CS代考 Lecture 2: Valuation, Atomic Prices, Complete and Incomplete Markets

Lecture 2: Valuation, Atomic Prices, Complete and Incomplete Markets
Economics of Finance
School of Economics, UNSW1

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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)?
Good weather
Bad weather
Present Value of a tree: PV = 0.285 · 63 + 0.665 · 48 = 49.875

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)
􏰏63􏰐 Good Weather q = 48 Bad Weather

Present value
The Present Value:
PV =p·q= 0.285 0.665 48 =49.875
Check the dimensions: (1 × states) · (states × 1) = (1 × 1)
􏰈 􏰇􏰆 􏰉􏰈 􏰇􏰆 􏰉 􏰈􏰇􏰆􏰉

MATLAB matrix operations
The following MATLAB commands will do the job:
>> p = [0.285 0.665] p=
0.2850 0.6650 >> q = [63 48]’ q=
>> PV = p*q
PV = 49.8750

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
 −49.875  NPV =􏰀 1.00 0.285 0.665 􏰁× 63 
NPV =−49.875+0.285×63+0.665×48=0 The net present value of a fairly priced investment is zero.

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?

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).

Riskless Securities
Good weather
Bad weather

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)
􏰏20􏰐 Good Weather
q = 20 Bad Weather 􏰀 􏰁􏰏20􏰐
The Present Value:
PV =p·q= 0.285 0.665 20 =0.285·20+0.665·20=19

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) ·20=0.95·20=19
sum of the atomic prices
• The discount factor for a date in question equals to the sum of appropriate atomic prices (prices of basic atomic securities)

The Discount Factor
E.g. df(1) = 0.95
pG = 0.285
Good weather
0.95 apples
pB = 0.665
Bad weather

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 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).

If your firm issues a stock:
The Firm promises to pay the holder all
the apples left over after the bondholder has been
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.

Principle of value additivity
What is the bond worth? What is the stock worth? Payment vectors are:
􏰏 63 􏰐 􏰏 20 􏰐 􏰏 43 􏰐 qfirm= 48 ; qbond= 20 ; qstock= 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.

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?

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
􏰏 40 􏰐 qbond × 2 = 40
in period 1. Net worth of the share is now 49.875 − 38 = 11.875

The shareholder’s payment structure is now:
11.875 apples
Shareholder
Good weather
Bad weather

Instead of:
30.875 apples
Shareholder II
Good weather
Bad weather

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.

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 Firm. • The riskless bond of the Firm.

securities:
􏰏20 43􏰐 Good Weather Q = 20 28 Bad Weather
Inferring Atomic Security Prices
Let Q {states × securities} be the payment matrix of the two
Let pS {1×securities} be a vector of security prices:
Bond Stock
pS = 􏰀19.0 30.875􏰁
Let n {securities×1} be a vector of portfolio holdings:
Bond Stock
􏰏1􏰐 number of Bonds (2×1) 2 number of Stocks

Inferring Atomic Security Prices
Let c {states×1} be the vector of payments in each state, then it must hold that
Q·n=c (states×securities) (securities×1) (states×1)
In our example the above identity reads as
􏰏20 43􏰐 􏰏1􏰐 Q·n= =
􏰏106􏰐 (2×2) (2×1) 20 28 2 76
where the vector of state-contingent payments is
􏰏106􏰐 Good Weather (2×1) 76 Bad Weather

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 = Q−1 · c (securities×1) (securities×states) (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.

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:
􏰏845􏰐 Good Weather (2×1) 620 Bad Weather
• The payoff matrix is Q is invertible since its determinant is
different from zero:
det(Q)=det 20 28 =20·28−20·43=−300̸=0

Desired Security
The portfolio vector that delivers the desired state-contingent payoffs is given by n = Q−1c
􏰏20 43􏰐−1 􏰏845􏰐 􏰏10􏰐 Bonds n==
(securities×1) 20 28 620 15 Stocks

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= pS · Q−1 (1×states) (1×securities) (securities×states)

Q−1 revisited
Recall that Qn = c and that n = Q−1c. Say you’d wish to find
a portfolio such that c = 0 . This portfolio is given by the
n=Q 0 = 0.0667 −0.0667 0 = 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
first column of Q−1. In our example
−1􏰏1􏰐 􏰏−0.0933 0.1433 􏰐􏰏1􏰐 􏰏−0.0933􏰐

Atomic Security Prices
Example: How much would it cost to get 845 GA and 620 BA? 􏰀 􏰁􏰏10􏰐
p=pS ·n= 19.0 30.875 15 =653.125 The prices of the atomic securities can be inferred from
−1 􏰏20 43􏰐−1
patom= pS · Q = 􏰀19.0 30.875􏰁 = 􏰀0.285
Using inferred prices of the atomic securities we can price c as follows
p = patom · c =
0.285 0.665 620 = 653.125
20 28 Good W.
0.665􏰁 Bad W.

> > Q = [20 43; 20 28];
>> ps = [19 30.875];
>> c = [845; 620];
>> n = inv(Q)*c
>> p = ps*n
>> p atom = ps*inv(Q) p atom =
0.2850 0.6650
Example in MATLAB

Another look
In our example c = 620 , so given Q, our problem is
20n1 + 43n2 = 845 20n1 + 28n2 = 620
We could use the first equation
n1= 20−20n2 and use the result in the second
20 20 −20n2 +28n2 =845−15n2 =620;n2 =15.

The Opportunity Set
Suppose you have a dollar.
What opportunity can you get from the market?

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􏰐 Good Weather 20/19 28/30.875 Bad Weather
Bond Stock
􏰏1.0526 1.3927􏰐 Good Weather
1.0526 0.9069 Bad Weather Bond Stock

Octave(Matlab)
The following script will do the job in the command prompt:
>> Q = [20 43; 20 28]; >> ps = [19 30.875]; >> vr = Q./[ps; ps];

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.

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:
􏰏20 43􏰐 Good Weather Q = 20 28 Bad Weather
pS {1*securities} is a vector of security prices:
Bond Stock
pS = 􏰀19.0 30.875􏰁
Bond Stock

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.

Opportunity set
• Let vrb = 1.0526 be the value relative for the bond;
• Let vrs = 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
􏰏1.3927􏰐 􏰏1.0526􏰐 = xs · 0.9069 +(1 − xs) · 1.0526

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.

Shorting securities
What about negative (short) positions in either security?

vratom = (2×2)
􏰏1/0.285 0/0.285
0/0.665􏰐 1/0.665 =
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:
−1 􏰏20 43􏰐−1
= 􏰀0.285 0.665􏰁 0 􏰐 Good W.
1. 503 8 Bad W. BW claim
pa = pS · Q = 􏰀19 30.875􏰁
􏰏3. 508 8 0

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.

The Opportunity Set
GW security
BW security 1 1.5038 2

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.

Arbitrage Opportunities

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

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