L E P M A

Transcription

L E P M A

The Complete Guide To CAS Exam 9
(Spring 2013)
http://www.actuarialtraining.com
Mark Hoffmann, Ph.D., FCAS
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CAS Exam 9 - Spring 2013
2
M. Hoffmann
Contents
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M
A Portfolio Theory and Equilibrium in Capital Markets
A.1 BKM 6: Capital Allocation . . . . . . . . . . . . . . . .
A.1.1 Risk Aversion . . . . . . . . . . . . . . . . . . . .
A.1.2 Mean-Variance Criterion . . . . . . . . . . . . . .
A.1.3 Capital Allocation . . . . . . . . . . . . . . . . .
A.1.4 Past Exam Problems . . . . . . . . . . . . . . . .
A.2 BKM 7: Optimal Portfolios . . . . . . . . . . . . . . . .
A.2.1 Two Risky Assets . . . . . . . . . . . . . . . . . .
A.2.2 Minimum Variance Portfolio . . . . . . . . . . . .
A.2.3 Optimal Risky Portfolio . . . . . . . . . . . . . .
A.2.4 Asset Allocation . . . . . . . . . . . . . . . . . .
A.2.5 The Markowitz Portfolio Selection . . . . . . . . .
A.2.6 The Separation Property . . . . . . . . . . . . . .
A.2.7 Diversification . . . . . . . . . . . . . . . . . . . .
A.2.8 Risk Pooling and Risk Sharing . . . . . . . . . . .
A.3 BKM 8: Index Models . . . . . . . . . . . . . . . . . . .
A.3.1 Single Index Models . . . . . . . . . . . . . . . .
A.3.2 The Optimal Risky Portfolio . . . . . . . . . . . .
A.4 BKM 9: The Capital Asset Pricing Model . . . . . . . .
A.4.1 The Capital Asset Pricing Model . . . . . . . . .
A.5 BKM 10: Arbitrage Pricing Theory . . . . . . . . . . . .
A.5.1 Arbitrage Pricing Theory . . . . . . . . . . . . . .
A.5.2 Expected Returns . . . . . . . . . . . . . . . . . .
A.5.3 The Chen, Ross, Roll Model . . . . . . . . . . . .
A.5.4 The Fama-French Three-Factor Model . . . . . .
A.6 BMK 11: Efficient Markets . . . . . . . . . . . . . . . . .
A.6.1 The Efficient Market Hypothesis . . . . . . . . . .
A.6.2 Implications of the EMH . . . . . . . . . . . . . .
CONTENTS
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M
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A
B Asset-Liability Management
B.1 BKM 15: The Term Structure of Interest
B.1.1 The Yield Curve . . . . . . . . .
B.1.2 Forward Rates . . . . . . . . . . .
B.1.3 Term Structure . . . . . . . . . .
B.1.4 Past Exam Problems . . . . . . .
B.2 BKM 16: Managing Bond Portfolios . .
B.2.1 Interest Rate Sensitivity . . . . .
B.2.2 Duration . . . . . . . . . . . . . .
B.2.3 Convexity . . . . . . . . . . . . .
B.2.4 Mortgage-Backed Securities . . .
B.2.5 Active Bond Management . . . .
B.2.6 Passive Bond Management . . . .
B.2.7 Financial Engineering . . . . . . .
B.3 Hull 4: Interest Rates . . . . . . . . . . .
B.3.1 Reference Rates . . . . . . . . . .
B.3.2 Interest Rates . . . . . . . . . . .
B.3.3 Zero Coupon Rates . . . . . . . .
B.3.4 Bond Pricing . . . . . . . . . . .
B.3.5 Bootstrapping Zero rates . . . . .
B.3.6 Forward Rates . . . . . . . . . . .
B.3.7 Forward Rate Agreements . . . .
B.3.8 Duration . . . . . . . . . . . . . .
B.3.9 Convexity . . . . . . . . . . . . .
B.3.10 Term structure of interest rates .
B.4 Hull 7: Swaps . . . . . . . . . . . . . . .
B.4.1 Currency Swaps . . . . . . . . . .
B.4.2 Comparative Advantage . . . . .
B.5 Feldblum: Asset Liability Matching . . .
B.5.1 Asset-Liability Matching . . . . .
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Event Studies . . . . . . . . . . . . . . . . . . . . . .
Market Efficiency . . . . . . . . . . . . . . . . . . . .
Past Exam Problems . . . . . . . . . . . . . . . . . .
12: Behavioral Finance . . . . . . . . . . . . . . . . .
Information Processing Errors and Behavioral Biases
Other Topics . . . . . . . . . . . . . . . . . . . . . .
Technical Analysis and Behavioral Finance . . . . . .
Past Exam Problems . . . . . . . . . . . . . . . . . .
S
A.6.3
A.6.4
A.6.5
A.7 BKM
A.7.1
A.7.2
A.7.3
A.7.4
M. Hoffmann
CONTENTS
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C Financial Risk Management
C.1 Hull 23: Credit Risk . . . . . . . . . . . . . . . . . . . . . .
C.1.1 Default Probabilities from Historical Data . . . . . .
C.1.2 Default Probabilities from Bond Prices . . . . . . . .
C.1.3 Asset Swaps . . . . . . . . . . . . . . . . . . . . . . .
C.1.4 Historical Defaults vs. Bond Prices . . . . . . . . . .
C.1.5 Default Probabilities from Equity Prices . . . . . . .
C.1.6 Derivatives Transactions . . . . . . . . . . . . . . . .
C.1.7 Credit Risk Mitigation . . . . . . . . . . . . . . . . .
C.1.8 Gaussian Copula . . . . . . . . . . . . . . . . . . . .
C.1.9 Credit VaR . . . . . . . . . . . . . . . . . . . . . . .
C.1.10 Past Exam Problems . . . . . . . . . . . . . . . . . .
C.2 Hull 24: Credit Derivatives . . . . . . . . . . . . . . . . . . .
C.2.1 Credit Default Swaps . . . . . . . . . . . . . . . . . .
C.2.2 Valuation of Credit Default Swaps . . . . . . . . . . .
C.2.3 Credit Indices . . . . . . . . . . . . . . . . . . . . . .
C.2.4 CDS Forwards and Options . . . . . . . . . . . . . .
C.2.5 Basket Credit Default Swaps . . . . . . . . . . . . . .
C.2.6 Total Return Swaps . . . . . . . . . . . . . . . . . . .
C.2.7 Asset-Backed Securities . . . . . . . . . . . . . . . . .
C.2.8 Collateralized Debt Obligation . . . . . . . . . . . . .
C.2.9 Role of Correlation in a Basket CDS and CDO . . . .
C.2.10 New Exercises . . . . . . . . . . . . . . . . . . . . . .
C.2.11 New Exercises - Solutions . . . . . . . . . . . . . . .
C.3 Coval, Jurek, Stafford: The Economics of Structured Finance
C.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
C.3.2 Manufacturing AAA-rated Securities . . . . . . . . .
C.3.3 The Challenge of Rating Structured Finance Assets .
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M. Hoffmann
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P
B.5.2 Cash Flow Matching . . . . . . . . . .
B.5.3 Duration Matching . . . . . . . . . . .
B.5.4 Loss Reserve Duration . . . . . . . . .
B.5.5 Past Exam Problems . . . . . . . . . .
B.6 Noris: Asset/Liability Management Strategies
B.6.1 The Market Value of a P&C Company
B.6.2 Managing Market Value Surplus . . . .
B.6.3 Duration Gap . . . . . . . . . . . . . .
B.7 Panning: Managing Interest Rate Risk . . . .
B.7.1 Franchise Value . . . . . . . . . . . . .
B.7.2 Sensitivity to Interest Rates . . . . . .
CONTENTS
C.6
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C.8
C.9
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C.5
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C.4
C.3.4 The Pricing of Systematic Risk in Structured Products
C.3.5 The Rise and Fall of the Structured Finance Market . .
C.3.6 Implications and Conclusions . . . . . . . . . . . . . .
C.3.7 Past Exam Problems . . . . . . . . . . . . . . . . . . .
C.3.8 Additional Exercises . . . . . . . . . . . . . . . . . . .
C.3.9 Additional Exercises - Solutions . . . . . . . . . . . . .
AAA: Report of the Life Liquidity Work Group . . . . . . . .
C.4.1 Background . . . . . . . . . . . . . . . . . . . . . . . .
C.4.2 Executive Overview . . . . . . . . . . . . . . . . . . . .
C.4.3 Liquidity Risk and its Causes . . . . . . . . . . . . . .
C.4.4 Management of Liquidity Risk . . . . . . . . . . . . . .
C.4.6 Additional Exercises . . . . . . . . . . . . . . . . . . .
C.4.7 Additional Exercises - Solutions . . . . . . . . . . . . .
Basel Committee on Banking Supervision: Liquidity Risk . . .
C.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
C.5.2 Liquidity Risk Management Challenges . . . . . . . . .
C.5.3 National Liquidity Regimes . . . . . . . . . . . . . . .
C.5.4 Initial Lessons from the Current Episode of Stress . . .
C.5.5 New Exercises . . . . . . . . . . . . . . . . . . . . . . .
C.5.6 New Exercises - Solutions . . . . . . . . . . . . . . . .
Basel Committee on Banking Supervision: Principles . . . . .
C.6.1 Principle 5 . . . . . . . . . . . . . . . . . . . . . . . . .
C.6.2 Principle 6 . . . . . . . . . . . . . . . . . . . . . . . . .
C.6.3 Principle 7 . . . . . . . . . . . . . . . . . . . . . . . . .
C.6.4 Principle 9 . . . . . . . . . . . . . . . . . . . . . . . . .
C.6.5 Principle 10 . . . . . . . . . . . . . . . . . . . . . . . .
C.6.6 New Exercises . . . . . . . . . . . . . . . . . . . . . . .
C.6.7 New Exercises - Solutions . . . . . . . . . . . . . . . .
Cummins: CAT Bonds and Other Risk-Linked Securities . . .
C.7.1 CAT Bonds . . . . . . . . . . . . . . . . . . . . . . . .
C.7.2 Sidecars . . . . . . . . . . . . . . . . . . . . . . . . . .
C.7.3 Catastrophe Equity Puts (Cat-E-Puts) . . . . . . . . .
C.7.4 Catastrophe Risk Swaps . . . . . . . . . . . . . . . . .
C.7.5 Industry Loss Warranties (ILW) . . . . . . . . . . . . .
C.7.6 Growth Impediments . . . . . . . . . . . . . . . . . . .
Butsic: Solvency Measurement for RBC Applications . . . . .
C.8.1 Expected Policyholder Deficit . . . . . . . . . . . . . .
C.8.2 EPD for Continuous Distributions . . . . . . . . . . . .
Culp, Miller, Neves: Value at Risk / Uses and Abuses . . . . .
M. Hoffmann
CONTENTS
M. Hoffmann
7
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265
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291
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341
341
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M
Provision
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S
D Rate of Return, Risks Loads, and Contingency
D.1 Feldblum: The Internal Rate of Return . . . . .
D.1.1 Summary . . . . . . . . . . . . . . . . .
D.1.2 Surplus Allocation . . . . . . . . . . . .
D.1.3 Example . . . . . . . . . . . . . . . . . .
D.1.4 Long Tail vs. Short Tail Lines . . . . . .
D.1.5 Insurance Risks . . . . . . . . . . . . . .
D.1.6 Past Exam Problems . . . . . . . . . . .
D.2 McClenahan: Insurance Profitability . . . . . .
D.2.1 Summary . . . . . . . . . . . . . . . . .
D.2.2 Past Exam Problems . . . . . . . . . . .
D.3 Ferrari: Return on Equity . . . . . . . . . . . .
D.3.1 Summary . . . . . . . . . . . . . . . . .
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Measurement
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A
C.9.1 Value at Risk . . . . . . . . .
C.9.2 Proctor & Gamble . . . . . .
C.9.3 Barings . . . . . . . . . . . .
C.9.4 Orange County . . . . . . . .
C.9.5 Metallgesellschaft . . . . . . .
C.9.6 Alternatives to VaR . . . . .
C.9.7 Past Exam Problems . . . . .
C.10 Stulz: Rethinking Risk Management
C.10.1 Managing Risk . . . . . . . .
C.10.2 Case Studies . . . . . . . . . .
C.10.3 Capital Structure . . . . . . .
C.10.4 The Tufano Study . . . . . .
C.11 Cummins: Capital . . . . . . . . . .
C.11.1 Capital Allocation . . . . . .
C.11.2 Capital Allocation Methods .
C.11.3 Myers & Read Allocation . .
C.11.4 Cost of Capital . . . . . . . .
C.12 Goldfarb: Risk-Adjusted Performance
C.12.1 Return on Capital Measures .
C.12.2 Risk Measures . . . . . . . . .
C.12.3 Sources of Risk . . . . . . . .
C.12.4 Capital Allocation . . . . . .
C.12.5 Performance Measurement . .
C.12.6 Pricing . . . . . . . . . . . . .
C.12.7 Multi-Period Commitment . .
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CONTENTS
S
D.8
A
M
D.7
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341
342
356
356
356
357
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360
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363
363
365
366
393
393
394
394
404
404
406
407
407
408
409
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417
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419
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421
422
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428
428
429
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430
430
431
E
D.6
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D.5
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P
D.4
D.3.2 Optimum Capital Structure . . . . . . . . . . . . . .
D.3.3 Past Exam Problems . . . . . . . . . . . . . . . . . .
Robbin: The Underwriting Profit Provision . . . . . . . . . .
D.4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . .
D.4.2 U/W Profit Provision in the Premium Formula . . .
D.4.3 Calendar Year Investment Income Offset Method . .
D.4.4 Present Value Offset Procedure . . . . . . . . . . . .
D.4.5 Calendar Year Return on Equity Method . . . . . . .
D.4.6 Present Value of Income over Present Value of Equity
D.4.7 Present Value Return on Cash Flow . . . . . . . . . .
D.4.8 Risk-Adjusted Discounted Cash Flow . . . . . . . . .
D.4.9 Internal Rate of Return on Equity Flows . . . . . . .
Roth: Surplus and Rate of Return without Leverage Ratios .
D.5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . .
D.5.2 Total Economic Return . . . . . . . . . . . . . . . . .
Bault: Risk Loads for Insurers . . . . . . . . . . . . . . . . .
D.6.1 Marginal Standard Deviation and Marginal Variance
D.6.2 CAPM and Leverage Ratios . . . . . . . . . . . . . .
D.6.3 Link between CAPM and Ruin Theory . . . . . . . .
D.6.4 CAPM and Utility Theory . . . . . . . . . . . . . . .
D.6.5 Remaining Problems . . . . . . . . . . . . . . . . . .
Kreps: Investment Equivalent Reinsurance Pricing . . . . . .
D.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
D.7.2 Single Payment at One Year . . . . . . . . . . . . . .
D.7.3 Swap Case . . . . . . . . . . . . . . . . . . . . . . . .
D.7.4 Option Case . . . . . . . . . . . . . . . . . . . . . . .
D.7.5 High Excess Layers and Minimum Premium . . . . .
D.7.6 Single Payment at Arbitrary Time . . . . . . . . . .
D.7.7 The Algorithm to Select a Risk Load . . . . . . . . .
D.7.8 Other Topics . . . . . . . . . . . . . . . . . . . . . .
Mango: Property Catastrophe Risk Load . . . . . . . . . . .
D.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
D.8.2 Marginal Surplus Method (MS) . . . . . . . . . . . .
D.8.3 Marginal Variance Method (MV) . . . . . . . . . . .
D.8.4 Build-Up Portfolios . . . . . . . . . . . . . . . . . . .
D.8.5 Build-Up with MS Method . . . . . . . . . . . . . . .
D.8.6 Build-Up with MV Method . . . . . . . . . . . . . .
D.8.7 Renewal with MS Method . . . . . . . . . . . . . . .
M. Hoffmann
CONTENTS
Renewal with MV Method . . . .
Renewal Additivity . . . . . . . .
General Covariance Share Method
Covariance Share Method . . . .
Shapley Value Method . . . . . .
Past Exam Problems . . . . . . .
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431
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A
M
P
L
E
D.8.8
D.8.9
D.8.10
D.8.11
D.8.12
D.8.13
M. Hoffmann
9
S
A
M
P
L
E
CONTENTS
10
M. Hoffmann
Chapter A
S
A
M
P
L
E
Portfolio Theory and Equilibrium in
Capital Markets
11
A.1. BKM 6: CAPITAL ALLOCATION
A.1
A.1.1
BKM 6: Capital Allocation
Risk Aversion
In our model world we assume that any investment is completely characterized by its uncertain return r with mean E(r) and standard deviation σ. By definition, risk-free investments
have σ = 0, and all risky investments have σ > 0. To measure how much investors care
about avoiding risk or taking risk, we assume a utility function of the form
1
U = E(r) − Aσ 2
2
In this context, a higher utility means a better investment. When the coefficient of risk aversion A is positive, utility is penalized by taking risk. For A = 0 the investor is indifferent
between the levels of risk, and for A < 0 utility actually improves with risk. The last case
would be someone who values risk as a desirable feature of an investment. Each value of
U determines a unique level curve of equivalent investments from a risk-return perspective.
Hence, any two investments with identical utility scores must be located on the same level
curve, and there is no preference for choosing one over the other. The certainty equivalent
of a risky investment is the risk-free rate rf such that (0,rf ) is located on the same utility
curve. In other words, the risky investment located at (σ,r) is connected through a level curve
to the risk-free investment at (0,r). An investor would be indifferent between (σ,r) and (0,rf ).
A.1.2
Mean-Variance Criterion
L
E
The Mean-Variance Criterion states that investors prefer opportunity A over opportunity
B whenever
E(rA ) ≥ E(rB ) and σA ≤ σB
M
P
with at least one strict inequality. An investment is preferred when returns are higher given
the same risk. Note that this criterion does not tells us what investment is preferred when
E(rA ) ≥ E(rB ) but σA > σB
Capital Allocation
A
A.1.3
S
Our investment universe consists of a risk-free asset F with return rf and a risky portfolio
P with return rp . The basic problem is how we allocate funds between these two choices.
The answer depends on our degree of risk aversion. The complete portfolio C is defined as
C = yP + (1 − y)F where y% is invested in P and (1 − y)% is invested in F . Its return is
then
rc = yrp + (1 − y)rf
12
M. Hoffmann
Notice that the variable rp is stochastic whereas rf is constant, and this makes rc a random
variable. Also notice that since F is risk free, the standard deviation of rc depends only on
the risky portfolio P . Expected value and standard deviation are precisely found as
E(rc ) = yE(rp ) + (1 − y)rf
and
σc = yσp
The Capital Allocation Line (CAL) shows the risk-return trade off, i.e. we graph E(r)
as a function of σ for portfolio C. The equation of this line is
E(rc ) = yE(rp ) + (1 − y)rf = rf + y (E(rp ) − rf ) = rf + σc
E(rp ) − rf
σp
Its slope is called the reward-to-variability ratio or Sharpe ratio, it expresses the marginal
return with respect to risk. The formula for the Sharpe-Ratio is
S=
E(rp ) − rf
σp
The point (0,rf ) on the CAL corresponds to a portfolio consisting only of the risk-free asset,
and the point (σp ,rp ) refers to a portfolio which is 100% invested in the risky asset.
Exercise (Capital Allocation). Suppose you allocate 30% of funds to the risky asset. The
risk-free rate is 5%. The expected return on the risky portfolio is 10% with a standard
deviation of 15%. What are the expected return and standard deviation of the complete
portfolio? What is the reward-to-variability ratio?
Solution. We have
Therefore,
E(rc ) = 0.3E(rp ) + 0.7rf = 0.3(10%) + 0.7(5%) = 6.5%
L
and
P
σc = yσp = 0.3(0.15) = 0.045
M
The reward-to-variability ratio is
0.10 − 0.05
E(rp ) − rf
=
= 0.333
σp
0.15
S
A
S=
E
rc = yrp + (1 − y)rf = 0.3rp + 0.7rf
Without borrowing we have 0 ≤ y ≤ 1, but it makes sense to interpret y > 1 or y < 0 as
borrowing or short selling. In that case we have to pay attention to the borrowing rate rfb ,
which is typically higher than the risk-free rate rf . Borrowing can be viewed as taking a
short position in a risk-free asset at rate rfb . As a result there is a kink on the CAL where
y = 1, and the slopes for the two regions y < 1 and y > 1 are different.
M. Hoffmann
13
Exercise (Borrowing). Suppose you borrow an additional 30% of your existing funds to
invest exclusively in a risky asset. The risk-free rate is 5%, and the borrowing rate is 8%.
The expected return on the risky portfolio is 10% with a standard deviation of 20%. What
are the expected return and standard deviation of the complete portfolio? What is the rewardto-variability ratio?
Solution. We have
rc = yrp + (1 − y)rf = 1.3rp − 0.3rfb
Therefore,
E(rc ) = 1.3E(rp ) − 0.3rfb = 1.3(10%) − 0.3(8%) = 10.6%
and
σc = yσp = 1.3(0.20) = 0.26
The reward-to-variability ratio is
S=
E(rp ) − rfb
0.10 − 0.08
=
= 0.10
σp
0.20
E(r)
CAL
C
Sharpe-Ratio (borrow)
rfb
Sharpe-Ratio
σ
E
rf
P
L
σc
A
M
In the previous discussion we knew the portion y invested in the risky asset beforehand.
Now, we will find out how to determine y based on the risk aversion of the investor. If the
coefficient A is known, we can maximize utility to the investor by taking y ∗ such that
S
U (y ∗ ) = max U (y)
y
1 2
= max E(rc ) − Aσc
y
2
1 2 2
= max rf + y (E(rp ) − rf ) − y Aσp
y
2
14
M. Hoffmann
At this point we apply calculus to find the maximum of the last expression, i.e. determine
y such that U (y) = 0 and U (y) < 0. Note that the specific utility function used here is
simply a downward parabola, and we readily see
y∗ =
E(rp ) − rf
Aσp2
Equivalently, we are looking for the utility curve which is tangent to the Capital Allocation
Line. This happens when the utility function is maximized under the constraint that its
argument is located on the CAL. Thus, the tangent point between the CAL and U contains
the optimum value of y.
A passive strategy takes a representative index of stocks as the risky portfolio thus avoiding
an active selection process. When paired with monthly T-bills we obtain a special CAL called
Capital Market Line (CML). In contrast, an active strategy makes specific selections of
stocks believed to yield returns in excess of the average market return. The advantages of a
passive strategy are:
• Cost reduction. Active strategies require research and higher transaction costs.
• Free-rider benefit. Active traders have already established fair prices in their pursuit
of profit.
A.1.4
Past Exam Problems
2003/4
E
You are given the following information about a risky portfolio that you manage, and a
risk-free asset.
L
• E(rp ) = 11%
P
• σp = 15%
M
• rf = 5%
S
A
a. Client A wants to invest a proportion of her total investment budget in your risky fund
to provide an expected rate of return on her overall or complete portfolio equal to 8%.
What will be the standard deviation of the rate of return on her portfolio?
b. Client B wants the highest return possible subject to the constraint that you limit his
standard deviation to be no more than 12%. Which client is more risk averse? Explain
why?
M. Hoffmann
15
Solution.
a. Solve 0.08 = E(rc ) = yE(rp ) + (1 − y)rf = y(0.06) + 0.05 for y. We obtain
y = 0.50
and the standard deviation is
σc = yσp = 0.50(0.15) = 0.075
b. When σc = 0.12 we have 0.12 = yσp = y(0.15) and hence y = 0.80. Since B invests a
larger portion in the risky portfolio than A, we know that A is more risk averse.
2004/3
You are given the following information.
• Expected return of the risky asset (E(r)) = 0.13.
• Variance of the risky asset (σ 2 ) = 0.01
• Risk-free rate (rf ) = 0.06
• Coefficient of risk aversion (A) = 5
• Utility function: U = E(r) − Aσ 2 . [modified from original to fit current textbook]
a. Calculate the expected return and standard deviation of a portfolio that is invested
40% in the risky asset and 60% in a risk-free asset.
E
b. Determine the optimal amount to invest in each asset to maximize the utility.
Solution.
L
a. We have y = 0.40 and hence
and
b. The optimal allocation is at the maximum value of
A
σc = yσp = 0.40(0.10) = 0.04
M
P
E(rc ) = y (E(r) − rf ) + rf = 0.40(0.13 − 0.06) + 0.06 = 0.088
S
U = E(r) − Aσc2 = y (E(r) − rf ) + rf − Aσp2 y 2
Solving U (y) = 0 yields
y∗ =
E(rp ) − rf
= 0.70
2Aσp2
16
M. Hoffmann
2005/4
You are given the following information.
Investment
1
2
3
Expected Return
12%
15%
20%
Standard Deviation
20%
30%
40%
Your utility formula is represented by U = E(r) − 0.3Aσ 2 . [modified from original]
a. Briefly explain which of these three investments a risk-neutral investor would select.
b. Identify which investment an investor with the utility function shown above and A = 2
would select.
c. Calculate the certainty equivalent of the investment selected in part b. above.
Solution.
a. A risk-neutral investor has A = 0 and therefore U = E(r). Such an investor decides
purely on expected return and would pick investment 3.
b. Compare the utilities
U = E(r) − 0.3Aσ 2 = E(r) − 0.6σ 2
Standard Deviation
20%
30%
40%
Utility
0.096
0.096
0.104
L
Expected Return
12%
15%
20%
P
Investment
1
2
3
E
of each investment.
M
Investment 3 has the highest utility.
S
A
c. The certainty equivalent of investment 3 is r = 0.104. This is the risk-free rate which is
on the same utility curve as the risky investment. In other words, this is the E(r)-axis
intercept of U .
M. Hoffmann
17
2006/4
Consider the following information about a risk portfolio and a risk-free asset.
• The risk premium of the risky portfolio is 15%.
• The reward-to-variability ratio of the risky portfolio is 0.75.
• The expected return on the risk-free asset is 3%.
Assume you can invest in some combination of the risky portfolio and the risk-free asset.
Determine the equation for the Capital Allocation Line under these assumptions and graph
the Capital Allocation Line. Label all items properly.
Solution. The equation of the CAL is
E(rc ) = rf + σc
E(rp ) − rf
= 0.03 + 0.75σc
σp
E(r)
CAL
P
E(rc )
C
L
E
E(rc )
Sharpe-Ratio
M
P
rf
A
σP
S
σC
σ
18
M. Hoffmann
2007/1
Consider the following information about a risk portfolio and a risk-free asset.
• A risky portfolio has an expected return of 16% and a standard deviation of 25%.
• The T-bill rate is 6%.
a. Suppose you invest 60% of your funds in the risky portfolio and 40% in a T-bill money
market fund. Calculate the expected value and the standard deviation of the rate of
return of the portfolio.
b. Determine the equation of the Capital Allocation Line (CAL) of the risky portfolio
and graph the CAL. Plot the position of the overall portfolio on the CAL graph. Label
all items properly.
Solution.
a. We have y = 0.60 and hence
E(rc ) = y (E(r) − rf ) + rf = 0.60(0.16 − 0.06) + 0.06 = 0.12
and
σc = yσp = 0.60(0.25) = 0.15
b. The equation of the CAL is
0.16 − 0.06
E(rp ) − rf
= 0.06 +
σc = 0.06 + 0.4σc
σp
0.25
E(r)
CAL
P
P
E(rc )
L
E
E(rc ) = rf + σc
A
M
C
E(rc )
Sharpe-Ratio
S
rf
σC
M. Hoffmann
19
σ
σP
2009/1
Given the following information regarding a risk-free asset and a portfolio of risky assets:
• The risk-free rate is 3%.
• The expected return on the risky portfolio is 11%.
• The standard deviation of the risky portfolio’s return is 25%.
• An investor has utility function U = E(r) − 13 Aσ 2 with risk aversion parameter A = 2.
This investor has a $50,000 budget for investing.
Calculate how much the investor should invest in the portfolio of risky assets in order to
maximize the investor’s utility.
Solution. We are given rf = 0.03, E(rp ) = 0.11, σp = 0.25, A = 2. Assume the investor
invests a portion y in risky assets. Then
E(rc ) = yE(rp ) + (1 − y)E(rf ) = 0.11y + 0.03(1 − y) = 0.08y + 0.03
σc2 = y 2 σy2 = 0.252 y 2
1
2
U = E(rc ) − Aσc2 = 0.08y + 0.03 − 0.252 y 2
3
3
The function U in terms of y describes a downward parabola which attains a global maximum
at a point y ∗ (the vertex) which can be found where
L
4
0 = U = − 0.252 y ∗ + 0.08
3
P
This means
∂U
=0
∂y
E
U =
and thus
3 0.08
= 0.96
4 0.252
Hence, the investor should invest 0.96($50,000) = $48,000 in risky assets to maximize
utility.
S
A
M
y∗ =
2010/2
• The return of a risk-free asset is 5%.
• An investment company offers a risky asset, with a Sharpe ratio of 0.2.
20
M. Hoffmann

L E P M A

Transcription

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