Transcript Uncertainty, Default and Risk

Chapter 6
Uncertainty,
Default, and Risk
Copyright © 2009 Pearson Prentice Hall. All rights reserved.
Chapter 6 Outline
6.1 An Introduction to Statistics
6.2 Interest Rates and Credit Risk (Default
Risk)
6.3 Uncertainty in Capital Budgeting
6.4 Splitting Uncertain Project Payoffs into
Debt and Equity
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6-2
Uncertainty, Default, and Risk
Introduction
•
What happens if we still have perfect markets, but we don’t have perfect
forecasts and thus have plenty of uncertainty?
•
The main impact of uncertainty is to make our decisions more challenging due to
forecast errors, but our decision rule, NPV, still works best.
•
With uncertainty, the quoted return may differ from the expected return.
• The quoted return is also called the stated or promised return.
•
Expected returns are lower than quoted returns because firms may default.
•
Before we discuss firms raising capital with debt or equity issues, we have
to talk about statistics.
•
Wait!….don’t go……it’s basic stats….you’ll be fine…
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Uncertainty, Default, and Risk
Introduction to Statistics
•
Expected Value --
the most important statistical concept
•
•
•
the average probability of an event
is computed over future outcomes infinitely
Random Variable --
such as ‘coin toss outcome’
•
•
the item that is yet to occur in the future
Notation for Expected Outcome of a Random Variable (has a tilde)
(c)  Expected value of random event "c"
•
If a coin toss of heads pays $1 and tails pays $2, compute the expected value
(c)  Expected value of coin toss = Prob(Heads) $1  Prob(Tails) $2
(c)  $1.50
•
Once tossed, the outcome is known and is no longer a random variable.
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Uncertainty, Default, and Risk
Histograms
•A histogram is a graph of the distribution of possible outcomes.
FIGURE 6.1 A Histogram for a Random Variable with Two Equally
Likely Outcomes, $1 and $2
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Uncertainty, Default, and Risk
Fair Bets
•
A Fair Bet is a bet that costs its expected value.
•
If the cost of the bet equals its expected value, then it is fair.
•
What is the expected value of a bet that has these payoffs?
•
•
In other words, you get what you pay for.
If the bet is made over and over, both sides come out even.
$4 with a 16.7% chance
$10 with a 33.3% chance
$20 with a 50% chance
(D)  Expected value of Dice Roll = Prob(1) $4  Prob(2, 3) $10  Prob(4,5,6) $20
(D)  16.7% $4  33.3% $10  50% $20
(D)  $14
• You would pay $14 if you wanted to break-even in the long-term.
•
Some bets are not fair.
• Vegas has spent a lot of time convincing you to take less than fair bets.
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Uncertainty, Default, and Risk
Variance and Standard Deviation
•
Risk is the most important characteristic to know after return.
•
Risk is the variability of outcomes around an expected value or mean.
•
Standard deviation is the most common measure of risk. It is the square
root of the average squared deviation from the mean, or sqrt(Variance).
•
Looking at our $14 expected value or mean, we note the following:
Outcomes
Deviations
Squared
Prob weights
Wt’d Squared
$4
-$10
$100
16.7%
$16.7
$10
-$4
$16
33.3%
$5.3
$20
+$6
$36
50%
$18
Variance = sum of the weighted squares
Standard deviation is the square root of variance
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(taken from $14 mean)
(investors agree here)
(sum = Variance)
= $40.00
= $ 6.32
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Uncertainty, Default, and Risk
Risk Neutrality -- A Lead into Risk Aversion
• For now, we assume risk neutral investors: they take fair bets.
• To a risk neutral investor, all fair bets are taken.
• They will take a certain $1 or a 50-50 chance to earn $0 or $2.
• Risk neutral investors are motivated by the payoff they expect, not risk.
• Risk averse investors will take the certain $1 over the 50-50 chance.
• Both alternatives have an expected value of $1, but risk averse
investors require a higher return than risk neutral investors to take a
fair bet.
• Financial markets provide an invaluable service by spreading risks.
• Individuals see a smaller level of risk (think of diversification) due to the
lower aggregate risk aversion in the market.
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6-8
Uncertainty, Default, and Risk
Interest Rates and Credit Risk (Default Risk)
•
Risk Neutral Investors Demand Higher Promised Rates
•
When faced with the possibility of default (an uncertain cash flow), a risk neutral investor should
charge a higher quoted rate or promised rate. This compensates them for the lower expected
return due to default risk.
•
If a borrower of $1M at a rate of 10% has a 50% chance of default and will either pay back
$750,000 or $1.1M, depending on default outcome, the lender sees an expected return lower
than the 10% promised return desired or needed by the lender.
Prob(Default) • Payment if Default + Prob(Solvent) • Payment if Solvent = (payout)
50% • $750,000 + 50% • $1,100,00 = $925,000 Expected Value
•
The lender should not extend credit since the expected value is a loss of 7.5% on the loan. The
lender needs to increase the quoted rate to raise the desired expected value to $1.1M. The
quoted rate needs to be 45%!
50% • $750,000 + 50% • $1,450,00 = $1,100,000 Expected Debt Value
•
The 35% return above the needed return of 10% is called the default premium.
•
Expected values and returns matter, not promised returns.
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Uncertainty, Default, and Risk
Default Example with Probability Ranges: Payoff Table
•
Borrower has a 98% probability of full repayment, a 1% chance of paying
back 50% of the loan, and a 1% chance of paying back nothing. Assume
this is a loan for $200 at a rate of 5%, what is the expected payoff?
Probability
98%
1%
1%
X
Cash Flow = Expected Value
$210
$205.80
$100
$ 1.00
$ 0
$ 0.00
Expected Payoff $206.80
Promised rate was 5% but payoff is only a 3.4% return.
If you can buy a safe government bond that pays 5%, do that!
•
What rate is needed as a quoted rate to equal a payoff of $210?
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Uncertainty, Default, and Risk
Default Example with Probability Ranges: Expected rate
•
Borrower has a 98% probability of full repayment, a 1% chance of paying back 50% of
loan, and a 1% chance of paying back nothing. Assume this is a loan for $200 and a
safe return is 5%. What rate is needed as a quoted rate to equal a payoff of $210?
•
Find the full-repayment cash flow first:
Probability
98%
1%
1%
X
Cash Flow = Expected Value
$ ?
$209.00
$100
$ 1.00
$ 0
$ 0.00
Expected Payoff
$210.00
Solving for the full-repayment cash flow, $209/.98 = $213.27.
•
The promised rate will now be 6.63%, for an expected return of 5%.
You can now lend to the borrower because the expected rate equals 5%.
(r)  Expected rate = Prob(1) (6.63%)  Prob(2) (50%)  Prob(3) (100%)
(r)  Expected rate = 98% (6.63%)  1% (50%)  1% (100%)  5%
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Uncertainty, Default, and Risk
Deconstructing Quoted Rates of Return: Time and Default Premiums
•
Earlier, the lender expected to earn 5%, but quoted 6.63%. The difference
of 1.63% is the default premium for credit risk.
Promised rate
6.63%
= Time premium + Default premium
=
5%
+
1.63%
•
Safe government bonds have no default premium and the quoted rate and
the expected rate (time premium) are the same (5%).
•
Risky corporate bonds have a risk premium for default, so the quoted rate
is greater than the expected rate.
•
Because lenders do not expect to earn every default premium they charge
in a risk neutral setting, the expected realized default premium is 0%.
(r)  Expected realized default premium = 98% (1.63%)  1% (55%)  1% (105%)  0%
Note the gains and losses are taken from a 5% return or loss of time premium.
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Uncertainty, Default, and Risk
Other Debt Premiums
•
In addition to the time premium and the default premium, there are:
•
Liquidity premiums compensate the lender for future costs to sell bonds.
It is payment for the inability to convert to cash.
•
Risk premiums compensate investors for their willingness to take risk.
It is payment for risk aversion.
•
These are important, but not as large as the time and default premiums.
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6-13
Uncertainty, Default, and Risk
Credit Ratings and Default Rates
•
Firms such as Moody’s, Fitch, Duff and Phelps, and Standard & Poor’s
provide quality ratings on the credit risk of bonds.
•
The usual grading scale is AAA to C
……and yes there’s grade inflation, everyone wants a high A.
•
Bonds are separated into two grades or groups:
•
Investment grade -
high-quality borrowers
0.3% chance of default in any year
•
Speculative or junk - low-quality borrowers
3.5% to 5.5% chance of default in an average year
•
Junk bond default rates rise in recessions to 10% and fall in booms to 1.5%.
•
The amounts recovered in default by lenders vary by bond grades.
•
The amounts recovered also vary in economic boom vs. bust cycles.
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6-14
Uncertainty, Default, and Risk
Credit Ratings
TABLE 6.1 Rating Categories Used by Moody’s and Standard & Poor’s
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6-15
Uncertainty, Default, and Risk
Cumulative Probability of Default by Original Rating
FIGURE 6.2 Cumulative Probability of Default by
Original Rating
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6-16
Uncertainty, Default, and Risk
Bond Contract Feature: Call Risk and Early Prepayment
•
Bonds have option features that allow the borrower to change the terms.
•
One option feature is the ability to prepay the note before it is due.
Why would you want this? To take advantage of lower rates.
Example:
If you borrow at 10% and then rates drop to 5%:
You should pay back original loan early and take a new loan at 5%.
If you borrow at 10% and then rates rise to 15%:
You should keep your original loan.
•
For the lender this is not a good deal and thus lenders charge higher rates.
•
Individuals prepay mortgages, and it is usually called refinancing.
•
Firms do this with bonds: Callable bonds pay higher interest than
noncallable bonds since there is an early prepayment option.
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6-17
Uncertainty, Default, and Risk
Differences in Quoted Bond Returns in 2002
TABLE 6.2 Promised Interest Rates for Some Loans in May 2002
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6-18
Uncertainty, Default, and Risk
Credit Default Swaps
•
The credit default swap (CDS) is an innovation in finance; it emerged in the 1990s. It
allows investors to trade directly on the credit risk of a firm.
•
Two counterparties bet on the credit outcome of a firm with bonds outstanding. Assume a
pension fund owns $10M of bonds and is interested in protection against default on the bonds.
•
A hedge fund wants to bet that the $10M in bonds does not have default risk and is the
counterparty to the pension fund’s credit default swap. The hedge fund is providing insurance
and collecting a fee to do so.
•
Pension fund pays $130,000 to the hedge fund for credit protection.
•
•
If the bonds default, the hedge fund owes the pension fund $10M.
If the bonds do not default, the hedge fund’s profit is $130,000.
•
This is the cost of default, so it is a form of credit premium.
•
By executing this swap, the pension fund collects the time premium, but not the default
premium.
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6-19
Uncertainty, Default, and Risk
Uncertainty in Capital Budgeting: State-Contingent Payoffs
•
To find the value of a project, managers construct a payoff table. It has
expected discounted cash flows and uses expected rates of return.
•
Example of PV with State-Contingent Payoff Tables
Expected Building Value:
Event
Probability
Tornado
20%
Sunshine
80%
Expected Value 20%(T) + 80%(S)=
•
Value
$ 20,000
$100,000
$ 84,000
PV (r=10%)
$18,181.82
$90,909.09
$76,363.64
If the discount rate is 10%, the PV of the expected value equals $76,363.64.
PV  20% ($18,181.82)  80% ($90, 909.09)  $76, 363.64
or
PV 
20% ($20, 000)  80% ($100, 000)
1.10
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 $76, 363.64
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Uncertainty, Default, and Risk
State Dependent Rates of Return
•
If you buy the building for the $76,363.64, what is your expected return?
If Sunshine:
Pay $76,363.64
Value $100,000
If Tornado (dramatic, eh?):
Pay $76,363.64
Value $20,000
•
Probability 80%
Return 30.95%
Probability 20%
Return -73.81%
The expected return is the probability-weighted average return.
 (r)  Expected return = Prob(S) (30.95%)  Prob(T) (73.81%)
 (r)  Expected rate = 80% (30.95%)  20% (73.81%)  10%
•
The expected return of 10% is your required cost of capital: you paid $76,363.64.
•
If you pay a different value than the asset’s calculated PV, you’ll change your return.
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6-21
Uncertainty, Default, and Risk
Splitting the Projected Payoffs into Debt and Equity
•
•
•
Debt and equity are state-contingent claims that we can value.
•
Once we know the expected payoffs, we can sell the payoffs to debt and equity
investors.
•
We have to pay the liability (debt) owners first.
•
The remaining cash flow is owned by the equity owners.
Loans
•
A mortgage is a non-recourse loan: the lender can take back the building but
cannot ask the borrower for any more cash.
•
This is a limited liability.
•
Most financial securities offer limited liability.
Shareholders can only lose the value of their stock, nothing more.
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6-22
Uncertainty, Default, and Risk
Loans
•
What if we borrow $25,000 to own the building worth $76,363.64?
Now the building has two owners: a mortgage owner and the residual owner.
•
The mortgage owner, the lender, has to determine an appropriate loan rate.
If the lender expects to earn 10%, the quoted rate will be higher.
•
To solve, find the promised payoff that will result in an expected return of 10%:
Quoted Probability Weighted Values
80% ($Promise) + 20% ($20,000)
80% ($Promise)
Promise = $23,500 / .80
=
=
=
=
Expected Value
25,000 + 10%
$23,500
$29,375 (17.50% more than $25,000)
•
•
•
If the sun shines, the promised return is 17.50% ($29,375 / $25,000 - 1).
If the tornado hits, the return is -20% ($20,000 / $25,000 - 1).
Therefore, the expected return is .80(17.50%) + .20(-20%) = 10.0%.
•
•
The loan rate will be 17.50% to offset the loss probability and its expected rate is 10%.
If the loss or default probability were 0%, then the quoted loan rate would be 10%.
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6-23
Uncertainty, Default, and Risk
Levered Equity
•
What does the equity owner expect if $25,000 is borrowed?
•
•
•
•
•
The equity owner has a building worth $76,363.64 and a mortgage of $25,000.
Net worth (equity) equals $51,363.64, which the owner paid in cash.
In a year the house will be worth $100,000 (Sunshine) or $20,000 (Tornado).
The equity owner will owe the lender $25,000 + $4,375 interest or the $20,000
house.
The equity owner will have either $70,625 (100,000 – 29,375) or nothing.
Owner’s Payoff Table
Expected Building Value:
Event
Probability
Tornado
20%
Sunshine
80%
Value
$
0
$70,625
Expected Value 20%(T) + 80%(S)=$ 56,500
•
PV (r=10%)
$
0.00
$51,363.64
$51,363.64
If the appropriate rate is 10%, the owner’s expected value equals $51,363.64,
which is $25,000 less than the total value of $76,363.64.
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6-24
Uncertainty, Default, and Risk
Levered Equity Rate of Return
•Once we know the expected payoffs, we can find the rate of return to equity.
•
The equity owner has a beginning net worth of $51,363.64, which will rise or fall:
If Sunshine, return is +37.50%: ($70,635 - $51,363.64) / $51,363.64
If Tornado, return is -100%:
($0 - $51,363.64) / $51,363.64
(r)  Expected return = Prob(S) (return if S)  Prob(T) (return if T)
(r)  Expected rate = 80% (37.5%)  20% (100%)  10%
Since the owner also used 10% cost of capital when determining his initial purchase
price, the owner expects to earn 10%. The real world could differ from expectations,
of course!
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6-25
Uncertainty, Default, and Risk
Debt and Equity Payoff Tables Summarized
TABLE 6.3 Payoff Table and Overall Values and Returns
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6-26
Uncertainty, Default, and Risk
Which is More Risky: Equity, Debt, or Full Ownership?
FIGURE 6.3 Three Probability Histograms for Project
Rates of Return
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6-27
Uncertainty, Default, and Risk
What Leverage Really Means – Financial and Operational
•
•
•
Debt is often called leverage. Equity is levered ownership with debt.
Leverage increases volatility, our home owner will earn either 37.5% or -100%.
Operational leverage is a trade-off between fixed and variable costs.
High fixed costs increase the volatility of earnings.
TABLE 6.4 Financial and Real Leverage
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6-28
Uncertainty, Default, and Risk
Many Possible Outcomes: Plot E(V) vs. Promised
FIGURE 6.4 Promised versus Expected Payoff for a Loan on the
Project with Five Possible Payoffs
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Uncertainty, Default, and Risk
Mistake: Do Not Discount a Promised Payoff with a Promised Rate of Return
•
We should always discount the expected payoff with the expected rate of
return. If we don’t, then we will make errors.
•
If a $100,000 bond promises 16% with a 50% chance of defaulting on its interest
payments, do not discount the promised cash flow by the promised rate.
•
If the risk-free rate is 10% and the credit premium is 2%, the promised rate is
12% and the PV of $100,000 plus 16,000 discounted at 12% is $115,195.
You would incorrectly believe the NPV is a positive $3,571.
•
The correct valuation is to find the PV of both $100,000 + E(Interest).
If we find the probability-weighted cash flow, we can use r = 10%.
•
NPV = -$100,000 + PV(100,000) + PV(50% of $16,000) = -$1,818
This is a bad investment using expected values discounted by the expected rate.
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