20
)
Good Weather
Bad Weather
The Present Value:
PV = p · q = (0.285 0.665)(20
20
)
= 0.285 · 20 + 0.665 · 20 = 19
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The Discount Factor
Definition: The discount factor (for a certain date) represents
the present value of a payment of one unit to be made with
certainty at the specified future date.
The tree example:
PV = p · q = (0.285 + 0.665)︸ ︷︷ ︸
sum of the atomic prices
· 20 = 0.95︸︷︷︸
df(1)
· 20 = 19
• The discount factor for a date in question equals to the
sum of appropriate atomic prices (prices of basic atomic
securities)
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The Discount Factor
E.g. df(1) = 0.95
0.95 apples
1 apples Bad weather
pB = 0.665
1 apples Good weather
pG = 0.285
t = 0 t = 1
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Financing Methods
Say you’d like to set up an apple firm which consists of an apple
tree, i.e., you need 49.875 apples to purchase the tree.
There are two ways to finance this investment, issue bonds or
issue stocks. Assume your firm issues a bond:
The Apple Tree Firm promises to pay the holder 20
apples at the end of the year, no matter what the
weather has been.
This way the holder does not bear any face value risk (though
other types of risk, e.g., default risk or interest rate risk, etc.,
remain).
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Stock
If your firm issues a stock :
The Apple Tree Firm promises to pay the holder all
the apples left over after the bondholder has been
paid.
This way the holder bear the risk of the apple production net
the issued bond payment, BUT is entitled a voting right.
The bond represents the ownership of the money, i.e., prior
claim; the stock represents the ownership of the firm, i.e.,
residual claim.
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Principle of value additivity
What is the bond worth? What is the stock worth? Payment
vectors are:
qfirm =
(
63
48
)
; qbond =
(
20
20
)
; qstock =
(
43
28
)
The values are:
p× qfirm= 49.875;
p× qbond= 19.000;
p× qstock= 30.875;
Note that p× qfirm = p× (qbond + qstock). This is called
Principle of value additivity.
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Shareholding Structure
We know from principle of value additivity that
• either issuing bond or issuing stock or any proportion of
each, it wouldn’t change apple prices p
• Neither does it change the apple tree production and firm’s
value p× qfirm
How does shareholding structure matter?
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Net worth
The key is shareholder’s risk! Consider the instead of issuing 1
bond, you issue 2 bonds which takes a value of
19× 2 = 38,
which presents a liability of
qbond × 2 =
(
40
40
)
in period 1. Net worth of the share is now
49.875− 38 = 11.875
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Shareholder
The shareholder’s payment structure is now:
11.875 apples
8 apples Bad weather
23 apples Good weather
t = 0 t = 1
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Shareholder II
Instead of:
30.875 apples
28 apples Bad weather
43 apples Good weather
t = 0 t = 1
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Take-outs
While avoiding diluting voting right, issuing more bond makes
shareholder’s net worth more risky.
• In practice, the way to finance an investment depends on
the director board’s attitude towards risks.
• An aggressive, dictating director board is more likely to
issue more unites of bonds, and bears more risk;
• An modest, cooperative director board however, is more
likely to issue less bonds, issue more stocks and share risks
with other shareholders;
• Usually it also involve more complex factors, e.g., the bond
market capacity, possibility of merger and acquisition, or
even political factors.
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Inferring Atomic Security Prices
• Until now we have assumed that dealers stand ready to
buy and sell basic atomic securities;
• Even though there are financial instruments that resemble
atomic securities (e.g. insurance policy) this assumption is
not very realistic;
• We will relax this assumption and consider the world in
which only two securities are traded on a regular basis:
• The common stock of the Apple Tree Firm.
• The riskless bond of the Apple Tree Firm.
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Inferring Atomic Security Prices
Let Q {states × securities} be the payment matrix of the two
securities:
Q
(2×2)
=
(
20 43
20 28
)
Bond Stock
Good Weather
Bad Weather
Let pS {1×securities} be a vector of security prices:
pS
(1×2)
=
(
19.0 30.875
)
Bond Stock
Let n {securities×1} be a vector of portfolio holdings:
n
(2×1)
=
(
1
2
)
number of Bonds
number of Stocks
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Inferring Atomic Security Prices
Let c {states×1} be the vector of payments in each state, then
it must hold that
Q
(states×securities)
· n
(securities×1)
= c
(states×1)
In our example the above identity reads as
Q
(2×2)
· n
(2×1)
=
(
20 43
20 28
)(
1
2
)
=
(
106
76
)
where the vector of state-contingent payments is
c
(2×1)
=
(
106
76
)
Good Weather
Bad Weather
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Obtaining a desired portfolio
Question: What portfolio n will provide a desired set of state
state-contingent payments c?
If the payoff matrix Q is invertible, then the answer is simple:
n
(securities×1)
= Q−1
(securities×states)
· c
(states×1)
Note: If a matrix Q satisfies the following conditions:
(i) Q is a square matrix i.e. its number of rows equals to its
number of columns;
(ii) Q is non-singular i.e. its rows/columns are linearly
independent;
then Q−1 exists.
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Obtaining a desired portfolio
Example: Suppose we wish to have 845 apples if the weather is
good and 620 if the weather is bad.
• The vector of state-contingent payments is:
c
(2×1)
=
(
845
620
)
Good Weather
Bad Weather
• The payoff matrix is Q is invertible since its determinant is
different from zero:
det(Q) = det
(
20 43
20 28
)
= 20 · 28− 20 · 43 = −300 6= 0
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Desired Security
The portfolio vector that delivers the desired state-contingent
payoffs is given by n = Q−1c
n
(securities×1)
=
(
20 43
20 28
)−1(
845
620
)
=
(
10
15
)
Bonds
Stocks
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Atomic Security Prices
To obtain payment c, we can buy a portfolio n = Q−1c. This
portfolio will cost us
p = pS · n =
[
pS ·Q−1
]
c
Therefore, we can infer atomic security prices from the prices
and payments of the traded securities:
patom
(1×states)
= pS
(1×securities)
· Q−1
(securities×states)
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Q−1 revisited
Recall that Qn = c and that n = Q−1c. Say you’d wish to find
a portfolio such that c =
(
1
0
)
. This portfolio is given by the
first column of Q−1. In our example
n = Q−1
(
1
0
)
=
( −0.0933 0.1433
0.0667 −0.0667
)(
1
0
)
=
( −0.0933
0.0667
)
What is the present value of n?
psn =
(
19.0 30.875
)( −0.0933
0.0667
)
19× (−0.0933) + 30.875× 0.0667 = −1.7727 + 2.0593 = 0.285
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Atomic Security Prices
Example: How much would it cost to get 845 GA and 620 BA?
p = pS · n =
(
19.0 30.875
)(10
15
)
= 653.125
The prices of the atomic securities can be inferred from
patom= pS ·Q−1 =
(
19.0 30.875
)(20 43
20 28
)−1
=
(
0.285 0.665
)
Good W. Bad W.
Using inferred prices of the atomic securities we can price c as
follows
p = patom · c =
(
0.285 0.665
)(845
620
)
= 653.125
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Python: Finding atomic prices
Q = np.array([[20,43],[20,28]])
ps = np.array([19,30.875])
c = np.array([845,620])
n=np.matmul(np.linalg.inv(Q),c.transpose())
patom=np.matmul(ps,np.linalg.inv(Q))
print(n)
print(patom)
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Another look
In our example c =
(
845
620
)
, so given Q, our problem is
20n1 + 43n2 = 845
20n1 + 28n2 = 620
We could use the first equation
n1 =
(
845
20
− 43
20
n2
)
and use the result in the second
20
(
845
20
− 43
20
n2
)
+ 28n2 = 845− 15n2 = 620;n2 = 15.
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The Opportunity Set
Suppose you have a dollar.
What opportunity can you get from the market?
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Value Relative
Definition: Value relative associated with a given state of
nature is the future payment per unit invested, that will be
received if that state occurs.
In our example the matrix for the value relatives is:
vr
(2×2)
=
(
20/19 43/30.875
20/19 28/30.875
)
Bond Stock
Good Weather
Bad Weather
=
(
1.0526 1.3927
1.0526 0.9069
)
Bond Stock
Good Weather
Bad Weather
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Python
The following script will do the job:
Q = np.array([[20,43],[20,28]])
ps = np.array([19,30.875])
pS=np.array([ps,ps])
vr=np.divide(Q,pS)
print(vr)
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Value relative and return
Value of relative is the percentage value of an ending value in
terms of initial value.
E.g., if the weather is good, the value relative of a GA is
1/0.285 = 3.5088. If the weather is bad, the value relative of
a GA is 0/0.285 = 0.
Return is value relative, net 100%
return = vr− 1
An atomic security returns -100% in all states except the one it
corresponds to.
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The Opportunity Set
Definition: The opportunity set is the set of feasible future
payoffs available with a wealth of one unit of present value.
Example: The opportunities for future apples for a present
apple invested:
Q {states*securities} is the payment matrix of the two
securities:
Q
(2×2)
=
(
20 43
20 28
)
Bond Stock
Good Weather
Bad Weather
pS {1*securities} is a vector of security prices:
pS
(1×2)
=
(
19.0 30.875
)
Bond Stock
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Derivative securities
By combining existing securities (the bond and the stock), one
can synthesize a security that does not exist (e.g. a
state-contingent claim).
The result is often termed a derivative security.
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Opportunity set
• Let vrb =
(
1.0526
1.0526
)
be the value relative for the bond;
• Let vrs =
(
1.3927
0.9069
)
be the value relative for the stock;
• Let xs denote a proportion of wealth invested in the stock,
then the value relative for the portfolio is given by
vrp = xs · vrs+(1− xs) · vrb
= xs ·
(
1.3927
0.9069
)
+(1− xs) ·
(
1.0526
1.0526
)
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How much can we get from one apple?
By choosing a portfolio that includes positive (long) positions in
the Bond and in the Stock with a total present value of 1 apple,
an investor can obtain any position on the line segment
connecting the two securities in the Figure.
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Shorting securities
What about negative (short) positions in either security?
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Opportunity Set Frontier
Suppose, one can take negative positions in a security as long as
investor’s overall portfolio does not lead to negative net
payments in any state of the world
• Then, an investor can obtain any point on the line through
Bond and Stock extended all the way to the axes (see the
next figure)
• Value relative of the atomic securities can be found as
follows:
pa = pS ·Q−1 =
(
19 30.875
)(20 43
20 28
)−1
=
(
0.285 0.665
)
vratom
(2×2)
=
(
1/0.285 0/0.665
0/0.285 1/0.665
)
GW claim BW claim
=
(
3. 508 8 0
0 1. 503 8
)
GW claim BW claim
Good W.
Bad W.
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Plotting Opportunity Set
If we add to vr the value relatives of the atomic securities we
obtain:
These are points we can plot in the space of GA and BA.
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The Opportunity Set
Bond
GA
BA1 2
1
BW security
GW security
Stock
2
3
45°
1.5038
3.5088
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The Opportunity Set: Remarks
• Taking a negative position in the Stock amounts to signing
a document of the form: “I promise to pay the holder
whatever the firm (tree) pays its stockholders”
• By combining (in the right proportions):
• a long position in the Bond with a short position in the
Stock one can construct a pure “Bad Weather Claim”
• a short position in the Bond with a long position in the
Stock one can construct a pure “Good Weather Claim”
• By combining existing securities (the Bond and the Stock)
one can synthesize a security that does not exist (e.g. a
Good Weather claim). The result is termed a derivative
security, since it is derived from the existing securities.
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Arbitrage Opportunities
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Arbitrage Opportunities (cont.)
• Any security not priced in accordance with the atomic
prices implied by the traded securities will present an
opportunity for arbitrage
• For example, imagine a security Z appears outside the
opportunity set frontier
• Draw a line through Z to the origin; Denote ZZ the point
where the line intersects the opportunity set frontier;
• Payments ZZ can be obtained by a portfolio of the Bond
and the Stock worth 1 PA;
• Sell ZZ short, and use the proceeds (1 PA) to buy Z;
• Z pays more that ZZ (per apple invested) in every state of
the world, hence we obtain arbitrage opportunity