Risk, Return and Portfolio Study Guide Part 4

SFM - COMPILER Question 31 : Nov 2012 - Paper – 8 Marks Mr. FedUp wants to invest an amount of Rs. 520 lakhs and had approached his Portfolio Manager. The Portfolio Manager had advised Mr. FedUp to invest in the following manner: Security Moderate Better Good Very Good Best Amount in Lakhs 60 80 100 120 160 Beta 0.5 1.00 0.80 1.20 1.50 You are required to advise Mr. FedUp in regard to the following, using Capital Asset Pricing Methodology: (i) Expected return on the portfolio, if the Government Securities are at 8% and the NIFTY is yielding 10%. (ii) Advisability of replacing Security 'Better' with NIFTY. Solution (i) Calculation of Expected Return on each Stock by CAPM Model According to CAPM, Expected Return = RF + β (RM – Rf) Moderate = 8 + 0.5 (10 – 8) = 9% Better = 8 + 1 (10 – 8) = 10% Good = 8 + 0.8 (10 – 8) = 9.60% Very Good = 8 + 1.2 (10 – 8) = 10.40% Best = 8 + 1.5 (10 – 8) = 11% (ii) Expected Return on the Portfolio = Wt Average Return on individual Stocks Total Investments = 60 + 80 + 100 + 120 + 160 = 520 80 100 60 Expected Return on the portfolio = 9 x 520 + 10 x 520 + 9.60 x 520 + 160 120 10.4 x 520 + 11 x 520 = 10.208% SFM - COMPILER 2013 Question 32 : May 2013 - RTP The following information is available in respect of Security X Equilibrium Return 15% Market Return 15% 7% Treasury Bond Trading at $140 Covariance of Market Return and Security Return 225% Coefficient of Correlation 0.75 You are required to determine the Standard Deviation of Market Return and Security Return. Solution (i) 𝐑𝐟 = 𝟕 𝟏𝟒𝟎 x 100 = 5% Applying CAPM 15% 10% β = 5% + β(15% - 5%) = β(10%) =1 β = Cov (r,m) 1 = 225 2 σm = 225 σm 2 σm 2 σm = √225 = 15 Standard Deviation of Market Return = 15 (ii) Cov (r,m) Cor. = 0.75 = σr = 20% σm σr 225 15σr Standard Deviation of Security Return = 20% Question 33 :May 2013 – RTP – Similar to - Question 21 - May 2011 - Paper – 5 Marks SFM - COMPILER Question 34 : May 2013 - Paper – 8 Marks On Jan 1, 2013 an investor has a portfolio of 5 shares as given below. Security Price No of Shares Beta A 349.60 5,000 1.15 B C D 480.50 593.52 734.70 7,000 8,000 10,000 0.40 0.90 0.95 E 824.85 2,000 0.85 The cost of capital to the investor is 10.5%per annum. You are required to calculate: (i) The beta of his portfolio. (ii) The theoretical value of the NIFTY futures for February 2013. (iii) The number of contracts of NIFTY the investor needs to sell to get a full hedge until February for his portfolio if the current value of NIFTY is 5900 and NIFTY futures have a minimum trade lot requirement of 200 units. Assume that the futures are trading at their fair value. (iv) The number of future contracts the investor should trade if he desires to reduce the beta of his portfolios to 0.6. No. of days in a year be treated as 365. Given: In (1.105) = 0.0998 e(0.015858) = 1.01598 SFM - COMPILER Solution (i) Calculation of Total Portfolio Security Price A 349.60 No of Shares 5,000 Total 17,46,500 B C D 480.50 593.52 734.70 7,000 8,000 10,000 33,63,500 47,48,160 73,47,000 E Total 824.85 2,000 16,49,700 1,88,54,860 (ii) Calculation of Portfolio Beta = Wt Average Beta of Individual Stocks 33,63,500 17,46,500 Portfolio Beta = 1.15 x 1,88,54,860 + 0.40 x 1,88,54,860 73,47,000 16,49,700 47,48,160 + 0.90 x 1,88,54,860 + 0.95 x 1,88,54,860 + 0.85 x 1,88,54,860 = 0.849 (iii) Calculation of Theoretical Value of Future Contract Cost of Capital = 10.5% p.a. Accordingly, the Continuously Compounded Rate of Interest ln(1.105) = 0.0998 For February 2013 contract, t= 58/365= 0.1589 Further F= Sert F= 5,900e(0.0998)(0.1589) F= Rs. 5,900e0.015858 F= Rs. 5,900X1.01598 = Rs. 5,994.28 (iv) When total portfolio is to be hedged: = Value of Stop position Requiring Hedging x Portfolio Beta Value of Future Contract 1,88,54,860 = 5,994 x 200 x 0.849 = 13.35 contracts (v) When total portfolio beta is to be reduced to 0.6: = Number of Contracts to be sold = Value of Portfolio (βt - βt) F x M x βf = 1,88,54,860(0.849 - 0.600) = 3.92 contracts i.e 4 contracts 5,994.28 x 200 SFM - COMPILER Question 35 : Nov 2013 - RTP Following data is related to Company X, Market Index and Treasury Bonds for the current year and last 4 years: Year Company X 2009 2010 2011 2012 2013 (Current Year) Average Share Price Dividend Per share Rs. 139 Rs. 7.00 Rs. 147 Rs. 8.50 Rs. 163 Rs. 9.00 Rs. 179 Rs. 9.50 Rs. 203.51 Rs. 10.00 Average Market Index Market Index Market Dividend Yield Return on Treasury Bonds 1300 1495 1520 1640 1768 3% 5% 5.5% 4.75% 5.5% 7% 9% 8% 8% 8% With the above data estimate the beta of Company X’s share. Solution (i) Calculation of Capital Gain for share X Share price has increased from 139 at the end of year 2009 to Rs. 203.51 at the end of year 2013 So the appreciation is in 4 periods (From end of 2009 to end of 2013) 139 (1 + r)4 = 203.51 203.51 Therefore r = 139 - 1 = 10% (ii) Average annual dividend yield (%) Year Dividend / Share Price 2009 2010 2011 2012 2013 Average = 7 / 139 x 100 8.5 / 147 x 100 9 / 163 x 100 9.5 / 179 x 100 10 / 203.51 x 100 Dividend Yield 5% 5.8% 5.5% 5.3% 4.9% 5+5.8%+5.5%+5.3%+4.9% = 5.35 5 Therefore Expected Return on the company’s stock = Capital Appreciation + Annual Dividend Yield = = 10% + 5.3% = 15.3% SFM - COMPILER (iii) Calculation of Capital Gain for Market Index Market Index has increased from 1300 at the end of year 2009 to Rs. 1768 at the end of year 2013 So the appreciation is in 4 periods (From end of 2009 to end of 2013) 1300 (1 + r)4 = 1768 1768 Therefore r = 1300  1/4- 1 = 8%   (iv) Average Annual Dividend Yield (%) = 3%+5%+5.5%+4.75%+5.5% = 4.75% 5 Therefore Expected Return on the market Index = Capital Appreciation + Annual Dividend Yield = = 8% + 4.75% = 12.75% (v) Average Annual Risk Free Rate = 7%+9%+8%+8%+8% = 8% 5 (vi) With the help of above information and using CAPM, we can calculate β Expected Return on Stock = Rf + β (RM – Rf 15.3% = 8% + β[12.75% - 8%] β = 1.54 SFM - COMPILER Question 36 : Nov 2013 – RTP The rates of return on the security of Company X and market portfolio for 10 periods are given below: Period Return of Security X (%) 20 22 25 21 18 –5 17 19 –7 20 1 2 3 4 5 6 7 8 9 10 Return on Market Portfolio (%) 22 20 18 16 20 8 –6 5 6 11 1. What is the beta of Security X? 2. What is the characteristic line for security X? Solution Period 𝑹𝒙 𝑹𝒎 𝒅𝒙 𝒅𝒙 𝟐 𝒅𝒎 𝒅𝒎 𝟐 𝒅𝒙 𝒅𝒎 1 2 3 4 5 6 7 8 9 10 20 22 25 21 18 –5 17 19 –7 20 22 20 18 16 20 8 –6 5 6 11 5 7 10 6 3 -20 2 4 -22 5 25 49 100 36 9 400 4 16 484 25 10 8 6 4 8 -4 -18 -7 -6 -1 100 64 36 16 64 16 324 49 36 1 50 56 60 24 24 80 -36 -28 132 -5 Total 150 120 1148 706 357 Return = ΣRx n 15 12 Variance Σd2 = n Variance Σd2 = n COVxm dxdm = n = 114.8 = 70.6 SD = Variance = 10.71 (ii) Characteristic line for security X = 35.7 COVxm SD = Βx = σ 2m Variance = 0.505 = 8.40 = α + β × RM Alpha (α) = 15 – (0.505 × 12) = 8.94% ∴Characteristic line for security X = 8.94 + 0.505 RM SFM - COMPILER Question 37 : Nov 2013 - Paper – 5 Marks A trader is having in its portfolio shares worth Rs. 85 lakhs at current price and cash Rs. 15 lakhs. The beta of share portfolio is 1.6. After 3 months the price of shares dropped by 3.2%. Determine: (i) Current portfolio beta (ii) Portfolio beta after 3 months if the trader on current date goes for long position on Rs. 100 lakhs Nifty futures. Solution Current portfolio Current Beta for share = 1.6 Beta for cash = 0 Current portfolio beta = 0.85 x 1.6 + 0 x 0.15 = 1.36 Portfolio beta after 3 months: Beta for portfolio of shares 1.6 = Change in the value of portfoli of shares change in value of market index 0.032 = change in value of market index Change in value of market portfolio (Index) = (0.032 / 1.6) x 100 = 2% Position taken on 100 lakh Nifty futures Long Value of index after 3 months = Rs. 100 lakh x (100 - 0.02) = Rs. 98 lakh Mark-to-market paid = Rs. 2 lakh Cash balance after payment of mark-to-market = Rs. 13 lakh Value of portfolio after 3 months = Rs 85 lakh x (1-0.032) + Rs. 13 lakh = Rs. 95.28 lakh Change in value of portfolio 100 - 95.28 = 100 = 4.72% Portfolio beta = 0.0472/0.02 = 2.36 SFM - COMPILER Question 38 : Nov 2013 - Paper – 8 Marks Mr Ram is holding the following securities: Particulars of Securities Equity Shares: Gold Ltd. Silver Ltd. Bronze Ltd. GOI Bonds Cost Dividends/Intere st Rs. 11,000 16,000 12,000 40,000 Market price Rs. 1,800 1,000 800 4,000 Beta Rs. 12,000 17,200 18,000 37,500 0.6 0.8 0.6 1.0 Average return of the portfolio is 14%, calculate: 1. Expected rate of return in each, using the Capital Asset Pricing Model (CAPM). 2. Risk free rate of return. Solution (i) Expected Rate of Return on market portfolio Securities Cost Dividends/Inte Market price rest Rs. Rs. Equity Shares: 11,000 1,800 12,000 Gold Ltd. 16,000 1,000 17,200 Silver Ltd. 12,000 800 18,000 Bronze Ltd. 40,000 4,000 37,500 GOI Bonds Total 79,000 7,600 Capital Gains Rs. 1,000 1,200 6,000 -2,500 5,700 Dividend + Capital Gains Return = x 100 = 16.84% Total Investment 0.6+0.8 + 0.6 + 1 = 0.50(Alternatively we can also calculate 4 wt average beta also) (ii) Average β = (ii) Calculation of Rf Average Return = Rf + β (RM – Rf) 14% = Rf + 0.5(16.84 – Rf) 14 = Rf + 8.42 – 0.5Rf Therefore Rf = 11.16% (iv) Calculation of Expected return of each security by CAPM Gold = 11.16 + 0.6 (16.84 – 11.16) Silver = 11.16 + 0.8 (16.84 – 11.16) = 14.568 = 15.704 Bronze = 11.16 + 0.6 (16.84 – 11.16) = 14.568 GOI = 11.16 + 1 (16.84 – 11.16) = 16.84 SFM - COMPILER Question 39 : Nov 2013 - Paper – 6 Marks Ram buys 10,000 shares of X Ltd. at a price of Rs. 22 per share whose beta value is 1.5 and sells 5,000 shares of A Ltd. at a price of Rs. 40 per share having a beta value of 2. He obtains a complete hedge by Nifty futures at Rs. 1,000 each. He closes out his position at the closing price of the next day when the share of X Ltd. dropped by 2%, share of A Ltd, appreciated by 3% and Nifty futures dropped by 1.5%. What is the overall profit/loss to Ram? Solution No. of the Future Contract to be obtained to get a complete hedge = 10,000 x 22.15 x 1.5 - 5,000 x 40 x 2 = 70 contracts 1,000 Thus, by purchasing 70 Nifty future contracts to be long to obtain a complete hedge. Cash Outlay = 10000 x Rs. 22 – 5000 x Rs. 40 + 70 x Rs. 1,000 = Rs. 2,20,000 – Rs. 2,00,000 + Rs. 70,000 = Rs. 90,000 Cash Inflow at Close Out = 10000 x Rs. 22 x 0.98 – 5000 x Rs. 40 x 1.03 + 70 x Rs. 1,000 x 0.985 = Rs. 2,15,600 – Rs. 2,06,000 + Rs. 68,950 = Rs. 78,550 Gain/ Loss = Rs. 78,550 – Rs. 90,000 = - Rs. 11,450 (Loss)