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Investment Management 333-693 Final Examination Seat No: Semester 1, 2010 Student No: (Do NOT write your name anywhere on this exam booklet) THE UNIVERSITY OF MELBOURNE FACULTY OF ECONOMICS AND COMMERCE EXAMINATION - SEMESTER ONE 2010 SUBJECT NUMBER: SUBJECT NAME: COMMON CONTENT: NONE EXAM DURATION: READING TIME: TOTAL MARKS: CONTRIBUTION TO FINAL MARKS: NO. OF PAGES: 333-693 INVESTMENT MANAGEMENT 2 HOURS 15 MINUTES 100 60% 19 (including cover page, 2 blank pages, 3 formula pages, and 1 calendar page) AUTHORIZED MATERIALS 1. This is a CLOSED BOOK exam. Students are NOT permitted to have any materials in their possession. 2. Non-programmable calculators are permitted. Calculators with the ability to enter and/or retrieve text are NOT permitted. 3. Students are NOT permitted to use a foreign language dictionary. INSTRUCTIONS TO INVIGILATORS 1. 2. 3. 4. Students should be provided ONLY with this booklet. Students are to write answers ONLY in the spaces provided in this booklet. Writing is NOT permitted during reading time. This booklet must be COLLECTED at the end of the exam. INSTRUCTIONS TO STUDENTS 1. 2. 3. 4. 5. Write ONLY your seat number and student number in the boxes at the top of this cover page. Do NOT detach any exam pages (EXCLUDING the attached blank, formula and calendar pages) from this booklet. Exams with missing pages will NOT be marked. Writing is NOT permitted during reading time. Answers must ONLY be written on question pages in this booklet. All other writing will NOT be marked. For full credit show ALL your calculations including formulas used. BAILLIEU LIBRARY 1. This paper may NOT be lodged with the library. Examiners Use Only Q1 (15) Q2 (15) Q3 (10) Q4 (10) Q5 (15) Q6 (20) 1 Q7 (15) Total (100) Investment Management 333-693 Final Examination Semester 1, 2010 Question 1 (15 marks) Both the CAPM and a single-factor APT model imply the same basic pricing relationship: assets with the same beta should have the same expected returns. In both models, mispriced assets become \"correctly\" via market forces. Compare this correction process via the CAPM versus this correction process via the APT. What are the key differences? You may use examples and graphs, if necessary. 2 Investment Management 333-693 Final Examination Semester 1, 2010 Question 2 (15 marks) You currently manage the following portfolio on behalf of an individual investor: Number of Units Current Price (per unit) Shares of BHP 500 $38.12 Shares of FGL 1 000 $5.40 20 $970.874 1 year zero coupon bonds, with par value of $1000 You estimate the following characteristics of the assets over the next year: BHP FGL Expected Return .06 .09 Standard Deviation .08 .12 The covariance of BHP and FGL is .00264. Your client's utility function is given by = 2 ! . What transactions* should you undertake to generate the optimal complete portfolio for your client? (*An example transaction would be \"Buy 10 shares of BHP\".) [Hint: First solve for the optimal complete portfolio in terms of weights/percentages.] 3 Investment Management 333-693 Final Examination Semester 1, 2010 Question 3 (10 marks) You observe the following rates: !" = 3.9% !" = 4.2% !" = 5.0% !" = 4.5% !" = 5.2% Do these rates make sense? Supposing you could borrow up to $100 million, how would you optimally invest given these rates? 4 Investment Management 333-693 Final Examination Semester 1, 2010 Question 4 (5 + 5 = 10 marks) a) What is the interest rate (% pa) on a zero-coupon bond currently trading at $93.127 and maturing in 3 years? b) What is the price per $100 par value of a commercial bill with 73 days to maturity at the settlement date if the quoted interest rate is 5.45%? 5 Investment Management 333-693 Final Examination Semester 1, 2010 Question 5 (15 marks) On 12 Jan 2010, you purchased an Australian government bond with a coupon rate of 8% pa (paid semiannually) and maturing on 15 April 2012. The bond dealer quoted a yield to maturity of 5.2% on that date (12 Jan). Today is 10 June 2010, and you are selling that bond to a dealer, who quotes you a yield of 4.8%. What is your holding period return in %pa, assuming you did not reinvest any coupons? 6 Investment Management 333-693 Final Examination Semester 1, 2010 Question 6 (4 x 5 marks = 20 marks) Consider the following two bonds paying semi-annual coupons, both with $1000 par value. Coupon (%pa) Yield to Maturity Maturity Bond A 5% 3.80% 2 years Bond B 3% 3.96% 3 years a) What is the price of each bond? Assume the settlement date is today. b) What is the duration of each bond? c) What is the convexity of each bond? d) Suppose you believe yields on both bonds will decrease by 0.5% over the next week. If you were going to hold a position for one week only, which bond should you buy? Justify your answer. 7 Investment Management 333-693 Final Examination Semester 1, 2010 Question 7 (15 Marks) Morningstar, a mutual fund monitoring firm, reports the following information: 5 year average annual return S.D. Alpha Beta Blackrock Large Cap Growth 2.73% 17.60% 1.16% 0.97 Dean Small Cap Value 4.32% 23.64% 3.14% 1.29 Invesco Diversified Dividend 2.44% 15.62% 1.38% 0.83 Over this 5 year interval, the market index had an average return of 0.30% and a standard deviation of 16.31%. Additionally, the average return on risk-free government bonds was 1.51%. For each of the following measures, rank the funds in order of best to worst: Sharpe Ratio Treynor Ratio Information Ratio END OF EXAMINATION 8 Investment Management 333-693 Final Examination Semester 1, 2010 FORMULA SHEET E (rP ) = wD E (rD ) + wE E (rE ) wMin ( D ) = 2 E Cov(rE , rD ) 2 2 D + E 2Cov(rE , rD ) 2 2 2 2 2 P = wD D + wE E + 2wD wE Cov(rD , rE ) , SP = E (rP ) rf P , TP = E (rP ) rf P y= Cov(rD , rE ) D E P 2 , M P = 1 M rF + M rP P P P , P = rP E (rP ), IRP = 2 U = E ( rP ) 1 A P , 2 wD = E (rp ) rf 2 A P ( E (r ) r ) ( E (r ) r ) Cov(r , r ) = ( E (r ) r ) + ( E (r ) r ) ( E (r ) r + E (r ) r ) Cov(r , r ) D D 2 E f 2 E f E E f 2 D f D D f E E f D E E (rc ) = yE (rp ) + (1 y )rf = rf + y[ E (rp ) rf ] c = y p Ri (t ) = i + i RM (t ) + ei (t ) 2 i2 = i2 m + 2 (ei ), wi0 = 2 Cov(ri , rj ) = i j m i , 2 (ei ) wi0 wi = n , 0 i w i =1 0 wA = 2 A / (eA ) , 2 E ( RM ) / M w* = A 0 wA 0 1 + (1 A ) wA S = S + A ( eA ) 2 P n n A = wi i , i =1 2 2 M n A = wi i , 2 (eA ) = wi2 2 (ei ) i =1 i =1 2 2 = ( w + w* A ) M + ( w* ( eA ) ) A A 2 P * E ( RP ) = ( wM + w* A ) E ( RM ) + w* A , A A * M 2 E (rM ) rf = A M E[ri ] rf = cov ( ri , rM ) E ( r )r = i E ( rM ) rf M f 2 M ri = E[ri ] + i F + ei wP rP wB rB = ( wP wB ) rB + wB ( rP rB ) + ( wP wB )( rP rB ) 9 2 Investment Management 333-693 1 d 0T = P0 = Final Examination , T (1 + z 0 T ) P0 = Par [1 d (1 r )] ( T * 1 + z 0T ) Semester 1, 2010 1T Par (1 + z 0T )T , Par = P0 z 0T 1 1T * z 0T , Par [1 d (1 r )] = P0 1 Par 365 Par n , P0 = P0 = Par 1 q 1 , s = n 360 P0 n 1+ s 365 1/T 1/ X 1/ X Par PX P0 PX z 0T = HPR0T = 1 , HPR0 X = 1 + 1 = 1 , P0 P0 P0 C C C C C + Par P0 = + + + ... + + 2 3 T 1 1 + z 01 (1 + z 02 ) (1 + z 03 ) (1 + z 0T )T (1 + z 0,T 1 ) Par + T (1 + ytm ) 1 C 1 Par P0 = C+ 1 + f /h n 1 n 1 ytm (1 + ytm ) (1 + ytm ) (1 + ytm ) 1 d 0T P0 = Par d 01c + d 02c + ... + d 0T (1 + c ) , ytmT = d 01 + d 02 + ... + d 0T P0 = f tT (1 + z 0T )T = t (1 + z 0t ) C 1 1 ytm (1 + ytm )T 1 ( T t ) (1 + z 0T )T = (1 + z 0t )t (1 + f tT )T t , d 0T = d 0t d tT , d tT = d 0T 1 , d 0t (1 + z 0T )T = (1 + z 0t )t 1 + E (rtT ) { T t 1T )} ( z 0T = (1 + z 01 ) 1 + E (r12 ) 1 + E (r23 ) ... 1 + E rT 1, T 1 T 2 D= 3 T C / (1 + i ) C / (1 + i ) C / (1 + i ) (C + Par ) / (1 + i ) 1+ 2+ 3 + ... + T , P P P P 2 D= P dP i 1 d P i 2 2 + i ) D + X ( i ) 2 ( P di P 2P di 1+ i 2C T (T + 1)(C + Par ) 1 6C 12C X= + + + ... + 2 2 3 2 P (1 + i ) 1 + i (1 + i ) (1 + i ) (1 + i )T rt = Pt + Dt T 1 , rT = (1 + rt ) 1, Pt 1 10 t CF (1 + i t)t t =1 P Investment Management 333-693 Final Examination Semester 1, 2010 n 1n TWR (total ) = (1 + rt ) 1, TWR ( mean ) = 1 + TWR (total ) 1 t =1 P + Dt rt = l n t , rT = T rt Pt 1 n 1 TWR ( total ) = rt , TWR ( mean ) = TWR (total ) n t =1 n D t 1 + DWR t + t =1 ( ) Pn (1 + DWR )n P0 = 0 FUM t = FUM t 1 + DIVt + CGt DISTt REDt + INFt FEESt rt = FUM t 1 + DIVt + CGt FEESt 1 FUM t 1 FUM t 1 + DIVt + CGt FEESt rt = l n FUM t 1 DIST + RED INF FUM n t t1 + DWR t t + 1 + DWR n FUM 0 = 0 ( ( ) ) t =1 rP rf rP rf S= , T= , P = rP rf + P (rM rf ) n P P IR = P , M 2 = rP* rM (eP ) rP ,t = + E rE ,t + L rL,t + S rS ,t M M F F I I rB = wB rB + wB rB + wB rB M M F F I I rP = wP rP + wP rP + wP rP wP rP wB rB = ( wP wB ) rB + wB ( rP rB ) + ( wP wB )( rP rB ) 11 Investment Management 333-693 Final Examination Public Holidays are circled. 12 Semester 1, 2010 Investment Management 333-693 Final Examination Semester 1, 2010 Question 1 The CAPM pricing relation is enforced by all investors shading their portfolios toward underpriced securities and away from overpriced securities. As investors are mean-variance optimizers, they seek maximum expected returns for any level of (market) risk, which in a CAPM world is measured by beta. The APT pricing relation is enforced by a small number of investors finding an arbitrage opportunity with mispriced portfolios, buying underpriced ones and selling overpriced ones. The key differences are 1. The APT is enforced by arbitrage, while the CAPM is enforced by risk-averse investors maximising expected returns. 2. The APT requires well-diversified portfolios, while the CAPM applies to all assets, including individual securities. 3. Strictly speaking, the APT does not require the identification of a market portfolio. (This is less of an issue in practical applications, however.) Question 2 First, solve for the risk free rate: (1000 - 970.874) / 970.874 = 3% Then, solve for the weight of BHP in the optimal risky portfolio: wBHP = ( (.06 - .03)*.12^2 - (.09 - .03)*.00264 ) / ( (.06 - .03)*.12^2 + (.09 - .03)*.08^2 - (.06 - .03 + .09 - .03)*.00264 ) = 47.3%, implying wFGL = 1 - .473 = 52.7% The corresponding optimal risky portfolio has E[rP] = .473*.06 + .527*.09 = .0758 var = = (.473*.08)^2 + (.527*.12)^12 + 2 (.473)(.527) .00264 = .00675 Her risk aversion parameter A is 4, so we can solve for her optimal investment in the risky portfolio as y = (.0758 - .03)/( 4 * .00675) = 170% This implies the weights of the assets in her optimal complete portfolio are bonds = -70%, BHP = .473 * 1.7 = 80%, FGL = 90% The client's wealth is 500 * 38.12 + 1000 * 5.40 + 20 (970.874) = 43,877.5 Her optimal dollar-value holdings of bonds: 43877.5 * (-.7) = -30,714 of BHP: 43877.5 * (.8) = 35,102 of FGL: 43877.5 * (.9) = 39,489.75 13 Investment Management 333-693 Final Examination Semester 1, 2010 This implies desired holdings of -30714/970.874 = -32 bonds, 35102/38.12 = 921 shares of BHP, and 39489.75/5.40 = 7313 shares of FGL. To achieve these holdings, she will need to sell 52 bonds, buy 421 shares of BHP, and buy 6313 shares of FGL. Question 3 Solve for the future rates from the zero rates: 1 + !" = 1 + !" ! 1 + !" ! 1.042! = !" = 4.5% 1 + !" 1.039 1 + !" ! 1.05! = = !" = 5.55% 1 + !" 1.039 The quoted forward rates do not match the forward rates implied by the zero rates. To take advantage of this mismatch, you could borrow $100m for one year at z01 and continue to borrow for two more years at f13. The total interest you will owe is 1.039 1.052! 1 = 14.9865% Simultaneously, you would invest that $100m for three years at z03. The interest you will earn is 1.05! 1 = 15.7625% This will be a risk-free $100m * (15.7625% - 14.9865%) = $775,954. Alternatively, rather than invest $100m for three years, you could invest the present value of the $100m * 1.148965 owed 3 years from today: 100 1.148965 = 99.252 1.05! This would yield an arbitrage profit 100m - 99.252m = $748,000 today. Question 4 a) 100 93.127 ! ! 1 = 2.4019% b) If you priced in Australia: 100 1 + 0.0545 73 365 14 = 98.92175 Investment Management 333-693 Final Examination Semester 1, 2010 Question 5 The initial settlement date is 15 Jan 2010, from which the bond will pay 5 more coupons. We can solve for h! = 16 + 30 + 31 + 31 + 28 + 31 + 15 = 182 ! = 16 + 28 + 31 + 15 = 90 Implying ! = 1 1 + .026 !" !"# 40 + 40 1 1 . 026 1.026 ! + 1000 = 1078.76 1.026! The upcoming settlement date is 16 June 2010, from which the bond will pay 4 more coupons. As the next coupon will be paid on 15 Oct 2010, we can solve for ! = 365 182 = 183 ! = 14 + 31 + 31 + 30 + 15 = 121 Implying ! = 1 1 + .024 !"! !"# 40 + 40 1 1 . 024 1.024 ! + 1000 = 1068.89 1.024! The holding period is 16 + 28 + 31 + 30 + 31 + 16 = 152 days. So the holding period return is 1068.89 + 40 = 1078.76 !"# !"# = 6.84% Question 6 a) year cf pv 1 50 48.16956 2 1050 974.5286 3 cf pv 30 28.85725 30 1030 27.75803 916.7236 Price 1022.698 973.3389 b) 48.16956 1 + 974.5286 2 = 1.9529 1022.698 28.85725 1 + 27.75803 2 + 916.7236 3 ! = = 2.912186 973.3389 ! = 15 Investment Management 333-693 c) Final Examination Semester 1, 2010 ! = 48.16956 2 + 974.5286 6 = 2.697 2 1022.698 1.038! ! = 28.85725 2 + 27.75803 6 + 916.7236 12 = 5.335 2 973.3389 1.0396! d) You could simply observe that both the duration and the convexity of B are much larger than those of A. Given that they have similar initial yields, B's price should be more sensitive to price changes than A. Alternatively, one could calculate the new prices explicitly: !"# !"# ! = 1033.67, ! = 988.39 and notice that the B has a higher percentage increase in price than A. Question 7 Sharpe Ratio: S(B) = (2.73 - 1.51) / 17.6 = .0693 S(D) = (4.32 - 1.51) / 23.64 = .1189 S(I) = (2.44 - 1.51) / 15.62 = 0.05954 S(D) > S(I) > S(B) Treynor Ratio: T(B) = (2.73 - 1.51) / .97 = 1.258 T(D) = (4.32 - 1.51) / 1.29 = 2.178 T(I) = (2.44 - 1.51) / .93 = 1.12 T(D) > T(B) > T(I) Information Ratio: You need to estimate the idiosyncratic risk ! for each fund: ! ! ! = ! ! ! + ! (! ) We have sd(e_B) = sqrt(var(B) - beta_B^2 - var(M)) = sqrt( .176^2 - .97^2 * .1631^2) = .0771 sd(e_D) = sqrt(var(D) - beta_B^2 - var(M)) = sqrt( .2364^2 - 1.29^2 * .1631^2) = .1078 sd(e_I) = sqrt(var(I) - beta_I^2 - var(M)) = sqrt( .1562^2 - .83^2 * .1631^2) = .0779 IR(B) = 1.16/7.71= .150 IR(D) = 3.14/10.87 = .289 IR(I) = 1.38/7.79 = .177 IR(D) > IR(I) > IR(B) 16 Student Number: 11111111111 THE UNIVERSITY OF MELBOURNE Department of Finance Faculty of Business and Economics Semester One Examination, 2012 FNCE90056 INVESTMENT MANAGEMENT 2 hours 15 minutes 50 10 (including cover page) SUBJECT NUMBER: SUBJECT NAME: EXAM DURATION: READING TIME: TOTAL MARKS: NO. OF PAGES: Authorised Materials: 1. 2. 3. This is a CLOSED BOOK exam. Students are NOT permitted to have any materials in their possession. Non-programmable calculators are permitted. Calculators with the ability to enter and/or retrieve text are NOT permitted. Students are NOT permitted to use a foreign language dictionary. Instructions to Invigilators: 1. 2. 3. 4. Students should be provided ONLY with this booklet. Students are to write answers ONLY in the spaces provided in this booklet. Writing is NOT permitted during reading time. This booklet must be COLLECTED at the end of the exam. Instructions to Students: 1. Write ONLY your student number in the boxes at the top of this cover page. 2. 3. 4. 5. 6. 7. Do NOT detach any pages (EXCLUDING the formula collection on pages 9-10) from this booklet. Exams with miSSing pages will NOT be marked. For full marks, it is necessary to answer ALL questions. For full credit show ALL your calculations including formulas used. Answers must ONLY be written on Question pages in this booklet. All other writing will NOT be marked. Space for rough work is given on pages 7-8. Examiners will NOT take into account anything written on these pages. The formula collection is given on pages 9-10. This paper may be lodged with the Baillieu Library. Examiners Use Only Ql (10) Q2 (10) Q3 (10) Q4 (12) Q5 (8) Total (50) Page 1 oflO OPEN BOOK EXAM Answer Six (6) out of Eight (8) Questions, each 10 marks Page 28 IQUESTION 1 3+4+3=10 marks I Assume that you are using a two-factor APT model to find the expected return on a welldiversified portfolio Q that promises an expected return of 18%. The relevant factor portfolios, their betas, and their factor risk premiums are shown in the table below. The assumed risk-free rate is 3.5%. Factor A B Factor Beta 1.4 0.8 Factor Risk Premium 12.0% -3.5% Marks (a) What is the expected return on the portfolio Q if it is fairly priced? Marks (b) What is the arbitrage strategy? (c) Given that you do not believe in a "free lunch", what should the risk-free rate be to exclude the arbitrage opportunity? Page 2 of 10 Marks IQUESTION 2 4+6=10 marks I Assume there are 4 default-free bonds with the following prices and future cash flows: " ... ,,,,,,,,,... Bond Price Today Cash Flows .,,~~~=,--~,,,,,,, Year 1 Year 2 Year 3 A $934.58 1000 0 0 B 881.66 0 1000 0 C 1,118.21 100 100 1100 D 839.62 0 0 1000 Marks (a) Do these bonds present an arbitrage opportunity? Marks (b) If so, how would you take advantage of this opportunity? If not, why not? Page 3 of 10 IQUESTION 3 2+2+4+2=10 marks I An investor wishes to be sure she has $20 million in 15 months time. At present, i-year and 2-year zero-coupon bonds are priced to yield 9.7% pa. The investor sets up a bond portfolio using the duration-matching principle. Three months after setting up the portfolio, the yields on both bonds increase to 10.2% pa and then remain at that level for a further 12 months. Assume that all months are of equal length, that all bonds have a par value of $100, and that investors may trade any number of bonds, including fractions of bonds. (a) Calculate the prices today of the one-year zero-coupon bond and the two-year zerocoupon bond. Show your calculations. Marks Marks (b) How much (in total) does the investor need to invest today? (c) How much should be invested today in the one-year bond? How much should be invested today in the two-year bond? Show your calculations. Page 4 of 10 Marks (d) How many one-year bonds should be bought today? How many two-year bonds should be bought today? Show your calculations. IQUESTION 4 Marks 3+3+6=12 marks (a) In the pure expectations model of the term structure, what assumption is made concerning investors' attitude to risk? Explain briefly. Why is this assumption frequently considered to be a weakness of the model? (b) Suppose that the pure expectations model of the term structure is correct. Today's 3-year zero rate is 8.9% pa and it is known that investors expect the 1-year zero rate at the start of Year 4 to be 9.5% pa. Calculate today's 4-year zero rate (in % pal Show your calculations. Page 5 of 10 I Marks Marks (c) Assume now that you believe in the liquidity premium theory of the term structure. You observe that today's term structure is upward-sloping. What inference{s) can you draw regarding investors' expectations of the future course of interest rates? Explain. IQUESTION 5 Marks 8 marks There are usually considered to be four sources of superior investment performance. List these sources and provide a brief description of each. END OF EXAMINATION Page 6 of 10 I Marks SPACE FOR ROUGH WORK Examiners will NOT take into account anything written on this page. Page 7 of 10 SPACE FOR ROUGH WORK Examiners will NOT take into account anything written on this page. Page 8 of 10 FORMULA SHEET E(rp) wDE(rD) + wEE(rE) = W . (D) = ai - Cov(rE' rD) 2 aD +aE - 2Cov ( rE,rD) Mill ? 222222 ap=wDaLJ+wEaE+ WLJW C( ovrLJ,rE) , Sp = E(rp)-r[ , Tp = ap E(rp) - r[ , a p = rp -E(rp), IRp f3p U E (rp ) - = + , Aa~ y= p= = ap ap Cov(rD,rE) aDaE M~ ~ [1- ~: 1"' + [ ~: 1Ii , E(rp)-rr 2 Aa p (E(rD)-rr )ai -(E(rE)-r[ )Cov(rD,rE) wD = (E(rD) - r[ )ai + (E(rE) - r[ )a; - (E(rD) - r[ + E(rE) - r[ )Cov(rD,rE) E(rJ = yE(rp) + (1- y)r( ac = = r( + ;{E(r;,) - r yap Ri (t) = a i + f3iRM (t)+ ei (t) a i2 = f32 am + a 2 (eJ, 2 i 0 Wi o W,t . = COV(I;, r) . ai 2(e;) , a =-- a A / a 2 (e A ) 2 ' E(RM)/a M = f3if3p,;, . 0 W=~ i ~WiO 11 0 * WA ' = , , [a, WA ( 0 1+ 1-f3A ) wA r Sp =SM + a(e A) 11 a A = ~wiai' f= E(Rp) = 11 11 f3A = ~Wif3f' a 2(e A) = ~ w~a2(eJ (w;r + w:tf3 A)E(RM) + w:a A' 2 a p = ( . + wAf3A WM f aM + (* (eA)f wAa 2 E(r"J - rr = Aa,~ E[fi] - r1 = wprp - wsrs = a 2 rM) [ E ( ) - rf ] = f3i [ (M - r[ ] 1'.11 E r) M fi = E[r;] + f3 iF + ei COY (fi, (Wp - WB)rB + Ws (rp - rB)+ (Wp - WB)(rp - rs) Page 9 of 10 FORMULA SHEET dOT = 1 o+Z~T 0- )] = Par[7t-d(l-r Po P _ T (1+Z 0T ) , Par , T (1 + ZOT ) , ZOT = d{ rpZOT = ( Parr' Po -1 (1- Par. Po ) Po = Par ( 1- qx_n_) , , ~ ( Par -I) x 365 , Po 360 Po n )lr -1 Par = tl 1+sx365 I/X l/X _I~(PX) -1, ZOT = HP~)T = (Parr -1 'HPRox~(I+Px-Po) Po Po III C C C C C+Par Po = - - + 2 + 3 + ... + T-l + T 1 + Z(J1 (1+Z 02 ) (l+Z 03 ) (1+Z 0 ,T-l) (1+Z0T ) 1 C [ 1 T + Par T Po = - 1 .y tm ( 1 + )' tm) ( 1 + Y t1 1 Po = ( ~J/h 1+ yt11l 1 C + - [ 1C 1 Par + Y /111 ( 1 + Y t;;-I ( 1 + J't~-I 1 III = Par [dolC + do2 c + '" + dllT (1+c)] [ ' f,-n (Y . = ( 1 + ZOT ) 1:r 1 + ZOI _1 , Y tllQ' = I-dOT dOl + d02 + ... + dOT x , (1 + ZOT f = (1 + ZOt Y (1 + 1:1' f- I ,daT -- dOt x dtT , dtT -_ dOT d at (1 + ZOT ZOT = {1 + ZOI )x f = (1 + ZOt )' X[1 + E (rlT )] T-I [1 + E ('12 )] X[1 + E (r23 )] x ... x [1 + E ~T-I,T )]}T -1 i D= C/(l+i) C/(1+1f C/(l+iY (C+Par)/(l+il xl+ x2+ x3+ ... + xT, D= P P P P 1=1 (r 2 !1P dP !1i 1 d P !1i ",,--.xD+X !1i-""-. -+---.? ( r-! 1 i P dz P 2P dz1+ z 1 [ 2C 6C 12C T ('1' + 1)(C + Por) X = 2P(1+if 1+i + (1+zf + (1+i)3 + ... + (l+i? Page 10 of 10 1 tC~ (1 + i Y P
