FIN 340 CH. 5 (Numerical)

Consider a Treasury bill with a rate of return of 5% and the following risky securities: Security A: E(r) = 0.15; variance = 0.0400 Security B: E(r) = 0.10; variance = 0.0225 Security C: E(r) = 0.12; variance = 0.1000 Security D: E(r) = 0.13; variance = 0.0625 The investor must develop a complete portfolio by combining the risk-free asset with one of the securities mentioned above. The security the investor should choose as part of her complete portfolio to achieve the best CAL would be __________.

(E(rp) - Rf) / σp (0.15 - 0.05) / 0.04 ^ 0.5 (0.1) / 0.04^0.5 0.1 / 0.2 = 0.5 (0.13 - 0.05) / 0.0625^0.5 0.08 / 0.25 = 0.32 0.5 slope is the highest number

Two assets have the following expected returns and standard deviations when the risk-free rate is 5%: Asset A E(rA) = 10%; σA = 20% Asset B E(rB) = 15%; σB = 27% An investor with a risk aversion of A = 3 would find that __________ on a risk-return basis.

A = (E(r) - Rf) / σ^2 E(r) = A x σ^2 + Rf E(rA) = 3 x 0.2^2 + 0.05 E(rA) = 3 x 0.04 + 0.05 E(rA) = 0.17 or 17% 17% > 10% so Asset A is not acceptable E(rB) = 3 x 0.27^2 + 0.05 E(rB) = 3 x 0.0729 + 0.05 E(rB) = 0.2687 or 26.87% 26.87> 15% so Asset B is not acceptable

The stock of Business Adventures sells for $45 a share. Its likely dividend payout and end-of-year price depend on the state of the economy by the end of the year as follows: Dividend Stock Price Boom $2.60 $55 Normal Economy $1.50 $48 Recession $0.06 $39 a. Calculate the expected holding-period return and standard deviation of the holding-period return. All three scenarios are equally likely. (Do not round intermediate calculations. Round your answers to 2 decimal places.) Expected Return: __________ Standard Deviation: ___________ b. Calculate the expected return and standard deviation of a portfolio invested half in Business Adventures and half in Treasury bills. The return on bills is 5%. (Do not round intermediate calculations. Round your answers to 2 decimal places.) Expected Return: ___________ Standard Deviation: ____________

A. Formula: Boom: (Boom Stock Price - Share Price + Boom Dividend) / Share price Boom: (55 - 45 + 2.6) / 45 Boom: 0.28 or 28% Normal Economy: (Normal Economy Stock Price - Share Price + Normal Economy Dividend) / Share Price Normal Economy: (48 - 45 + 1.5) / 45 Normal Economy: 0.1 or 10% Recession: (Recession Stock Price - Share Price + Recession Dividend) / Share Price Recession: (39 - 45 + 0.6) / 45 Recession: -0.12 or -12% E(HPR) = ((1/3) x 0.28) + ((1/3) x 0.1) + ((1/3) x (-0.12)) E(HRP) = 0.08667 or 8.67% Var(HRP) = [(1/3) × (0.28 − 0.08667)^2] + [(1/3) × (0.100 − 0.08667)^2] + [(1/3) × (−0.1200 − 0.08667)^2] = 0.026756 σ = √(0.026756) σ = 0.16357 or 16.36% B. E(r)= (Half invested in Business Adventures x E(HRP)) + (Half invested in Treasury Bills x Rf) E(r)= (0.5 x 0.0867) + (0.5 x 0.05) E(r)= 0.0684 or 6.84% σ= 0.5 x 0.1636 σ= 0.0818 or 8.18%

You are considering investing $1,000 in a complete portfolio. The complete portfolio is composed of Treasury bills that pay 5% and a risky portfolio, P, constructed with two risky securities, X and Y. The optimal weights of X and Y in P are 60% and 40%, respectively. X has an expected rate of return of 14%, and Y has an expected rate of return of 10%. To form a complete portfolio with an expected rate of return of 11%, you should invest __________ of your complete portfolio in Treasury bills.

E(rc)= Expected Rate of Return on Complete Portfolio 0.11 = (0.14 x 0.6 + 0.1 x 0.4) x (1 - Wf) + 0.05 x Wf 0.11 = (0.124) x (1 - Wf) + 0.05 x Wf 0.11 = 0.124 - 0.124Wf + 0.05Wf 0.11 - 0.124 = -0.124Wf + 0.05Wf -0.014 = -0.074Wf -0.014 / -0.074 = Wf Wf= 0.189189 or 19%

You have $500,000 available to invest. The risk-free rate, as well as your borrowing rate, is 8%. The return on the risky portfolio is 16%. If you wish to earn a 22% return, you should __________.

E(rc) = E(rp) x Wp + Rf x Wf 0.22 = 0.16 x Wp + 0.08 x (1 - Wp) 0.22 = 0.16Wp + 0.08 - 0.08Wp 0.22 - 0.08 = 0.16Wp - 0.08Wp 0.14 = 0.08Wp 0.14 / 0.08 = Wp Wp = 1.75 Wf = 1 - 1.75 Wf = -0.75 1.75 x 500,000 = 875,000 875,000 - 500,000 = 375,000

You are considering investing $1,000 in a complete portfolio. The complete portfolio is composed of Treasury bills that pay 5% and a risky portfolio, P, constructed with two risky securities, X and Y. The optimal weights of X and Y in P are 60% and 40%, respectively. X has an expected rate of return of 14%, and Y has an expected rate of return of 10%. The dollar values of your positions in X, Y, and Treasury bills would be __________, __________, and __________, respectively, if you decide to hold a complete portfolio that has an expected return of 8%.

E(rc) = Expected Rate of Return of Complete Portfolio E(rx) = Expected Rate of Return of X E(ry) = Expected Rate of Return of Y Wx = Optimal weight of X Wy = Optimal Weight of Y Rf= Risk Free Rate (T-bill) E(rc)= (E(rx) x Wx + E(ry) x Wy) x Wp + Rf x Wf 0.08 = (0.14 x 0.6 + 0.1 x 0.4) x (1 - Wf) + 0.05 x Wf 0.08 = 0.124 x (1 - Wf) + 0.05Wf 0.08 = 0.124 - 0.124Wf + 0.05Wf 0.08 = 0.124 - 0.124Wf + 0.05Wf 0.08 = 0.124 - 0.074Wf -0.044 = -0.074Wf -0.044 / -0.074 = Wf Wf = 0.594595 or 59.46% T-bills: 0.5946 x 1000 T-bills: 595 Security X: (1 - 0.5946) x (0.6) x 1000 Security X: 243 Security Y: (1 - 0.5946) x (0.4) x 1000 Security Y: 162

You invest $10,000 in a complete portfolio. The complete portfolio is composed of a risky asset with an expected rate of return of 15% and a standard deviation of 21% and a Treasury bill with a rate of return of 5%. How much money should be invested in the risky asset to form a portfolio with an expected return of 11%?

E(rc) = Expected Rate of Return of complete portfolio E(rp) = Expected Rate of Return of portfolio σc= Standard Deviation of complete portfolio Rf= Risk Free Asset (T-bill) Wr= What percent should be invested in Risky Asset Wrm= How much money should be invested in Risky Asset I(cp)= Investment in a complete portfolio Formula: E(rc) x Wr + Rf x (1 - Wr) = E(rp) 0.15 x Wr + 0.05 x (1 - Wr) = 0.11 0.15Wr + 0.05 - 0.05Wr = 0.11 0.15Wr -0.05Wr = 0.11 - 0.05 0.1Wr = 0.06 Wr= 0.06 / 0.1 Wr= 0.6 or 60% Wrm= Wr x I(cp) Wrm= 0.6 x 10,000 Wrm= 6,000

An investor invests 70% of her wealth in a risky asset with an expected rate of return of 15% and a variance of 5%, and she puts 30% in a Treasury bill that pays 5%. Her portfolio's expected rate of return and standard deviation are __________ and __________ respectively.

E(rc)= Expected Rate of Return of complete portfolio σc= Standard Deviation of complete portfolio √= Square root RA= Risky Asset E(r)= Expected rate of return V= Variance Rf= Risk Free Asset (T-bill) E(rc)= (RA x E(r) + Rf x V) E(rc)= (0.7 x 0.15 + 0.3 x 0.05) E(rc)= 0.12 or 12% σc= (RA x √(V)) σc= (0.7 x √(0.05)) σc= 0.1565 or 15.65% Answer: 12% ; 15.65%

Assume that you manage a risky portfolio with an expected rate of return of 15% and a standard deviation of 39%. The T-bill rate is 6%. Your client chooses to invest 70% of a portfolio in your fund and 30% in a T-bill money market fund. a. What are the expected return and standard deviation of your client's portfolio? (Round your answers to 1 decimal place.) Expected Return: __________ Standard Deviation: ___________ b. Suppose your risky portfolio includes the following investments in the given proportions: Stock A: 23% Stock B: 32% Stock C: 45% What are the investment proportions of your client's overall portfolio, including the position in T-bills? (Round your answers to 1 decimal place.) c. What is the reward-to-volatility ratio (S) of your risky portfolio and your client's overall portfolio? (Round your answers to 4 decimal places.)

E(rc)= Expected Rate of Return on Complete Portfolio y= Proportion y E(rp)= Expected Rate of Return on Portfolio (15%) Rf= Risk Free Rate (T-bill) (6%) σc= Standard Deviation of Complete Portfolio σp= Standard Deviation of Portfolio (39%) A. E(rc)= y x E(rp) + (1 - y) x Rf E(rc)= (0.7 x 0.15) + (0.3 x 0.06) E(rc)= 0.123 or 12.3% σc= y x σp σc= (0.7 x 0.39) σc= 0.273 or 27.3% B. Security Investment Proportions T-bills: 30% Stock A: (0.7 x 23%)= 16.1% Stock B: (0.7 x 32%)= 22.4% Stock C: (0.7 x 45%)= 31.5% C. S= (E(rp) - Rf) / σp S= (0.15 - 0.06) / 0.39 S= 0.2308 or 23.08% S= (E(rc) - Rf) / σc S= (0.123 - 0.06) / 0.273 S= 0.2308 or 23.08%

Your investment has a 40% chance of earning a 15% rate of return, a 50% chance of earning a 10% rate of return, and a 10% chance of losing 3%. What is the standard deviation of this investment?

E(rp)= 0.4 x 0.15 + 0.5 x 0.1 + 0.1 x (-0.03) E(rp)= 0.06 + 0.05 - 0.003 E(rp)= 0.1070 or 10.7% σ(rP) = 0.4 × (0.15 − 0.107)^2 + 0.5 × (0.10 − 0.107)^2 + 0.10 × (−0.03 − 0.107)^2 σ(rP) = 0.00074 + 0.000025 + 0.001877

You manage an equity fund with an expected risk premium of 13.8% and a standard deviation of 52%. The rate on Treasury bills is 3.6%. Your client chooses to invest $120,000 of her portfolio in your equity fund and $30,000 in a T-bill money market fund. What are the expected return and standard deviation of your client's portfolio? (Round your answers to 2 decimal places.) Expected Return: __________ Standard Deviation: ___________

E(rp)= Expected Rate of Return for portfolio E(rf)= Expected Rate of Return for fund σp= Standard Deviation of portfolio Rf= Risk Free Rate (T-bill) (3.6%) RP= Risk Premium (13.8%) σef= Standard Deviation of equity fund (52%) E(fi)= Equity fund investment (120,000 / (120,000 + 30,000)) or 80% Rfi= Risk Free Investment (T-bill) (30,000 / (120,000 + 30,000)) or 20% Formula: E(rf)= Rf + RP E(rf)= 0.036 + 0.138 E(rf)= 0.174 or 17.4% E(rp)= (E(fi) x E(rf)) + (Rfi x Rf) E(rp)= (0.8 x 0.174) + (0.2 x 0.036) E(rp)= 0.1392 + 0.0072 E(rp)= 0.1464 or 14.64% σp= E(fi) x σef σp= 0.8 x .52 σp= 0.4160 or 41.6%

Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either $50,000 or $150,000, with equal probabilities of 0.5. The alternative riskless investment in T-bills pays 4%. A. If you require a risk premium of 10%, how much will you be willing to pay for the portfolio? (Round your answer to the nearest dollar amount.) B. Suppose the portfolio can be purchased for the amount you found in (a). What will the expected rate of return on the portfolio be? (Do not round intermediate calculations. Round your answer to the nearest whole percent.) C. Now suppose you require a risk premium of 14%. What is the price you will be willing to pay now? (Round your answer to the nearest dollar amount.)

ECF= Expected Cash Flow Ep= Equal Probability (0.5) Vp= Value of the Portfolio RR= Required Return Rf= Risk Free Rate (T-bill) A. Formula: (0.5 x 50,000) + (0.5 x 150,000) = 100,000 X x (1 + 0.14) = 100,000 1.14x = 100,000 x = 100,000 / 1.14 x = 87,719.2982 or 87,719 B. Formula: E(rp)= (100,000 - 87,719) / (87,719) E(rp)= 0.14 or 14% C. Formula: RR= Rf + E(rp) RR= 4% + 14% RR= 18% X x (1 + 0.18) = 100,000 1.18x = 100,000 x= 100,000 / 1.18 x= 84,745.76 or 84,746

Consider the following two investment alternatives: First, a risky portfolio that pays a 20% rate of return with a probability of 60% or a 5% rate of return with a probability of 40%. Second, a Treasury bill that pays 6%. If you invested $50,000 in the risky portfolio, your expected profit would be _________.

EP= Expected Profit V0= Value at the beginning (50,000) V1= Value at the end E(rp)= Expected Rate of Return of portfolio Rr0= Rate of Return at the beginning (20%) Rr1= Rate of Return at the end (5%) P0= Probability at the beginning (60%) P1= Probability at the end (40%) Formula: EP= V0 x E(rp) - Find E(rp) first E(rp)= (Rr0 x P0 + Rr1 x P1) E(rp)= (0.2 x 0.6 + 0.05 x 0.4) E(rp)= 0.14 or 14% EP= 50,000 x (0.14) EP= 7,000

You invest $1,000 in a complete portfolio. The complete portfolio is composed of a risky asset with an expected rate of return of 16% and a standard deviation of 20% and a Treasury bill with a rate of return of 6%. A portfolio that has an expected value in 1 year of $1,100 could be formed if you __________.

EV= Expected Value (1,100) I(c)= Investment in complete portfolio (1,000) E(rc)= Expected Rate of Return of Complete Portfolio (16%) Rf= Risk Free Rate (T-bill) (6%) σc= Standard Deviation of complete portfolio Formula: EV= Wp x I(c) x (1 + E(rc)) + (1 - Wp) x I(c) x (1 + Rf) 1,100= Wp x 1,000 x (1 + 0.16) + (1 - Wp) x 1000 x (1 + 0.06) 1,100= Wp x 1,000 x 1.16 + (1 - Wp) x 1000 x 1.06 1,100= 1,160Wp + 1,060 - 1,060Wp 1,100= 100Wp + 1,060 1,100-1,060= 100Wp 40 = 100Wp Wp= 0.4 or 40%

Treasury bills are paying a 4% rate of return. A risk-averse investor with a risk aversion of A = 3 should invest entirely in a risky portfolio with a standard deviation of 24% only if the risky portfolio's expected return, E(rQ) is at least __________.

Formula: A = (E(rq) - r1) / σ^2 3 = (E(rq) - 0.04) / 0.24^2 3 = (E(rq) - 0.04) / 0.0576 3 x 0.0576 = E(rq) - 0.04 0.1728 = E(rq) - 0.04 0.1728 + 0.04 = E(rq) E(rq) = 0.2128 or 21.28%

You put up $50 at the beginning of the year for an investment. The value of the investment grows 4% and you earn a dividend of $3.50. Your HPR was __________.

HPR= Holding Period Return Pn= Value at the start of the holding period ($50) Iv= Value of the investment grows... (4%) D= Dividend ($3.50) Formula: HPR= (Pn x (1 + Iv) + D - Pn) / (Pn) HPR= (50 x (1 + 0.04) + 3.50 - 50) / (50) HPR= (50 x 1.04 + 3.50 - 50) / (50) HPR= (52 + 3.50 - 50) / (50) HPR= 5.5 / 50 HPR= 0.11 or 11%

The price of a stock is $55 at the beginning of the year and $50 at the end of the year. If the stock paid a $3 dividend and inflation was 3%, what is the real holding-period return for the year?

P0 = Price of stock at the beginning (55) P1 = Price of stock at the end (50) D= Dividend (3) NR= Normal Return on Stock r= Real Return i= Inflation NR= (50 - 55 - 3) / 55 = -0.0364 or - 3.64% Real Return: r= ((1 + NR) / (1 + i)) - 1 r = (1 + (-0.0364) / (1 + 0.03) - 1 r = (0.9636) / (1.03) -1 r = 0.0645 or 6.45%

Suppose you pay $9,400 for a $10,000 par Treasury bill maturing in 6 months. What is the effective annual rate of return for this investment?

Pa = Par ; Face value (10,000) Pv= Paid value (9,400) M= T-bill Maturity (6 months) Ef(ar)= Effective annual rate of return for this investment Ef(ar)= (Pa / Pv)^ (1 Year / M) - 1 Ef(ar)= (10,000 / 9,400) ^ (12 - 6) - 1 Ef(ar)= (1.0638) ^ (2) - 1 Ef(ar)= 1.1317 - 1 Ef(ar)= 0.1317 or 13.17%

A portfolio with a 22% standard deviation generated a return of 17% last year when T-bills were paying 6.0%. This portfolio had a Sharpe ratio of __________.

R= Return Rf= Risk free rate (T-bill) σp= Standard Deviation of portfolio Sr= Sharp Ratio Formula: Sr = (R - Rf) / σp Sr = (0.17 - 0.06) / 0.22

Consider the following two investments alternatives: First, a risky portfolio that pays a 15% rate of return with a probability of 40% or a 5% rate of return with a probability of 60%. Second, a Treasury bill that pays 6%. The risk premium on the risky investment is _________.

RP= Risk Premium E(rp)= Expected Rate of Return of Portfolio Rf= Risk free rate (T-bill) Rr0= Rate of Return at the beginning (15%) Rr1= Rate of Return at the end (5%) P0= Probability at the beginning (40%) P1= Probability at the end (60%) RP= E(rp) - Rf - Find E(rp) first E(rp)= Rr1 x P1 + Rr2 x P2 E(rp)= 0.15 x 0.4 + 0.05 x 0.6 E(rp)= 0.09 or 9% Rp= 0.09 - 0.06 Rp= 0.03 or 3%

Assume that you manage a risky portfolio with an expected rate of return of 15% and a standard deviation of 31%. The T-bill rate is 5%. Your risky portfolio includes the following investments in the given proportions: Stock A 26% Stock B 33% Stock C 41% Your client decides to invest in your risky portfolio a proportion (y) of his total investment budget with the remainder in a T-bill money market fund so that his overall portfolio will have an expected rate of return of 14%. Required: A. What is the proportion y? (Round your answer to 1 decimal place.) Proportion y: ___________ B. What are your client's investment proportions in your three stocks and in T-bills? (Round your intermediate calculations and final answers to 2 decimal places.) T-Bills: ___________ Stock A: ___________ Stock B: ___________ Stock C: ____________ C. What is the standard deviation of the rate of return on your client's portfolio? (Round your intermediate calculations and final answer to 1 decimal place.) Standard Deviation _______________ % per year

y= Proportion y E(rc)= Expected Rate of Return for overall portfolio (14%) E(rp)= Expected Rate of Return for portfolio (15%) Rf= Risk Free Rate of Return (5%) σp= Standard Deviation of portfolio (31%) σc= Standard Deviation of Complete Portfolio A. Formula: E(rc)= y x E(rp) + (1 - y) x Rf - Solve for y y x (E(rp) + (1 - y) x Rf = E(rc) y x 0.15 + (1 - y) x 0.05 = 0.14 0.15y + 0.05 - 0.05y = 0.14 0.15y - 0.05y = 0.14 - 0.05 0.1y = 0.09 y= 0.09 / 0.1 y= 0.9 or 90% A. Answer: 90% B. T-bill = 1 - y T-bill = 1 - 0.9 T-bill = 0.1 or 10% Stock A = 0.9 x Stock A% Stock A = 0.9 x 26% Stock A = 23.4% Stock B = 0.9 x Stock B% Stock B = 0.9 x 33% Stock B = 29.7% Stock C = 0.9 x Stock C% Stock C = 0.9 x 41% Stock C = 36.9% B. Answer: T-bill= 10% Stock A= 23.4% Stock B= 29.7% Stock C= 36.9% C. Formula: σc = y x σp σc = 0.9 x 0.31 σc = 0.279 or 27.9% per year

You invest $1,000 in a complete portfolio. The complete portfolio is composed of a risky asset with an expected rate of return of 16% and a standard deviation of 20% and a Treasury bill with a rate of return of 6%. __________ of your complete portfolio should be invested in the risky portfolio if you want your complete portfolio to have a standard deviation of 9%.

σc0= Beginning Standard Deviation of complete portfolio σc1= Ending Standard Deviation of complete portfolio Wp= Percentage of complete portfolio to be invested in risky portfolio Formula: σc0 = Wp x σc1 0.09 = Wp x 0.2 Wp= 0.09 / 0.2 Wp= 0.45 or 45%

You invest $2,300 in a complete portfolio. The complete portfolio is composed of a risky asset with an expected rate of return of 17% and a standard deviation of 20% and a Treasury bill with a rate of return of 9%. __________ of your complete portfolio should be invested in the risky portfolio if you want your complete portfolio to have a standard deviation of 15%.

σc0= Standard Deviation of Complete Portfolio at the beginning (20%) σc1= Standard Deviation of Complete Portfolio at the end (15%) y= Variable Formula: σc1 = y x σc0 0.15 = y x 0.2 y = (0.15 / 0.2) Y= 0.75 or 75%