Economics 434Economics 434

Financial Markets Professor Burton

University of VirginiaFall 2014

October 21, 2014

Answer to Question 2 on First Mid Term

October 21, 2014

Imagine that there are only four assets in the world. Three are publicly traded stocks (ABC, MNO, and XYZ) and the fourth asset is the risk free asset.

Price of One Share Shares Outstanding

ABC Stock $ 10 1,000

MNO Stock $ 8 1,000

XYZ Stock ? 1,000

Imagine that we are in a CAPM equilibrium with the prices and outstanding shares given in the above table with a risk free rate of 3 percent. Note that the market capitalization of ABC is $ 10,000.

If a person we know has a portfolio worth $ 100 with half of his total portfolio in the risk free asset, while one fourth of his total portfolio (including the risk free asset) is held ABC stock.

a. How much XYZ stock does he own? Answer: He owns 2.5 shares at $ 2 per share ($ 5 worth) which is 20 percent of the market basket (because since 10 percent of the

market basket has to be XYZ stock and hence 10 percent of his risky asset portfolio must also be XYZ stock)

b. What is the price of XYZ stock?Answer: $ 2 per share (see reasoning from above)

c. If the ABC stock has a beta of 1.2 and MNO has a beta of .5, what is the beta of XYZ stock?Answer: Beta of ABC = 2.00,( so that ½ times 1.2 plus 4/10 times .5 plus .1 times 2.00 = 1.00 the market beta).

d. If the expected return of MNO is 5 percent, what is the expected return of ABC stock?Answer: Since risk free rate is 3 %, the excess return of MNO is 2 %. The beta of MNO is .5, so the excess return of the market must be 4 %. Given a beta of 1.2, ABC must have an excess return of 5 percent, which means its expected return must be 8 percent.

Today

Tomorrow

s1

s3

s2

And, we may not have any idea what the probabilities of s1, s2, s3 may be!!

October 21, 2014

Fundamental Theorem of Finance

• The Assumption of No Arbitrage is True

• If and only if

• There exist positive state prices (one for each state) that represent the price of a security has a return of one dollar in that state and zero for all other states

October 21, 2014

Significance of FT• A new security’s value can be determined by multiplying its payoffs in

each state by the state price and adding these products up• The price of the risk-free asset which produces one unit next period

regardless of which state occurs is simply: q1 + q2 + q3

• State prices are generally not the same for every state: higher for ‘bad states,’ lower for ‘good states’– An asset that pays x in a good state (and in no other state) has a lower price than an

asset that pays x in a bad state (and in no other state)– Diversification and risk management is about buying (owning) securities that pay off in

bad states

October 21, 2014

Calculating the Risk Free Rate?(using state prices q1, q2,q3)

• By investing q1 + q2 + q3, we can earn one in the next period, regardless of which state actually occurs

• Total Return = 1 – (q1 + q2 + q3)

• Cost = q1 + q2 + q3

• So• r = (1/(q1 + q2 + q3)) - 1 or

• 1 + r = 1 divided by (q1 + q2 + q3)

October 21, 2014

Discounted Expected Value(using ith security’s returns)

October 21, 2014

Discounted Expected Value = π1Ri1 + π2Ri2 + π3Ri3

1+r

Insert: (1+r) = 1/(q1 + q2 + q3)

And insert: πi = qi/(q1 + q2 + q3)

Discounted Expect Value = q1 Ri1 + q2 Ri2 + q3 Ri3Result:

Which is equal to Pi (from the definition of state price)

Price Equals Expected Value Discounted

October 21, 2014

π1Ri1 + π2Ri2 + π3Ri3

1+rPi =

But the probabilities are not “true” probabilities, but are, instead, “risk-adjusted” probabilities

Martingales and the EMH

• EMH = Efficient Market Hypothesis

• Definition of a martingale process: E[Xt] = Xs where t>=s

• In words, the expected future value of a (random) variable is equal to whatever it’s current value is.

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Example of a Martingale Process

October 21, 2014

Imagine flipping a coin and keeping track of the number of heads and tails.

Everytime you flip heads add one to your total; everytime you flip tails subtract onefrom your total.

When you start the total is zero. What is the expected number of heads minus tails at any future time after any particular number of flips?

Answer: zero

What if first flip is heads? Then our total is plus one. Now imagine that you willcontinue to flip a specific but unknown number of times. What will be the totalof heads minus tails after you have flipped this specific number of times?

Answer: one

Our total of heads minus tails coin flips is a martingale process

Imagine more than period(where r is the same for all future periods)

October 21, 2014

Define

Using this definition, Xt is a martingale, meaning that:

Xt = Eπ [Xt+s]

Where the Eπ is the expectation using “risk-adjusted”probabilities

Default Free Securites (Sovereign Debt)

• US Treasuries ($ 13 trillion outstanding)– Bills (less than one year in original maturity)– Notes (ten years or less, longer than one year)– Bonds (greater than ten years at issuance)

• Random facts– Bills are called coupon and/or discount issues: 3

mo, 6 mo, year bills. Year assumed to be 360 days.– Notes and bonds are “coupon” issues; pay fixed

coupons twice yearlyOctober 21, 2014

Naming conventions

• Bills named by their maturity date– “12/15 14” for example

• Notes and bonds are named by: (i) their coupon rate; and (ii) their maturity date– “14s of Nov 11” was originally issued in mid-

November of 1981 with a 14 coupon. It matured on November 15, 2011. (Assume $ 100,000 principal. Then, it paid $ 7,000 on May 15th and Nov 15th starting May 15th 1982, ending Nov 15th, 2011

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October 21, 2014