FIN2101 BUSINESS FINANCE II MODULE 1- RISK-RETURNANALYSIS QUESTION 1 Discuss the measurement of portfolio risk and ind icate how and why it differs from measuring risk for a single security QUESTION 2 A friend is considering some investments and is confused. He asks you"What is this notion of probabilities and what does it tell me? " Provide an answer for him Your friend comes back and says "All right, I understand probabilities, now what confuses me are the notions of expected return and standard deviation. What do they tell me? " Once again, help your friend out QUESTION 3 You are given the following information about the possible returns from an investment Possible returns Probabilities 0.12 0.09 0.60 0.06 025 Calculate the investment's expected return and the standard deviation of the return QUESTION 4 You are considering a portfolio of two assets-A and B- and have obtained the following dat Return on a Return onb probability 0.10 0.16 0.3 0.15 0.10 0.5 0.05 0.04 The returns on the two assets have a correlation coefficient of 0.3. What is the expected return and standard deviation of a portfolio consisting of 50% of A and 50% of B?
July 2003 FIN2101 BUSINESS FINANCE II MODULE 1 - RISK-RETURN ANALYSIS QUESTION 1 Discuss the measurement of portfolio risk and indicate how and why it differs from measuring risk for a single security. QUESTION 2 A friend is considering some investments and is confused. He asks you "What is this notion of probabilities and what does it tell me?". Provide an answer for him. Your friend comes back again and says "All right, I understand probabilities, now what confuses me are the notions of expected return and standard deviation. What do they tell me?". Once again, help your friend out. QUESTION 3 You are given the following information about the possible returns from an investment: Possible Returns Probabilities 0.12 0.15 0.09 0.60 0.06 0.25 Calculate the investment's expected return and the standard deviation of the return. QUESTION 4 You are considering a portfolio of two assets - A and B - and have obtained the following data: Return on A Return on B Probability 0.10 0.16 0.3 0.15 0.10 0.5 0.05 0.04 0.2 The returns on the two assets have a correlation coefficient of 0.3. What is the expected return and standard deviation of a portfolio consisting of 50% of A and 50% of B?
QUESTION 5 Over a period of three years, Security X had returns of 10%, 14% and-3%. For the same three years, Security Y had returns of 12%, 10% and 5% (a) What is the standard deviation of returns for these two securities? (b) What is the covariance of returns between these two securities (c) What is the correlation of returns? QUESTION 6 Referring to the data in Question 5 above, calculate the standard deviation of returns for a two-asset portfolio comprising 40% of funds invested in Security X and the remain ing 60% invested in Security Y QUESTION 7 Hilda Hornbill has invested 60% of her money in Share A and the remainder in Share B. She assesses their prospects as follows Share a Share b Expected Return 0.15 0.20 Standard Deviation 0.20 Correlation Between Returns What is the expected return and standard deviation of Hilda s portfolio? QUESTION 8 Bernadetta Bloggs has invested 55% of her money in Share X and the remainder in Share Y She assesses their prospects as follows Share X Expected Return 0.10 0.15 Standard Deviation 0.20 0.28 Correlation between returns 0.3 What is the expected return and standard deviation of her portfolio?
July 2003 QUESTION 5 Over a period of three years, Security X had returns of 10%, 14% and –3%. For the same three years, Security Y had returns of 12%, 10% and 5%. (a) What is the standard deviation of returns for these two securities? (b) What is the covariance of returns between these two securities? (c) What is the correlation of returns? QUESTION 6 Referring to the data in Question 5 above, calculate the standard deviation of returns for a two-asset portfolio comprising 40% of funds invested in Security X and the remaining 60% invested in Security Y. QUESTION 7 Hilda Hornbill has invested 60% of her money in Share A and the remainder in Share B. She assesses their prospects as follows: Share A Share B Expected Return 0.15 0.20 Standard Deviation 0.20 0.22 Correlation Between Returns 0.5 What is the expected return and standard deviation of Hilda's portfolio? QUESTION 8 Bernadetta Bloggs has invested 55% of her money in Share X and the remainder in Share Y. She assesses their prospects as follows: Share X Share Y Expected Return 0.10 0.15 Standard Deviation 0.20 0.28 Correlation Between Returns 0.3 What is the expected return and standard deviation of her portfolio?
FIN2101 BUSINESS FINANCE II SOLUTIONS TO TUTORIAL QUESTIONS MODULE 1- RISK-RETURNANALYSIS
July 2003 FIN2101 BUSINESS FINANCE II SOLUTIONS TO TUTORIAL QUESTIONS MODULE 1 - RISK-RETURN ANALYSIS
July 2003 QUESTION 1 For a single investment, risk is measured by the standard deviation of the probabilit distribution of the expected returns. A portfolio's risk cannot be calculated by way of a simple weighted average of the risk of its individual assets, as some of the riskiness of one asset may be offset by the riskiness of another The corre lation coefficient. a measure of the relation between rates of return on two assets is, therefore, very important in determining the risk of a portfolio QUESTION 2 When an outcome or benefit from an investment opportunity is known with certainty, its probabil ity of occurrence is one, and deviations from that value are not expected. Perhaps the closest opportunity to this situation is a government secur ity, for example, Australian Sav ings Bonds(AsB). If an ASB is held to maturity, the probability of investors not receiving either their interest payments or return of principal is so small that it can be regarded as zero conversely, the probability of earning the promised interest rate is one When an investment is risky, however, it means that the outcome could take on any number of possibilities The likelihood of the occurrence of each outcome is measured by its probability. The probability of each occurring is less than one and greater than zero The expected value or mean outcome/return is a weighted average of the possible outcomes calculated. We weight each outcome ki by the probability of that outcome occurring Pri and then sum of the weighted outcomes k=∑k×P The expected value k is a probability weighted average value for the possible outcomes We regard an event as risky because the exact outcome is not known in ad vance even though each possible outcome and its probability of occurrence is known. We define risk in terms of the variability of the outcomes-the greater the variability the greater the risk To measure risk we use the standard devia tion(or variance) of the returns. This provides with a measure of the variabl ity of the outcomes about mean which reflects probability of each occurring. The formula for standard deviation(of a single asset)is a-∑k-P
July 2003 QUESTION 1 For a single investment, risk is measured by the standard deviation of the probability distribution of the expected returns. A portfolio’s risk cannot be calculated by way of a simple weighted average of the risk of its individual assets, as some of the riskiness of one asset may be offset by the riskiness of another. The correlation coefficient, a measure of the relation between rates of return on two assets, is, therefore, very important in determining the risk of a portfolio. QUESTION 2 When an outcome or benefit from an investment opportunity is known with certainty, its probability of occurrence is one, and deviations from that value are not expected. Perhaps the closest opportunity to this situation is a government security, for example, Australian Savings Bonds (ASB). If an ASB is held to maturity, the probability of investors not receiving either their interest payments or return of principal is so small that it can be regarded as zero; conversely, the probability of earning the promised interest rate is one. When an investment is risky, however, it means that the outcome could take on any number of possibilities. The likelihood of the occurrence of each outcome is measured by its probability. The probability of each occurring is less than one and greater than zero. The expected value or mean outcome/return is a weighted average of the possible outcomes calculated. We weight each outcome ki by the probability of that outcome occurring Pri and then sum of the weighted outcomes: = = n i 1 i Pri k k The expected value k is a probability weighted average value for the possible outcomes. We regard an event as risky because the exact outcome is not known in advance even though each possible outcome and its probability of occurrence is known. We define risk in terms of the variability of the outcomes - the greater the variability, the greater the risk. To measure risk we use the standard deviation (or variance) of the returns. This provides us with a measure of the variability of the outcomes about the mean which reflects the probability of each occurring. The formula for standard deviation (of a single asset) is: ( ) n i=1 i 2 k = ki - k Pr
QUESTION 3 k=∑kxP =(0.120.1)+(0.09k0.60+(0.060.2 0.0180.0540.015 =0.080r8.7% △(-x x Pr (0.120.080.15+(0.090.08)0.60(0.060.08)0.25 =√0.00016336.0000059.00018225 =√0.000351 =0.01871.8735%
July 200 3 QUESTION 3 ( ) ( ) ( ) = 0 .0 8 7o r 8 .7 % = 0 .0 1 8+ 0 .0 5 4+ 0 .0 1 5 = 0 .1 2 0 .1 5 + 0 .0 9 0 .6 0 + 0 .0 6 0 .2 5 k = k P r ni=1 i i ( ) ( ) ( ) ( ) 0 .0 1 8 7 3 5o r 1 .8 7 3 5 % 0 .0 0 0 3 5 1 0 .0 0 0 1 6 3 3 50 .0 0 0 0 0 5 40 .0 0 0 1 8 2 2 5 0 .1 2- 0 .0 8 7 0 .1 5 0 .0 9- 0 .0 8 7 0 .6 0 0 .0 6- 0 .0 8 7 0 .2 5 k - k P r 2 2 2 ni 1 i 2 k i === + + = + + = =
QUESTION 4 kA=(0.10×0.3+0.150.9+(0.050.2 0.030.0750.0l 0.115 kB=(0.160.3+(0.10k0.9+(0.040.2 0.0480.050.008 0.106 -x) ,=√0.100.1103+(0,150.110.5+(0.050.11)0.2 0.000065.0006125.000845 √0.001525 =0.03905 =√0.160.1060.3+(0.100.1060.5+(0.040.1060.2 =√0.00087480.0000180.0008712 =√0.001764 0.042
July 200 3 QUESTION 4 ( ) ( ) ( ) ( ) ( ) ( ) = 0 .1 0 6 = 0 .0 4 8+ 0 .0 5+ 0 .0 0 8 k = 0 .1 6 0 .3 + 0 .1 0 0 .5 + 0 .0 4 0 .2 = 0 .1 1 5 = 0 .0 3+ 0 .0 7 5+ 0 .01 k = 0 .1 0 0 .3 + 0 .1 5 0 .5 + 0 .0 5 0 .2 k = k P r BA ni=1 i i ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 .0 4 2 0 .0 0 1 7 6 4 0 .0 0 0 8 7 4 80 .0 0 0 0 1 80 .0 0 0 8 7 1 2 0 .1 6- 0 .1 0 6 0 .3 0 .1 0- 0 .1 0 6 0 .5 0 .0 4- 0 .1 0 6 0 .2 0 .0 3 9 0 5 0 .0 0 1 5 2 5 0 .0 0 0 0 6 7 50 .0 0 0 6 1 2 50 .0 0 0 8 4 5 0 .1 0- 0 .1 1 5 0 .3 0 .1 5- 0 .1 1 5 0 .5 0 .0 5- 0 .1 1 5 0 .2 k - k P r 2 2 2 k 2 2 2 k ni 1 i 2 k i BA === + + = + + === + + = + + = =
QUESTION 4 (Continued) k (0.5×0.11)+0.5×0.105 =0.05750.053 =0.1105 os,-Ww1o1+ w202+2w,w2.9,02 √0s×00390+5×042)+(2×050.5030.03908042 =√0.00038126.0004410.00024651 √0.00106826 0.0327
July 200 3 QUESTION 4 (Continued) ( ) ( ) = 0 .1 1 0 5 = 0 .0 5 7 5+ 0 .0 5 3 = 0 .5 0 .1 1 5+ 0 .5 0 .1 0 6 k = w k n j =1 p j j ( ) ( ) ( ) 0 .0 3 2 7 = 0 .0 0 1 0 6 8 2 6 5 = 0 .0 0 0 3 8 1 2 5 + 0 .0 0 0 4 4 1+ 0 .0 0 0 2 4 6 0 1 5 = 0 .5 0 .0 3 9 0 5+ 0 .5 0 .0 4 2 + 2 0 .5 0 .5 0 .3 0 .0 3 9 0 50 .0 4 2 = w + w + 2 w w r 2 2 2 2 1 2 1 , 2 1 2 22 22 21 2 k 1 p =
QUESTION 5 Note that the question involves historical data and ex post analysis is therefore required Step 1-Calculate the arithmetic mean returns for the two securities ∑k kx0.10+0.140.03 0.07 y=012+010+005 0.09 Step 2- Calculate the standard deviation of the returns for the two securities (k-k (0.100.0y+(0.140.07+(-0.030.0 0.00090.00490.01 =√0.0079 =0.08888 (0.13009+(010009+05009 0.00090.00040.0016 =√0.0013 =0.0360555
July 2003 QUESTION 5 Note that the question involves historical data and ex post analysis is therefore required. Step 1- Calculate the arithmetic mean returns for the two securities 0 . 0 9 3 0 . 1 2 0 . 1 0 0 . 0 5 k 0 . 0 7 3 0 . 1 0 0 . 1 4- 0 . 0 3 k n k k Y X n i 1 i = + + = = + = = = Step 2 - Calculate the standard deviation of the returns for the two securities ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 .0 3 6 0 5 5 5 0 .0 0 1 3 2 0 .0 0 0 9 0 .0 0 0 1 0 .0 0 1 6 3 -1 0 .1 2- 0 .0 9 0 .1 0- 0 .0 9 0 .0 5- 0 .0 9 0 .0 8 8 8 8 0 .0 0 7 9 2 0 .0 0 0 9 0 .0 0 4 9 0 .0 1 3 -1 0 .1 0- 0 .0 7 0 .1 4- 0 .0 7 - 0 .0 3- 0 .0 7 n -1 k - k 2 2 2 k 2 2 2 k n i 1 2 i k Y X = = + + = + + = = = + + = + + = = =
QUESTION 5 (Continued) Step 3- Calculate the covariance of returns between the two securities ∑…-k]x-k2 Co ku,k)=i=l (0.100.0(0.2-0.09)+(0.140.00(0.10-009+(-0.030.000.050.09 0.00090.000平0.004 0.0028 Step 4- Calculate the correlation coefficient of returns 0.0028 0.088880.03605 0.8737 QUESTION 6 =√w1G1+w22+2w1w2r1,2O1a2 0.4×0.088+0.6×0.036055(2x0,40.6×0,873800888036055 =√0.0012640.0004680.001344 =√0.003076 =0.05546
July 2003 QUESTION 5 (Continued) Step 3 - Calculate the covariance of returns between the two securities ( ) ( )( ) ( )( ) ( )( ) 0 .0 0 2 8 2 0 .0 0 0 9 0 .0 0 0 7 0 .0 0 4 3 -1 0 .1 0- 0 .0 7 0.1 2 0.0 9 0 .1 4- 0 .0 7 0.1 0 0.0 9 - 0 .0 3- 0 .0 7 0 .0 5- 0 .0 9 n -1 k - k k - k Co v k , k n i 1 2 2 ,i 1 1 ,i 1 2 = + + = − + − + = = = Step 4 - Calculate the correlation coefficient of returns ( ) 0 .8 7 3 7 0 .0 8 8 8 80 .0 3 6 0 5 5 5 0 .0 0 2 8 Co v k , k r 1 2 k k 1 2 1 , 2 = = = QUESTION 6 ( ) ( ) ( ) 0 .0 5 5 4 6 0 .0 0 3 0 7 6 0 .0 0 1 2 6 40 .0 0 0 4 6 80 .0 0 1 3 4 4 0 .4 0 .0 8 8 8 8 0 .6 0 .0 3 6 0 5 5 5 2 0 .4 0 .6 0 .8 7 3 7 0 .0 8 8 8 80 .0 3 6 0 5 5 5 w w 2 w w r 2 2 2 2 1 2 1 , 2 1 2 2 2 2 2 2 1 2 k 1 p = = = + + = + + = + +
QUESTION 7 =w. (0.6×0.1+(04×0.20 =0.090.08 0.17 12+w2a2+2ww2r1.g1o √0.6×0,20)+(04×0,22)+(2×0.604×0.5×0 200.2 =√0.01440.0077440.01056 =√0.032704 =0.1808 QUESTION 8 k=∑Wxk (0.550.10+0450.1 =0.0550.0675 0.1225 ok,=vwia1+w202+2 w,w2r,g,O2 √0.53×026)+(0.45×0.28)+(2×0.550,450,3×0.200,28 √0.012+0.0158760008316 036292 =0.1905
July 2003 QUESTION 7 ( ) ( ) = 0 .1 7 = 0 .0 9+ 0 .0 8 = 0 .6 0 .1 5+ 0 .4 0 .2 0 k = w k n j =1 p j j ( ) ( ) ( ) 0 .1 8 0 8 = 0 .0 3 2 7 0 4 = 0 .0 1 4 4+ 0 .0 0 7 7 4 4+ 0 .0 1 0 5 6 = 0 .6 0 .2 0 + 0 .4 0 .2 2 + 2 0 .6 0 .4 0 .5 0 .2 0 0 .2 2 = w + w + 2 w w r 2 2 2 2 1 2 1 , 2 1 2 2 2 2 2 2 1 2 k 1 p = QUESTION 8 ( ) ( ) = 0 .1 2 2 5 = 0 .0 5 5+ 0 .0 6 7 5 = 0 .5 5 0 .1 0+ 0 .4 5 0 .1 5 k = w k n j =1 p j j ( ) ( ) ( ) 0 .1 9 0 5 = 0 .0 3 6 2 9 2 = 0 .0 1 2 1+ 0 .0 1 5 8 7 6+ 0 .0 0 8 3 1 6 = 0 .5 5 0 .2 0 + 0 .4 5 0 .2 8 + 2 0 .5 5 0 .4 5 0 .3 0 .2 0 0 .2 8 = w + w + 2 w w r 2 2 2 2 1 2 1 , 2 1 2 2 2 2 2 2 1 2 k 1 p =
