Maintained Assumptions

• still assume perfect market.
• allow uncertainty

Statistics and Random Variable

• A random variable is a variable whose outcome is random.
• We know the distribution, but we don’t know the outcome.

Fair bet

• When expected value of the variable is 0, it is considered to be fair bet.

Extrapolation works reasonably well for short - term standard deviation, but almost always poorly on mean.

Wealth effect

If someone offered you bet of $1000 if head, -$1000 if tail, I would have to pay you some money (premium) for you to play. This is because people are “risk - aversive”.

One of the main feature of the financial market is that $1000 is spread to 1000 people, so that they become risk - neutral. $1, -$1 bet is risk - neutral comparative to $1000, -$1000 bet.

Default (“credit”) Risk

Most loans have credit risk, in that the borrower can default. If the probability of borrower failing to pay you is high, default risk is also high.

Q1: Can the US government default? Do Treasury securities have any default risk?

No. US government has “Zero” default risk. US $ acts as “key currency”

Q2: If you were to land me $200 for return of $210 next year, with 1% of default, what is the expected rate of return?

$! \begin{aligned} 1\% : \$ 50 & \rarr \$ 0.5 \\ 99\% : \$ 250 & \rarr \$ 207.9 \\ \$ 208.4 & \rarr 4.2\% \end{aligned} $!

Q3: Would you prefer to make this loan or 5% government bond?

Obviously, you would take government bond because they have no default risk. Therefore, Promised rate of return Must be greater than 5% in order to be equally priced. In this case, it has to be 5.81%. (as there are 1% of default)

Promised rate of return is also called Quoted interest rate and they are NOT expected rate of return.

However, considering default risk will not be enough as there are also other risks and considerations such as :

• liquidity risk,
• risk - premia
• imperfect - market - risk

Important Generalization of Premium Decomposition

In a risk - neutral world,

Quoted Interest rate $$\geq$$ Expected Interest rate

Quoted Interest rate = Time Premium + Default Premium

(Time premium = risk - free govnt bond’s yield)

Expected Interest rate = Time Premium

How do we calculate default risk?

Large corporation have credit ratings, too, ranging from AAA to F

Typical AAA firm has ~0% probability of default over 10 years B firm has 20% probability of one non - payment over 5 years C firm has 50% probability of one non - payment over 6 ~ 8 years

Most of the yield spread (corporate bonds yield - (risk - free yield(treasury bond))) is due to chance of default.

CDS: Credit default swaps

You can buy insurance against default, called credit swaps. This market is over - the - counter. Sellers are often hedge funds who want to speculate on default. Buyers are often mutual funds or pension funds who want to reduce their risk exposure.

The expected rate of return on 1 year Treasuries is 5%

Q4 : What is the appropriate price for this project?

$! PV = \frac{E(payoff)}{1 + E(R)} = \frac{\$ 95}{1.05} = \$ 90.48 $!

What is the rate of return on the project in the good sate?

Good : $100 $$\rarr$$ 100/90.48 - 1 = 10.52 %

In the Bad state: - 44.74 %

What is the expected rate of return on the project? 5%

Levered Equity (stock) + Risk - free Bond

You can finance the project in one of two ways:

• Buy the whole thing with your saving
• Buy it with a mortgage and a smaller sum from your savings account. Then you just own the residual equity called “Levered Equity”

In this case, assume that 50 was a loan(= bond). This meaning that $50 will be promised as a return.

What should be the price of this loan and leveled equity? Bond would be 50/1.05 because market is perfect.

Expected payoff is $95, therefore levered equity will be $45/1.05.

You can see that if you promise $50 as bond, you finance 48%. In a good case, you get $50 (out of $42.86 you invested) In a bad case, you get nothing, making -100% lost.

Full project ownership is less riskier than levered Equity, and more riskier than Bond ownership.

Limited Liability

You are only responsible for what you invested, no more. It made it possible for owners to hand control to specialists and not worry for their entire holdings.

What if bond owner lands more than $50? then the risk of the bond holding becomes non - zero.