Spring 2005 Exam 8 Sample Solutions Question 1: 1a. Fisher equation The nominal interest rate equals the real interest rate plus expected inflation ra

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30 Spring 2005 Exam 8 Sample Solutions Question 1: 1a. Fisher equation The nominal interest rate equals the real interest rate plus expected inflation rate. Fisher argues that holding the real rate constant, the nominal rate would increase 1 unit for each unit that expected inflation rate is increased. 1b. It is because the real interest rate is neither constant nor observable in time.

31 Question 2: 2a. Price weighted average 1 = Price weighted avg 2 = % change = 1 = b. Value wtd avg 1 = 50(30) + 26(150) + 80(75) = 11,400 Value wtd avg 2 = 75(30) + 13(150) + 80(75) = 10, %Change = 1 = c. The price weighted average is influenced more heavily by the stocks w/ higher share prices and their rates of return are positive or zero. So overall is positive. The value-weighted average is influenced by the market value holding each stock have the most of the stock w/ lowest price, and it also had a negative rate of return. So overall is negative.

32 Question 3: 3a. 1,000 = $1.70/ 1,080 = $1.85/ 1998 = % b. 1,000 3c The investor cannot hedge the price of the stocks using currency forwards. Alternatively, (1,080)(1.85) 3a. Total RoR = 1 (1,000)(1.70) RoR = 17.5% 3b. 1,080, final value of pounds. 3c. Would require that you know exactly how many pounds to sell forward at the beginning of the year. This amount is not known until the end of the year. Alternatively, 3a. Paid $1700 Got Stocks 1000 Now has stocks 1080 Is worth 1,080 x 1.85 = $1,998 Return = (1,998/1,700) 1 = 17.5% 3b. If he knew the stock s return, he would have sold forward If he didn t, he could have sold forward the amount invested plus the return = 1000e r f T 3c. In order to set up a perfect hedge, you must know the amount to be returned i.e., if you knew the stock would return 1080, you could hedge exactly the right amount.

33 Question 4: 4a. A risk neutral investor would select investment 3 because it has the highest expected return of 20%. Since investor is risk neutral he only cares about maximizing expected return and his investment decision is not impacted by level of risk (i.e. standard deviation) of investment. 4b. u = E(r) -.003Aσ 2 where A=2 u 1 = (2)(20) 2 = 9.6 u 2 = (2)(30) 2 = 9.6 u 3 = (2)(40) 2 = 10.4 Investor would select investment 3 because it has the highest utility of 10.4 based on investors utility function where A=2. 4c. Certainty equivalent would be 10.4 because certainty equivalent is expected return on risk free investment with same utility as investment 3. Since certainty equivalent is risk free, it has standard deviation of 0 and its expected return equals its utility value, which is 10.4.

34 Question 5: 5a. X = amount in A, Y = amount in B. Investor solves Min σ 2 p such that 0.15X + 0.2Y (1-X-Y) = μ x,y Min X 2 σ 2 A + Y 2 σ 2 B + 2XYcov(A,B) such that 0.15X + 0.2Y (1-X-Y) = μ x,y La Grange: L =.04x y 2 + 2xy(.1)(.2)(.15) + λ(μ -.15x -.2y x +.05y) =.04x y xy + λ(μ -.10x -.15y -.05).08x +.006y =.10λ FOC:.045y +.006x =.15λ.08x +.006y =.10λ.08x +.6y = 2λ.594y=1.7λ y = 3.20λ.006x =.006λ x = λ x 1 y 3.2 = =.238 = =.762 x + y 4.2 x + y 4.2 So 23.8% in X, 76.2% in Y This has variance σ 2 p =.04(.238) (.762) (.238)(.762) =.0164 σ p = 12.8% Alternatively, 5a. For optimal risky portfolio, weight in A. W a ( E( ra) rf ) σb^2 ( E( rb) rf ) σaσbρ = ( E( ra) rf ) σb^2 + ( E( rb) rf ) σb^2 ( E( ra) rf + E( rb) rf ) σaσbρ 2 ( ) (0.15 ) (.2.05) = 2 2 (.15.05) (.15 ) + (.2.05) (.2 ) ( ) =.24 weight in B W B = =.76

35 σ 2 p = W 2 A σ 2 A + W 2 B σ 2 B + 2W A W B σa σbρ 2 =.24.2 = σ p = b. If short sell is allowed and A and B are perfectly negatively correlated. There is arbitrage opportunity of creating a zero standard deviation portfolio with return higher than the risk free rate so that investors would like to take unlimited portfolio in it by borrowing at the risk free rate. That is optimal portfolio. Therefore, standard deviation of optimal risky portfolio is 0. 5c. Since the two assets are perfectly correlated, asset b dominates asset a, and short sales are not allowed then all will be invested in asset b. Thus the standard deviation will be 15.

36 Question 6: E(r p ) Stocks Short Sales Gold σ p As shown in the graph, gold is apparently inferior to stock, however, it may provide diversification benefits if it is independent of (or negatively correlated with) the stock performance. A risky portfolio including gold maybe preferable when considering the complete portfolio.

37 Question 7: 7a. E(r) = r f + β[e(r m ) - r f ] where r f :risk free rate β: beta E(r m ) - r f : market risk premium The situation is not possible because the higher beta of portfolio x would result in a higher expected return unless the market risk premium is negative which is not possible. 7b. β = Cov(r, r m )/ σ 2 m where r m : return on market σ 2 m: variance associated with the market return Cov(r, r m ): Covariance between the return Associated with the asset and the Market portfolio. If the situation is possible than as stated in (a), beta is higher for portfolio y. This implies that Cov(r y, r m ) is higher than Cov(r x, r m ). This is independent from the S.D. of the return. It is possible.

38 Question 8: 8. (r x -r f ) (r m -r f ) (R - Quarter r x R m R f R x R m (R m - AvgR m ) 2 x AvgR x )(R m - Avg R m ) E[R x ] = = E[R m ] = = Cor( X, M ) βx = = = 2. 5 Var( M ).00025

39 Question 9: 9a. Investor prefer liquidity so they must receive some extra expected return to compensate for illiquidity. 9b. 10 = r f + β(r m - r f ) + illiquidity prem 10 = (6-3) + illiquidity prem illiquidity prem = 2.5%

40 Question 10: Roll s Critique 10a. Single testable hypothesis of CAPM: The market portfolio is mean-variance efficient 10b. CAPM not testable unless the exact composition of the market portfolio is known and used in the test. 10c. A proxy such as the S+P 500 may be inefficient or subject to benchmark errors. 10d. Linear relationship between expected return and beta is not independently testable.

41 Question 11: 11a. Historical equity risk premiums in the U.S. seem too high given their associated risk. 11b. 1. The equity premium puzzle is largely related to the 2 nd half of the 20 th century. During this period, there were large capital gains above what would have been expected. These capital gains are not expected to persist into future periods. 2. The puzzle arose in the US market, which has long been one of the most successful in the world. Thus, the higher equity risk premiums may indicate survivorship bias, since the analysis does not include poorer performing markets or markets that have failed. This would lead to upward bias by not including such markets.

42 Question 12: 12. 5% = 3% + β A (6% - 3%) β A = 2/ β = (2 / 3) 6.25 = = B E(r B ) = 3% (6%-3%) = 15.5% The expected return for stock B should increase 1.5% to 15.5%.

43 Question 13: 13. Interest: I assume same % rate for each mortgage A = (600k/(600k+900k + 750k))(150,000) = 40,000 B = (900k/(600k+900k+750k))(150,000) = 60,000 C = (750k/(600k+900k+750k))(150,000) = 50,000 Total Principal payments = 350, ,000 = 750,000 A gets the first 600,000 because its outstanding principal is 600,000 B gets the remaining 150,000 which is less than 900,000 outstanding C gets 0 principal Totals: A = 40, ,000 = 640,000 total B = 60, ,000 = 210,000 total C = 50, = 50,000 total I assumed A got principal first, then B, then C because they are labeled A-B-C.

44 Question 14: 14a. Liquidity preference theory implies that forward rate must be higher than the expected short rate in the future i.e. f n > E(r n ) This is due to the assumption that bond investors are dominantly short-term; to attract them to hold long-term bonds, the yield rate must be higher as the maturity date becomes later. Therefore, f n >E(r n ). fn Yield curve E(r n )

45 Question 15: 15a. [Invoice Price] = [Flat Price] + [Accrued Interest] 1000 [Flat Price] = = $ days [Accrued Interest] = (08/ 01 08/ 05) 2 180days =$0.89 [Invoice Price] = = *Corp Bonds Use 30/360 day counts 15b. Est. Spread is the basis point spread of the bond YTM over T s.

46 Question 16: 16. Higher current ratio (CA/CL) is good for a company s solvency. High current ratio leads to lower bankruptcy probability. In Altman s Discriminant Analysis model, High Z score means good solvency. So the added ratio should receive a positive coefficient.

47 Question 17: [y(t ) y*(t )]T 17a. Q(T) =1 e Q (1) = 1 e =.003 Q (2) = 1 e =.0119 ( )(1) ( )(2) Prob. default in year 2 = = b. hazard rate in year 2 = Prob(default in year 2)/Prob(survival to year 2) =(.0089)/(1-.003) =.0089 (there is rounding here)

48 Question 18: 18. Callable bonds must have higher yield to maturity to compensate investor for the possibility that the bond be called. Under callable bonds the capital gains are limited due to the call provision. Also the higher the call price, the lower the probability that the bond will be called and also the lower the yield-to-maturity required. Therefore, the yield-to-maturity will be A > B > C

49 Question 19: 19a. V t = value of company s asset at time t E t = value of company s asset at time T D = value of debt at maturity of the debt agreement T = maturity time of debt agreement Firms Equity Position at time T: 1. if V t > D Equity = E t = V t - D 2. if V t < D Equity = 0 (default) Equity position is like a call option on asset V t with strike price D Value of Equity at time 0 by BS formula E o = V o N(d 1 ) - De -rt N(d 2 ) (*) ln(vo/d) + (r ((σ ^2)v /2)T Where d 2 = σ T r = risk free rate σ v = volatility of V t (*) Alternatively: also by Black-Scholes-Merton Differential Equation = σ E E o = N(d 1 )σ v V o - (**) where σ E E = volatility of S T solve for σ v and of V o from (*) and (**) (Since E o and σ E observable) calculate risk-neutral default probability = N(-d 2 ) Merton proposed a model where a company s equity is an option on the assets of the company. ET = max(vt-d,0) Since this can be seen as an option, then we can use option pricing (Black-Scholes and Ito s Lemma) to price it. The Black-Scholes formula and Ito s Lemma provide two equations that can be solved simultaneously to obtain the current value of the company s assets and the volatility of those assets. (All other variables in these formulas are observable or are estimatable.) Once these parameters are found, then the probability of default is determined by the Black-Scholes model [N(-d2)]. However, the default probabilities show significant difference from observed default probabilities. Fortunately, these default probabilities can be useful for estimating actual default probabilities.

50 Question 20: 20a. Assume yields are annual Value of zero at Beginning of year = 400. Yield = (1000/400) 1/10 = % This will be used for imputing interest Expected value at end of year = 400( ) = The actual value = (1000)(1.08) -9 = There is of imputed interest and = of capital gain After Tax Income = (38.38(1-.35)) + (61.87(1-.2)) = After tax return = 74.44/400 = 18.61% 20b. If not sold, the investor only pays tax on the imputed interest (.35)(38.38) = in taxes

51 Question 21: 21a. f = (S-I)e rt S-I = current price - PV(Coupons) S-I = e -.1(.5) - 35e -.1(.1) = f = e.1(1) = b. The current price is higher than the theoretical price so we want to be short the forward contract (sell high) and long the bond. To make an arbitrage profit, we would sell the forward, borrow 850 and buy the bond. Net cash flow today = 0. In one year we deliver the bond to the person who bought the forward from us, collect 875, and pay back borrowed money of 850e.1(1) = with money from forward and coupons of 35e.1(.5) + 35 = = riskless arbitrage profit = = 7.39.

52 Question 22: 22a. Margin is required to ensure parties to position (exchange, brokers, investor) against default losses. 22b. 5,000 bushels * 5 contracts = 25,000 bushels Margin required = 2,000 * 5 = 10,000 Maintenance margin = 1,500 * 5 = 7,500 price change greater than value: 25,000 * $2.40 = 60,000-2,500 25,000 * X = 57,500 x = $2.30 per bushel 2.3/2.4 = -4.17% change in price 22c. value: 25,000 * 2.40 = $60, ,000 25,000 * X = 62,000 x = $2.48 per bushel

53 Question 23: 23a. For platinum, let S o = 100, then one year forward price is F o = S o e (r+u)t =100e ( )*1.0 = So the equivalent cash interest rate of platinum loan is = > Therefore the interest on platinum loan is too high compared to cash loan.

54 Question 24: 24. Since it is 18 years and 5 months, we regard it as 18 years and 3 months. t=0 t=3/12 t=18 t=18(3/12) (in years) The value at 3 months from now = 100 * 8% * (1/2) * (1/ ) * (1/1.03) +100 * 1.03 i i=1 Alternatively: = = $ = $ The discount rate for 3 months is 1.03 = PV / = $ The accrued interest is $4/2 = $2 therefore price = $ = $ The conversion factor = /100 = round down to nearest 3 months 18 years and 3 months at delivery PV = (N=37, pmt = 4, i = 3%, FV=100 )* = * = accrued interest = 3/6 * 4 = 2 conversion factor = ( )/100 = 1.220

55 Question 25: 25. Cost of bankruptcy is value of forward contract to ABC This is the cost of a Swiss bond less the cost of a U.S. bond 7 years from now (including coupon paid in 7 years) B U.S. = 8%(10M) + 8%(10M) 1.09 = M B swiss = 0.80[4%(13M) + 4%(13M) 1.04 = M + 8%(10M) 8%(10M) +10M %(13M) 4%(13M) +13M + ] Thus cost of default = B swiss B U.S. =10.816M M =0.269M or 269,000 Alternatively, ABC receives 4% swiss on 13M pays 8% US on 10M if bankruptcy is declared before the end of year 7, 4 payments are lost amt ABC would have end of year 7 = (.04)(13M)(.8) = 416,000 present value at year 7 = 520(.8)+ = 10, f 1 =.8(1.09/1.04) =.838 f 2 =.8(1.09/1.04) 2 =.879 f 3 =.8(1.09/1.04) 3 =.921 (520)(.838) (520)(.879) 13520(.921) ^2 1.09^3 present value of payments (800/1.09) + (800/ ) + (10800/ ) = 10, Therefore ABC lost 10,815,662 10,546,871 = $268,791 Alternatively,

56 Receive.04(13M fr) = $520,000 fr. Pay =.08(10,010,010) = $800,000 f(x 7 ) = 0.80 f(x 8 ) = 0.80(1.09/1.04) =.8385 f(x 9 ) =.8788 f(x 10 ) =.9210 Pay fr F(x) $ value Change 7 800, , , , , , , , , , , , ,800,000 13,520, ,452,202 1,652,202 PV = (384,000) + (363,980/1.09) + (343,024/ ) + (1,652,202/ ) PV = $269,160 loss

57 Question 26: 26. S o = 54.6 σ = E(r) = 0.15 k = 75 Alternatively, Alternatively, Prob. of being exercised = N(d 2 ) ln( So / k) + ( r + (( σ ^2) / 2)) t d1 = σ t ln(54.6/ 75) + ( ((0.7071^2) / 2))2 = = d 2 = d1 σ t = = N( )= linear using interpolation in normal table. lns T ~ N(lnS o + (N (1/2)σ 2 )T, σ T ) lns T ~ N(ln(54.60) + (15% - ((1/2)(70.71%) 2 ) * 2, 70.71% 2 ) lns T ~ N(3.8, 1) lnst ln Pr(lnS T > ln75) = > 1 1 ln = 1 - N ( ) = 1 N(0.5175) 1 N(0.51) = N(0.52) = N(0.5175) = (3/4) * * (1/4) = Pr(lnS T > ln75) = = Prob (Stock price >75) Which to get stock price of 75 it is a return of 75/54.5 = So (1/T)ln(S T /S o ) ~ N(M (σ 2 /2), σ/ t ).1587 = (1/2)ln(75/54.6) ~ N(.15 ( / 2 ),.7071/ 2 )

58 .7071^ (.15 ) 1- ( 2 ) = 1 - (.5174) = =.3024 Probability it will be exercise is 30.24%

59 Question 27: 27. VHH + VHL + 2C V H = (,100) max 2(1 + rh) VHL + VLL + 2C r L = (, 100 ) max 2(1 + rl) 100 = V HH 6 = C V H = 100 r M =.0727 V =? R = = V HL 6 = C V L = r L = = V LL 6 = C VH+ VL + 2c (6) V o = = = (1 + r) 2(1.0595) V o =

60 Question 28: Prob = P S o u S o Prob = 1 - P S o d Alternatively, At T risk-neutral expected value of stock is P(S o u) + (1 - P)( S o d) This must equal S o e rt since stock value grows by risk-free rate S o e rt = P(S o u) + (1 - P)( S o d) e rt = P * u + d P * d e rt = P(u d) +d e rt d = P(u d) P = (e rt d)/(u - d) 28. p = (e rt d)/(μ d) f = e -rt (pf u + (1-p)f d ) buy Δ shares & sell 1 call for a risk neutral investment ΔS o μ fu = ΔS o d fd same value of position if up or down movement Δ = fu fd Soμ Sod initial investment = S o Δ - f value at time 1 = ΔS o μ f u = ΔS o d f d earn risk free rate S o Δ - f = e -rt (ΔS o μ f u ) fu fd S fu fd -rt o ( ) f = e (Soμ( Sou sod ) fu) Sou Sod fu fd fu fd = u( ) e -rt fu(e -rt ) + f u d u d rt rt ( fu fd) ue + ( fu fd) + fue ( u d) f = u d

61 rt rt rt e d u e = e fu + fd u d u d = e -rt (pf u + (1 p)f d ) e rt d p = u d 1 P = u d e u d rt + d (the d s in the numerator cancel out)

62 Question 29: 29. Own 1 share. Buy 1 put, k = 20, p = 2.5 Sell 1 call, k = 40, p = 1.5 Payoff Stock Price Stock Long Put Short Call Total S T < 20 S T 20 - S T < S T < 40 S T 0 0 S T S T 40 S T S T 40 Payoff S t

63 Question 30: 30a. Collect the data of past n days, find the stock price daily stock return, µ i = lns i /S i-1, then estimate standard deviation of the stock return by S 2 = n 1 (ui ui) 2, assuming 252 trading days per year. n 1 i=1 S i is the closing stock price today S i-1 is the closing stock price of previous day 30b. Implied volatilities are the volatilities that make an option price equal to its Black Scholes value. We need an iterative technique because σ can t be isolated in the B-S formula.

64 Question 31: 31a. # of shares per call = 100 K = 65 T = 8/12 σ = 30% S o = 61 r f = 6% rt 1 call = SoN( d1) Ke N( d 2) = 61(.5106) 65e -.06(8/12) ( ) = 5.32; so 100 * 5.32 = $532 d ln(61/ 65) + (.06 + (.3 =.3 8 /12 / 2))(8 /12) 2 1 = N(.0265) = N(.02) +.65(N(.03) N(.02)) = ( ) =.5106 d 2 = d1 r t = /12 = N(-.2184) = = N(.2184) = N(.21) +.84(N(.22) - N(.21)) = ( ) =

65 31b mos 6 mos 8 mos Is DIV 3mo > K(1 e -(3/12).06 ) 1 > 65( ) 1 > Yes, so may be optimal exercise at t = 3 mos Is DIV 6mo > 65(1 e -(2/12).06 ) 1 > Yes, so may be optimal exercise at t = 6 mos At t = 6 mos value of 1 call is S o N(d 1 ) Ke -rt N(d 2 ) Alternatively, 31a. No dividends = No early exercise d ln(61/ 65) + (.06 + (.3 =.3 8 /12 / 2))(8 /12) 2 1 = d 2 = /12 = N(d 1 ) =.65(.5120) +.35(.5080)) =.5106 N(d 2 ) =.85(.4129) +.15(.4168)) =.4135 C =.5106(61) (65)e -.06(8/12) = * 100 = per option 31b. Using Black s Approximation: If no early exercise, So = 61 1e -.06(3/12) 1e -.06(6/12) = ln(59.04 / 65) + (.06 + (.3 / 2))(8/12) d 1 = = /12 d 2 = N(d 1 ) =.65(.4562) +.35(.4602) =.4576 N(d 2 ) =.15(.3594) +.85(.3632) = C =.4576(59.04) (65)e -.06(3/12) = 4.37 If exercised at t = 6 mths, S o = 61 1e -.06(3/12) = 60.01

66 2 ln(60.01/ 65) + (.06 + (.3 / 2)(.5) d 1 = = d 2 = N(d 1 ) =.87(.4483) +.13(.4522) =.4488 N(d 2 ) =.08(.3632) +.92(.3669) =.3666 C =.4488(60.01) (65)e -.03 = 3.81 If exercised at t = 3 mths, S o = 61 ln(61/ 65) + (.06 + (.3 d =.3 1/ 4 d 2 = / 2)(1/ 4) = 2 1 N(d 1 ) =.84(.4013) +.16(.4052) =.4019 N(d 2 ) =.84(.3446) +.16(.3483) =.3452 C =.4019(61) (65)e -.06(.25) = Since the value is highest when it is not exercised early, it should not be exercised early.

67 Question 32: 32. Black-Scholes-Merton differential equations do not contain any variables that are affected by risk preference. Alternatively, When moving from a risk-neutral to a risk-averse world, changes in growth rate are offset by changes in the discount rate. When inspecting the Black-Scholes-Merton equation, all the variables in the equation are independent of risk preference, the only variable related to risk preference is u (expected return) which is not in the equation. So the equation applies to all expected returns including risk-free rate. So the equation obtained in a risk-neutral is also valid in a risk-averse world.

68 Question 33: 33a. Interest rate swap swap assets for better on, can be tax advantaged so better than restructuring and triggering capital gains. 33b. Issuing puts instead of buying/selling stocks lowers taxes and transaction costs. 33c. Swap assets, lower costs than buying and selling the assets. (lower transaction costs) Do not have to do accounting on new assets, since swapped.

69 Question 34: par = F = 1000 coupon = 6% semiannually y =.05 D* = Modified duration = MacaulayDuration 1+ y / k Macaulay duration = = = (1) 30(1.5) 1030(2) (.5) a. D* = = CFtn(n +1) 34b. Convexity = P(1 + y /k) 2 k 2 (1 + y /k) n 1 30(2)(1) = (2)(3) 30(3)(4) 1030(4)(5) ( )(1.025) (4) = (19, ) = c. (dp)/p = -D*(dy) + (1/2)C(dy) 2 dy = -.01 D* = C = (dp)/p = -(1.868)(-.01)+(1/2)(4.491)(-.01) 2 =.0189 %change in price is %

70 Question 35: 35a. -transform u/w cash flows into marketable securities -transfer u/w risk to the capital markets 35b. Reinsurance does not transform u/w cash flows into a marketable security. Reinsurance contracts are not traded Reinsurance does not transfer u/w risk to the capital markets, it is only transferring risk to another insurance company.

71 Question 36: 36a. S o = K = 50 r f =.05 σ =.20 T = 20/52 Δ = N(d 1 ) 1 = N(.0805) 1 = = ln( So / K) + ( r + ( σ / 2)) T d1 = σ T 2 ln(49.16 / 50) + (.05 + (.2 / 2))(20 / 52) d 1 =.2 20 / 52 =.0805 Buy.4679(100,000) = 46,790 shares 36b. Δ 1 = N(d 1 ) 1 = N(.1) 1 = = Δ 2 = N(d 1 ) 1 = N(.05) 1 = = At time 0, bought 46,790 shares (49.16) = 2,300,196.4 cost At end of wk 1, want.4602(100,000) = 46,020 shares Therefore since have 46,790 shares Buy 48,010 46,020 = 770 shares 770(49.33) = At end of wk 2, want.4801(100,000) = 48,010 shares Therefore since have 46,020 shares Buy 48,010 46,020 = 1,990 shares 1,990(49.09) = 97, Cumulative Cost of hedge = 2,300,196.40e.05(2/52) -37,984.10e.05(1/52) +97, = 2,364,292.57

72 Question 37: 37a. σ AL = 100,000(.007) = 700 σ 2 = 400,000(.002) = 800 σ Z,AL =.8(700)(800) = 448, σ Port = (448,000) = day, 99%ile VaR = ( 15 )(2.33) = 12,844 37b. with perfect correlation, VaR would be ( )( 15 )(2.33) diversification is worth ( )( 15 (2.33)) = c. Fit a quadratic model where δp = ΔδS + (1/2)Γ( δs) 2 Find mean + variance and assume follows normal rules

73 Question 38: 38a. VaR calculates the amounts of money that can be expected to be lost over the next N days with X% certainty. Shortfall Rick sets a level below which the company s assets cannot fall and produces estimates of risk accordingly. 38b. Shortfall risk penalizes bigger shortfalls more than smaller ones.

74 Question 39: 39. Companies in distress have no need for risk management. Any benefits from activities such as hedging would go to debt holders. If they take risks, there is a chance that there would be enough of a gain to escape financial distress. Any downside risk would affect bondholders.

75 Question 40: 40a. Managers behave opportunistically and fail to realize the owner s objective of maximizing value, adverse solution, and moral hazard arise, thus informational and agency costs reduce return. 40b. Investment income on insurer s securities are taxed twice once at corporate level, once via return to shareholders through dividends thus reduce the return. 40c. By limiting the amount of certain types of investments that insurance companies can hold, you create a situation where portfolios may not be optimal thus lowering returns. Also, RBC gives regulation authority to take over a company and seize its assets. This risk will lower returns as well.

76 Question 41: RBC: VaR: Advantage: The RBC method does capture most of the elements of risk. Disadvantage: It excludes other important categories of risk, like derivative risk and interest rate risk. Advantage: Based upon traditional actuated theories, like probability of ruin and maximum probable loss. Disadvantage: Does not reflect diversification across lines of business.

77 Question 42: a) RAROC = (Premium PV (Expected claims) PV (Expenses) + interest on capital)*(1-tax) / PV (Economic Capital) = ( )*(1-.34) / 2250= 12% Where PV (Economic Capital) = (0.95)*(5000) 2500 = No, RAROC does not exceed hurdle. b) Available Equity Capital = Statutory Surplus + Unrealized gains on bonds + Discounting adjustment to reserves + Adjustment for reserve conservatism + real Estate appreciation = ( ) +0 = Yes, available capital exceeds required capital of c) Nakada s study suggests the P&C industry has enough capital to support catastrophe risk to a high degree of solvency, but that it does not price this risk to earn an adequate return on that capital for its shareholders. For catastrophe prone lines of business, available capital is adequate, but the return on that capital is well below 10%.

78 Question 43: a. Company B -- a higher P/E ratio implies the ROE on projects invested in with the retained funds is higher than the required return, k. b. Company A -- a lower P/E ratio implies a higher required rate of return, k. One may argue that a low P/E means that it is a stable cash cow with a lower rate of return. c. You could look at the price-to-cash-flow ratio because it does not rely on accounting practices. Note other answers could be given full credit as long as the argument was solidly made for the response.

79 Question 44: a) D 0 =.80 k =.15 g = b*roe =.18*.6 =.108 P = D 0 (1+g)/(k-g) =.8*(1.108) / ( ) = $21.10 per share b) Dividend at end of year 1 =.8*1.108 =.8864 Dividend at end of year 2 =.8* =.9821 Revised dividends (after year 2) = 2* *.6 = Revised g =.15*.4 =.06 P =.8864/ / [(1.4732*1.06)/( )]/ = $18.86 per share c) The competitive information would immediately be reflected, thus decreasing the price per share.

80 Question 45: Due to the one-time shift in claim reserve levels, historical loss development patterns may not provide good guidance on expected future development. Industry development patterns (e.g., from aggregate industry-wide schedule P) may provide additional guidance. The actuary would need to determine the credibility of internal and external data and perhaps weight the two together somehow when making final selections.