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Choosing an Investment Portfolio

Chapter Twelve. Choosing an Investment Portfolio. Chapter Outline. What is Optimal Portfolio ? The Process of Personal Portfolio Selection Portfolio Selection of Financial Assets Efficient Diversification with (1 risky +1 riskless)

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Choosing an Investment Portfolio

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  1. Chapter Twelve Choosing an Investment Portfolio

  2. Chapter Outline • What is Optimal Portfolio ? • The Process of Personal Portfolio Selection • Portfolio Selection of Financial Assets • Efficient Diversification with (1 risky +1 riskless) • Finding the composition of the portfolio (known targeted return for the portfolio) • Calculate Portfolio’s  ( SD) • Calculate reward-to-risk-ratio • Efficient Diversification with (2 risky +1 riskless 1- Finding the composition of the two risky assets • First (Try and error) • The Direct Approach • 2- Finding the composition of the two risky + riskless

  3. The Process of Personal Portfolio Selection • Correlation of (-1) is not realistic so min variance rather then zero variance. • What is the optimal portfolio that fits everybody • Many portfolios ( Examples of Portfolio Selection) : • How much should you in stocks, bonds and other securities? • Should you buy a house (or rent it)? • Which type of insurance to buy? • Should you start working after the high school or should you go to university? • How can you mange liabilities? • No single portfolio strategy is best for all people. • The life cycle : age, family members , permanent income,….etc. • Planing Horizon : short or long , life time… • Risk Tolerance : capacity bearing and attitude

  4. Portfolio Selection of Financial Assets • If we focus on Bonds and stocks • Risk-Return Trade off fits everybody , no matter preference, age ……, • Two steps for portfolio optimization: • Find the optimal combination of risky assets • Mix that with riskless • What is Riskless asset • The riskless rate is the interest rate on t - bills maturing in 3 month. • Governmental bonds for longer period in a special unit of account if held to maturity

  5. Portfolio Selection for 1 risky and 1 riskless • Step 1: Finding the composition of the portfolio • E (R)P = w E(RS) + 1-w (Rf) = w E(RS)+RF-w(RF)→ • = Rf+ wE( RS) -Rf (1) • Either E( R)P or w should be given. • You have $ 100 000 and riskless asset gives 6% and a risky asset E (R )VOLVO = 14% and  = 20%. You require 9% return on the portfolio. What is the composition of each? • 0.09= Rf+ wE( RS) -Rf • = .06 + w(.14-.06) • 0.09= .06 +. 08w •  w = (.09 –06)/.08 = .375 • The portfolio consists of 37.5% risky and 62.5% riskless

  6. Efficient Diversification with (1risky +1 riskless) • Step 2: Calculate Portfolio’s  ( SD) • When we combine risky and riskless the  will be: • General form • Riskless has σ2 =0 & =0. Then, P = w S = .2w (2) • We know the value of w = .375 • Then P = .2(.375) = .075 SD • Step 3: Calculate reward-to-risk-ratio of a portfolio with 1 riskey and 1 riskless • Solve for (w) in equation (2): w = /S • Use this in equation .(1): E(R ) = R f + wE( RS) –R f • (3) • Where slope (extra return offered for each unit of extra risk an investor is willing to bear) or Reward-to-Risk Ratio. RF is the intercept • 0.06 + 0.40(.075)=0.09

  7. Efficient Diversification with (1risky +1 riskless) • Now let the target be 11% • Use equation (1): .11 = .06 + .08w  • w = (.11-.06)/.08 = .623 • The portfolio has 62.5% risky assets and 37.5% riskless • Find ( σ) Use Eq.(2): • Higher targeted return implies higher risk • If we have another risky asset which one to choose?Rf=6% • Assume that asset 2 (ABB) has; E(R) =.08 and =.15 • Remember Asset 1 (VOLVO) has E(R) =.14 and =.20 • Volvo: 0.40 better

  8. Efficient Diversification with (1risky +1 riskless) • The Market portfolio can be described by Capital Market Line (CML), the Best (Risk-reward) available to all investors • Everybody would try to get above CML, but competition will derive E( R)P to the LINE. Thus, A>B, if E(RA)> E(RB) • E(R ) • .16 CML • .14 * S Trade-off line • .12 * J • .10 *H • .08 *G • .06*F • .04 • .02 • 0 • 0 .05 .10 .15 .20 .25 

  9. Efficient Diversification with (2 risky +1 riskless) • To find the efficient portfolio in this case; two steps: • Step One: Finding the composition of the two risky assets • Two ways: • First (Try and error): • The formula for the mean E( R): • E (R) = w E(R1) + 1-w E(R2) (4) • We substituted Risky asset (2) for riskless asset • The variance according to Equation (2) • (5) • Ex: consider a portfolio (C) with 2 risky assets. We have: • E(R)1= .14 E(R)2=.08 σ1= .20 σ2 = .15 ρ=0

  10. Efficient Diversification with (2 risky +1 riskless • If we start with the composition (25% asset 1& 75% asset 2) • The expected return: • According to eq. (4): E (R)= w E(R1)+w2 E (R2) • = .25 E(R1) + .75 (R2) = .25 (.14) +.75 (.08) = .095 • The variance eq.(5): = .252 (.22) + .752(.15)2 + 0 = .01515625 •  =  .01515625 = .1231 • Is it optimal? We don't know. Try all combinations

  11. Efficient Diversification with (2 risky +1 riskless • Try all combinations of the compositions. We have :

  12. Efficient Diversification with (2 risky +1 riskless • Step Two: Finding the composition of the two risky + riskless • Now combing riskless + risky asset(1) + Risky asset(2). • We have ( RF=.06. E(R1)=.14, 1=.20 and E(R2)=.08 & 2=.15 & ρ=0 • The Direct Approach: • The formula the portfolio proportion is • w1 = 69.23% and w2= 30.77%

  13. Efficient Diversification with (2 risky +1 riskless • Now the new portfolio (RT ) has: • E(RT)=w1E(R1)+w2E(R2)= 69.23(.14)+30.7(.08)=12.2 % • σ2T : • = (.6923)2 (.2)2 + (.3077)2(.15)2 + 0 • = (0,479)(0,04)+( 0,0947)(0,0225)= 0,0192+ 0,002= 0,0213 • T =√ 0,0213 = .146 • Return- to - riskratio: • The slope reward to risk ratio) = .42 • Earlier with 1 risky & 1 riskless was .40.

  14. Efficient Diversification with (2 risky +1 riskless • Now we know composition between (asset 1 + asset 2). • What is the composition of both against riskless? • IF we have targeted Expected return (use eq.(1)). • Here (w) is the proportion of risky assets & (1-w) is the proportion of riskless. • Otherwise depending on risk preference • A person may: 50% on the two risky and 50% on riskless. • The weight of all the three :

  15. Efficient Diversification with (2 risky +1 riskless • The expected return now: • E( RE) = RF +.5E( RT)-RF) = 0.6+.5(.122-.06) = .091 • To be compared with 9% (asset 1 + riskless) • The variance • E = .5 (T)= .5 (.146) = .073

  16. Example- Achieving a Target Expected Return • You have $100 000 to invest. You want E( R) = 10%. • You may choose between two portfolios : • 1- A portfolio (S) consists of ( Asset 1 + riskless) with : • Asset 1: E (Rs) = 14% and s =.20. (RF) = 6%. • Find the composition and  for such portfolio. • 2- Another portfolio (T) with ( asset 1 + asset 2 + riskless) • the Optimal composition of risky assets(69.2% asset1 &30.8 of asset2) . • Optimum risky E(RT) = .122 and T=0.146. • How much should you invest in risky & riskless? • Find also  for such a portfolio.

  17. Example- Achieving a Target Expected Return • 1- Portfolio (S) • The composition: Solve for (w) in Eq(1): • E (R) = w E(RS) + 1-w (Rf) ➱ • E (R ) = RF + w (E(RS) – RF)) • = 0.06 + w(.14 – 0.06) = 0.06 + w(0.08) • 0.10 = 0.06 + w(0.08) ➱w = 0.04/0.08=0.5 • Then 50% should be invested in asset 1 and 50% in riskless • SD for the portfolio • We know that = w s = .5 (.20) = .10 or 10%

  18. Example- Achieving a Target Expected Return • 2- Portfolio (T) : • The composition between risky and riskless: Solvefor w in(1): • E (R ) = w E(RT) + 1-w(RF) • = Risky + Riskless • = Rf+ wE( RS) -Rf • E (R ) = .122w + .06(1-w) setting E( R) =.10 • 0.10 = .06 + w(.122-0.06) so w = (.10-.06)/.062= .65 • 65% of the $100 000 must be invested in the optimal combination of risky asset and 35% in riskless. • SD for the portfolio • We know that = w T = .65 (.146) =.095 • The composition • Weight in riskless 35% • Weight in risky 1 .65(.69.2%) 45% • Weight in risky 2 .65(30.8%) 20% • Total 100%

  19. Summary • Perfect diversification(σ= 0 ) requires: • Optimal Weight & ρ=-1) • But, ρ=-1 is not realistic, • Calculating the optimal Weights that give minimum variance : • Case I: Portfolio Selection for 2 risky assets

  20. Summary • Case II: Portfolio Selection for 1 risky+ 1 riskless • E (R)P = w E(RS) + 1-w (Rf) = w E(RS)+RF-w(RF)→ • = Rf+ wE( RS) -Rf (1) • If we know E(R)P solve for (w) • SD: P =  S w and . Using that in (1): • Return-to-risk • We choose among the different assets based in return –to-risk-ratio w = /S

  21. Summary • Case III: Portfolio Selection for 2 risky+ 1 riskless • Stage (1): Optimal risky • Weight of assets (1) • The rest of (1) is the weight of asset (2) • E(RT)=w1E(R1)+w2E(R2) both risky • σ2T • Return-to-risk ratio

  22. Summary • Stage II: Mix with riskless • Re - write : E (RE ) = w E(RT) + 1-w(RF) • = Rf+ wE( RT) -RF (1) • Solve for (w), if we know E(R)P • Here (w) is the proportion of risky assets & (1-w) is the proportion of riskless. • Otherwise risk preference : • If 50% on the two risky and 50% on riskless • E( RE) = RF +.5E( RT)-RF) • E= w T

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