Practice Exam I - Solutions

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1 Practice Exam I - Solutions (Exam 9, Spring 2018)

2 1. a. We have y = 0.55 and hence E(r c ) = y(e(r p ) r f )+r f = 0.55( )+0.03 = and σ c = yσ p = 0.55(0.10) = b. The equation of the CAL is E(r c ) = r f +σ c E(r p ) r f σ p = σ c = σ c 0.10

3 2. a. The reward-to-variability (Sharpe) ratio is S p = E[r p] r f σ p = = b. We have E[r c ] = 0.3E[r p ] + 0.7r f = = and σ c = yσ p = Hence, S c = E[r c] r f = = σ c = Note that this is in general the case: S c = S p, as can be proven: S c = E[r c] r f σ c = ye[r p]+(1 y)r f r f yσ p = y(e[r p] r f ) yσ p = S p

4 3. a. The risk-free rate is given as r f = 3%. We will find the weight on stock X together with optimal portfolio return and standard deviation, and then solve for A. Recall the formula for optimal portfolio weights: w X = (E(r X ) r f )σ 2 Y (E(r Y) r f )ρσ X σ Y (E(r X ) r f )σ 2 Y +(E(r Y) r f )σ 2 X (E(r X)+E(r Y ) 2r f )ρσ X σ Y The details are: Quantity X Y E(r) E(r) r f σ σ ρ w E(r p ) σp σ p Now, with w X = 0.36 we find y = w X /w X = Then A = E(r p) r f y σ 2 p = 12.73

5 b. E(r) Indifference Curve CAL E(r p ) E(r c ) Complete Portfolio P Efficient Frontier r f σ σ c σ p

6 4. Identify w A = 0.40,w B = 0.35,w f = 0.25,E(r A ) = 0.15,E(r B ) = 0.10,r f = 0.05,β A = 1.2,β B = 0.9,σ(e A ) = 0.40,σ(e B ) = 0.30,σ M = 0.25 a. We have E(r p ) = w A E(r A )+w B (r B )+w f (r f ) = 0.40(0.15)+0.35(0.10)+0.25(0.05) = b. β p = w A β A +w B β B = 0.40(1.2)+0.35(0.9) = c. This is β 2 pσ 2 M = (0.795)2 (0.25) 2 = d. We get σ 2 (e p ) = w 2 A σ2 (e A )+w 2 B σ2 (e B ) = (0.40) (0.30) = e. The total variance is β 2 pσ 2 M +σ2 (e p ) =

7 5. a. Stock Z has greater systematic risk since its graph has a steeper slope than Y. b. Stock Y has higher firm-specific risk since we see a higher degree of variation around the regression line. c. Stock Y s intercept is consistent with CAPM, since CAPM predicts α = 0. The intercept for Z is not consistent, since it is not zero. Any slope is consistent with CAPM, there are no restrictions.

8 6. On January 1, 2014 the stock is worth its initial value plus CAPM return and minus the dividend paid the day before (ignore overnight interest). S = S 0 (1+r f +β(e[r M ] r f )) D = 42( ( )) 1.00 = 42.22

9 7. First, find the theoretical expected return on this portfolio. E(r A ) = r f +β 1 (E(r 1 ) r f )+β 2 (E(r 2 ) r f ) = ( )+0.5( ) = This is 0.5% lower than the market s expectations. Therefore, go long portfolio A and take a short position in the factor portfolios. In this way, we are immunized against the actual outcomes of F 1 and F 2, and obtain 0.5% risk-free profit.

10 8. The actual returns of the stocks are given by r A = E(r A )+β A,GDP F GDP +β A,IR F IR r B = E(r B )+β B,GDP F GDP +β B,IR F IR After substituting the values given, we find r A = ( )+1.2(IR 0.035) 2r B = ( )+0.8(IR 0.035) Subtract twice the first equation from the second and solve for IR =

11 9. 1. Technical analysis is without merit. Technical analysis is the search for patterns in the history of stock prices. Finding such patterns would violate the weak form of the EMH. Technical analysis can be fruitful when stock prices respond slowly to changing market conditions and arbitrages opportunities exist for extended periods of time. 2. Resistance levels (support levels) are illogical. Resistance levels are prices above which or below which the market does not believe they are selling. Such prices are mainly based on psychology. When investors buy stock at $x, and subsequently they fall to $x-a, then $x serves as a break-even point which influences the investor s decision. He or she may be inclined to sell once the stock price is back at $x. 3. Fundamental Analysis is mostly without ground. Fundamental analysis determines stock values by taking the discounted present value of all future payout to shareholders. Hence, an evaluation of future performance is necessary to find current values. Since the basis for such evaluation is all available current information, a successful fundamental analysis violates the strong form of the efficient market hypothesis. One party s fundamental analysis can be successful only if it is superior to the analysis that the rest of the market has produced. 4. Active portfolio management is rarely profitable. Trading costs and portfolio size usually prevent active portfolios from beating passive portfolios which simply buy and hold a broad index of stock.

12 10. a. Framing. Decisions are based on how the situation is presented. Mental accounting. People tend to slice their total funds into groups and take a different risk for each groups. Regret avoidance. Making a bad decision with an unconventional choice is regretted more than a bad decision with a common choice. b. Forecasting errors. People give too much weight to recent experience, and their forecasts are generally too extreme. Overconfidence. People are overly convinced about the accuracy of their forecasts. Conservatism. Investors respond slowly when they need to change their beliefs. Sample size neglect. People tend to take too small a sample as representative for the whole population.

13 11. a. Yields are expected future spot rates. b. Short term bonds are more liquid than long term bonds. The later demand a liquidity risk premium. c. Yields depend only on supply and demand in the respective maturity markets.

14 12. a. This yield-to-maturity for the four-year bond is (1+y 4 ) 4 = 1,000/ = y 4 = 6.33% b. The one year forward rate one year from today is determined by Hence, f 2 = 7.31%. 1+f 2 = (1+y 2) 2 1+y 1 = 1,000/ ,000/ = =

15 13. a. With a discount factor of v = e y = the current price of the bond is P = 6v +6v v 3 = b. Duration is c. Convexity is D = (6v(1)+6v2 (2)+106v 3 (3)) = C = (6v(12 )+6v 2 (2 2 )+106v 3 (3 2 )) = d. The new yield is y = 7.1% which gives a discount rate of v = e y = Then P P = P 7.1% P 7.0% P 7.0% = = = % e. With y = 0.1% we have approximately P y+ y P [ 6v +6v v ] D y C( y)2 = (0.001)+ 1 2 (8.2677)(0.001)2 = %

16 14. a. The price of this bond is the present value of all future payments, P = 3e (0.05)0.5 +3e (0.058)1.0 +3e (0.064) e (0.068)2.0 = b. The bond s yield-to-maturity is the discount rate that produces a bond price equal to its market price. To determine the bond yield, we need to solve for y where First, find r where 3e 0.5y +3e 1.0y +3e 1.5y +103e 2.0y = (1+r) 1 +3(1+r) 2 +3(1+r) (1+r) 4 = and then solve for y using the relation r = e 0.5y 1. We obtain 6.76%.

17 15. Since the swap is for one year we know that payments are at t = 4/12 and t = 10/12 years with today being t = 0. The prior 6-months LIBOR rate r 0 = 10.3% is equivalent to r 0 = 2 ( e r 0/2 1 ) = 10.57% with semi-annual compounding. Hence the first payment at t=4/12 is r 0 L = 10.57% (100) = We can look at the floating rate bond concentrated at t = 4/12, since the remaining future payments are worth par at t = 4/12. Discount factors are the same for both fixed and floating bond. We obtain the following table. Month LIBOR Fix Float Discount PV(B (i) fix ) PV(B(i) fl ) % % Sum Therefore, the value of the swap to party X is V swap = B fix B fl = = $5.03M

18 16. a. The reduction in duration may be too little. Duration of assets may have to be taken below zero to get the duration of total economic value to the desired level. Benefits of the strategy are not clear to regulators, who may perceive the actions taken as speculative investments. It would raise solvency concerns and could impact financial ratings. b. A firm can take target return-on-surplus as a linear function of two parameters a and b. Then it can adjust the coefficients a and b while keeping k fixed, thus lowering asset durations without changing the portfolio composition. A disadvantage is that this strategy holds for only small changes in interest rates.

19 17. This is Panning s approach. We have cr = 0.80, P = 250, E = 75, L = 130, k = 0.15, y = 0.03 and then which implies Hence, F = d = cr 1+y = ( P E L ) d 1+y 1 d = T = C +F = =

20 18. a. This process involves two steps: Pooling. A set of credit sensitive assets is assembled in a special purpose vehicle (SPV). The SPV isolates the credit risk of its liabilities from the balance sheet of the originator. Without prioritization, this is known as a pass-through securitization. The credit rating of a pass-through security is identical to the average credit rating of the underlying securities. Tranching. A capital structure of prioritized claims a.k.a. tranches is issued against the underlying collateral pool. b. Denote bond payouts by X 1,X 2. The payout distribution of Y = X 1 +X 2 is: (1 q) 2, k = 2 P(Y = k) = 2q(1 q), k = 1 q 2, k = 0 Write Y = Z 1 +Z 2 where Z 1 is the senior tranche. Then Z 1 defaults if Y = 0, hence P(Z 1 = 0) = P(Y = 0) = = The expected payout of the senior tranche is E[Z 1 ] = (0) (1) = 0.984

21 19. a. The insurer pays a premium to the writer of a Cat-E-Put and receives the option to issue preferred stock at a the strike price when a specific event occurs. In this way the insurer would be able to raise capital after a catastrophic event which depresses stock prices. b. Advantage: Since no SPR or similar structure is needed to issue Cat-E-Puts, the transaction costs are relatively low. Disadvantages: Cat-E-Puts are not collateralized and hence pose credit risk. Issuing preferred stock would push a distressed insurer s share price further down. c. Indemnity trigger. Payout is based on the insurer s actual losses. This is preferred by insurers and reinsurers, because it minimizes basis risk, i.e. bond payout being smaller or greater than actual loss experience. Index trigger. Payout is based on an index, which is not related to the insurer s actual losses. Such trigger is preferred by investors, because it minimizes moral hazard issues such as inflating losses to increase bond payout. Three indices are used: Industry loss indices. Industry wide losses related to a specific event exceed a threshold. Modeled loss indices. An index calculated through a catastrophe model of a leading catastrophe modeling firm. Parametric indices. A certain physical measure of an event exceeds a threshold, e.g. a California earthquake of 7.0 magnitude. Hybrid trigger. This is a combination of both indemnity and index triggers.

22 20. a. Probability of ruin does not consider the severity of ruin. b. Butsic suggests to use Expected Policyholder Deficit which is the expected deficit as a percentage of expected losses. c. EPD considers the severity of ruin and can be applied to stochastic assets or liabilities,

23 21. a. To determine the assets needed to achieve an EPD ratio of 0.01 we first compute expected losses E[L] = 7,000(0.4)+12,000(0.3)+17,000(0.3) = 11,500 Now we want which means E[L A] + = 11,500(0.01) = 115 (7,000 A) + (0.4)+(12,000 A) + (0.3)+(17,000 A) + (0.3) = 115 Assume that only the scenario with the largest loss leads to deficit. Then (17,000 A)(0.3) = 115 which gives A = 16,617 which is consistent with our assumption. Consequently, the ratio of capital to incurred loss required is c = C L = A E[L] E[L] = 16,617 11,500 11, 500 = b. Two perfectly correlated risks with the above distribution can be treated as a single risk with twice the severity. Hence, the same calculation as before leads to twice the assets needed, thus leaving the capital to loss ratio invariant, i.e. c = c. In this case, we have to consider the joint distribution of L and L given by p ij = p i p j. Take indices 1,2,3 corresponding to losses in increasing magnitude. Expected losses are additive, so E[L+L ] = E[L]+E[L ] = 23,000 The scenario with the largest loss is (3,3) with probability p 33 = p 3 p 3 = = 0.09 and loss 34,000. Assuming this is the only scenario creating a deficit we find (34,000 A)(0.09) = 23,000(0.01) = 230 which gives A = 31,444 which is consistent with our assumption, since the next largest loss scenario is 29,000. The capital to loss ratio is now c = C L = A E[L] E[L] = 31,444 23,000 23, 000 = 0.367

24 22. Proportional. Determine the proportion of the stand-alone risks to the sum over all risk. This percentage is applied to the overall risk. Stand-alone risk can be measured according to any of the risk measures available (VaR, CTE, EPD) Incremental (Merton-Perold). Determine the amount contributed by each risk type when added to the group. Afterwards, use proportional allocation based on these amounts. Marginal (Myers-Read). This method looks at the derivative of the EPD put option with respect to capital. Capital is chosen to equate these derivatives across all risk types. Co-Measures (Kreps, Ruhm-Mango). Find the aggregate risk distribution with corresponding stand-alone components, e.g. through simulation and numbering all scenarios. Then, for a chosen risk measure such as VaR or CTE we look at its risk components at a given percentile and allocated overall risk in accordance with the observed proportions.

25 23. We have EVA A = 400,000 2,500,000(0.12) = 100,000 EVA B = 900,000 4,200,000(0.19) = 102,000 Therefore, both lines add value for the insurer. Line B performs better.

26 24. a. Create the cash flow table 1/1/ /31/ /31/ /31/2015 Premium 120,000,000 Paid Loss 26,000,000 26,000,000 26,000,000 UEPR 120,000,000 Loss Reserve 52,000,000 26,000,000 Required Surplus 60,000,000 26,000,000 13,000,000 Investment Income 9,000,000 3,900,000 1,950,000 Net Income 9,000,000 3,900,000 1,950,000 Change in Required Surplus -60,000,000 34,000,000 13,000,000 13,000,000 Equity Cash Flow -60,000,000 43,000,000 16,900,000 14,950,000 b. The NPV (in millions) is found from NPV d = 60+43v +16.9v v 3 where v = (1+d) 1. This produces NPV 0.25 = 7.13 and NPV 0.10 = 4.92 c. Use the cash flow keys of a financial calculator to solve 0 = v+13v v 3 where 1/v = 1+IRR. We find IRR 0.15.

27 25. Group Meaning of Profit Investors and insurers desired outcome Policyholders - stock insurer markup Policyholders - mutual insurer not a concern regulator measure of solvency

28 26. First, identify the variables involved. a. Total return on equity: Statutory Capital and Surplus 100 S Equity in Unearned Premium Reserve 2 Underwriting Profit/Loss (after-tax) 5 U Investment Income 10 I Earned Premium 120 P Unearned Premium Reserve 50 Paid Losses 30 T S = I +U S = = b. The percentage of non-invested assets from insurance operations will tend to rise. Increase in risk to owners equity leads to more conservative investment policy.

29 27. First, identify the variables involved. Note that surplus S includes the equity in the unearned premium reserves. a. Insurance leverage factor: Total Assets 80 A Statutory Capital and Surplus 20 S Equity in Unearned Premium Reserve 10 Underwriting Profit/Loss (after-tax) -2 U Investment Income 8 I Earned Premium 50 P 1+ R S = (20+10) = = 2.67 b. Insurance Exposure: c. Total return on equity: T S = I A ( 1+ R S ) P S = = U P P S = 8 2 (2.67) = 0.20

30 28. a. We need to solve for y where 0 = PV(x,y) = n x j (1+y) j j=0 The equity flowsef j = INC j (SCHNG) j are found as: Stat Stat Change in Equity t Income Surplus Surplus Flow 0 5,000 5,000-5, ,000-2,000 2, ,500-1,500 1, ,500 1,300 Then we set up the IRR equation with v = (1+y) 1 : 0 = v +1520v v 3 The cash flow function keys of a financial calculator provide i 3%. b. The choice of Stat or GAAP should not distort the solution significantly, since more and less conservatively treated items cancel on average.

31 29. Extend the table to find covariances share contributions. Event p X Y X+Y Cov(X,Y) CovShareX CovShareY ,000 7,500 39,500 4,704,000 7,621,671 1,786, ,000 5,000 18,000 1,274,000 1,840, , ,000 11,000 16,000 1,600,500 1,000,313 2,200, ,000 6,000 20,000 1,241,100 1,737,540 74, ,000 25,000 35,000 1,243, ,714 1,776,786 E[...] 1, Total 10,063,350 12,910,460 7,216,240 Var[...] 27,503,700 8,754,875 The Covariance Share renewal risk loads are given by R CS X = λ(var(x)+covshare(x,y)) = (27,503,700+12,910,460) = 1,576 R CS Y = λ(var(y)+covshare(y,x)) = (8,754,875+7,216,240) = 623

32 30. The conditional exceedance probabilities by scenario are as follows: Cond. Ex. Scenario p Severity Neither peril 86.4% - Wind only 9.6% % EQ only 3.6% 1, % 26.5% Wind + EQ 0.4% 1, % 2.9% Therefore, 99% VaR is found at $1,000. Wind contributes 250/1, 250 = 20% to the combined Wind and Earthquake severity. The by layer split across perils is then given by: Layer Size Wind1 EQ1 Wind+EQ Wind2 EQ2 Wind EQ Total The capital allocation to peril is Wind = 193/1,000=19.3% Earthquake = 807/1,000=80.7%.

33 31. a. The premium-to-surplus ratio implies $18 million total premium, with $10(1.1) = 11 million coming from Line A. Hence, (18 11)/1.08 = 6.5 million are coming from Line B. Based on the surplus allocation percentages, we find: Line A Line B Investment Total Allocation % 13.60% 84.30% 2.10% % Surplus allocated $ 1,223,892 $ 7,587,192 $ 188,916 $ 9,000,000 Premium $10,000,000 $ 6,481,481 Profit % 10% 8% 4% Return on Premium $ 1,000,000 $ 518,519 $ 360,000 $ 1,878,519 Return on Surplus 81.7% 6.8% 190.6% 20.9% b. 1. The two lines may not be separable, e.g. homeowners property and liability coverage. 2. Regulatory requirements could make it difficult to change or exit from Line B. 3. A major shift in focus could be severely disruptive to underwriting. c. Purchase reinsurance to balance the portfolio.

34 32. The timing and accounting notation view of the policy are: Timing Notation t Equity Income Accounting Notation Period QBOP QEOP IEOP Therefore, the policy contribution diagram looks like this: Balance Sheet Account Growth - Policy Contribution Year 1 Year 2 Year 3 Year 4 Policy BOY EOY BOY EOY BOY EOY BOY EOY Total with corresponding income diagram: Income Account Growth - Policy Contribution Year 1 Year 2 Year 3 Year 4 Policy BOY EOY BOY EOY BOY EOY BOY EOY Total The rate of return for year i is given by Total Year i EOY Income divided by Total Year i BOY Balance: Year ROE 12.0% 14.7% 14.7% 14.7%

Notes on: J. David Cummins, Allocation of Capital in the Insurance Industry Risk Management and Insurance Review, 3, 2000, pp

Notes on: J. David Cummins, Allocation of Capital in the Insurance Industry Risk Management and Insurance Review, 3, 2000, pp Notes on: J. David Cummins Allocation of Capital in the Insurance Industry Risk Management and Insurance Review 3 2000 pp. 7-27. This reading addresses the standard management problem of allocating capital