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LECTURE 5 : PORTFOLIO THEORY

LECTURE 5 : PORTFOLIO THEORY. (Asset Pricing and Portfolio Theory). Contents. Principal of diversification Introduction to portfolio theory (the Markowitz approach) – mean-variance approach Combining risky assets – the efficient frontier

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LECTURE 5 : PORTFOLIO THEORY

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  1. LECTURE 5 :PORTFOLIO THEORY (Asset Pricing and Portfolio Theory)

  2. Contents • Principal of diversification • Introduction to portfolio theory (the Markowitz approach) – mean-variance approach • Combining risky assets – the efficient frontier • Combining (a bundle of) risky assets and the risk free rate – transformation line • Capital market line (best transformation line) • Security market line • Alternative (mathematical) way to obtain the MV results • Two fund theorem • One fund theorem

  3. Introduction • How should we divide our wealth ? – say £100 • Two questions : • Between different risky assets (s’s > 0) • Adding the risk free rate (s = 0) • Principle of insurance is based on concept of ‘diversification’  pooling of uncorrelated events  insurance premium relative small proportion of the value of the items (i.e. cars, building)

  4. Assumption : Mean-Variance Model • Investors : prefer a higher expected return to lower returns • ERA ≥ ERB Dislike risk • var(RA) ≤ var(RB) or SD(RA) ≤ SD(RB) Covariance and correlation : Cov(RA, RB) = r SD(RA) SD(RB)

  5. Portfolio : Expected Return and Variance Formulas (2 asset case) : Expected portfolio return : ERp = wA ERA + wb ERB Variance of portfolio return : var(Rp) = wA2 var(RA) + wB2 var(RB) + 2wAwBCov(RA,RB) Matrix notation : Expected portfolio return : ERp = w’ERi Variance of portfolio return : var(Rp) = w’Sw where w is (nx1) vector of weights ERi is (nx1) vector of expected returns of individual assets S is (nxn) variance covariance matrix

  6. Minimum Variance ‘Efficient’ Portfolio • 2 asset case : wA + wB = 1 or wB = 1 – wA var(Rp) = wA2sA2 + wB2sB2 + 2wA wBrsAsB var(Rp) = wA2sA2 + (1-wA)2sB2 + 2wA (1-wA) rsAsB • To minimise the portfolio variance : Differentiating with respect to wA • ∂sp2/∂wA = 2wAsA2 – 2(1-wA)sB2 + 2(1-2wA)rsAsB = 0 • Solving the equation : wA = [sB2 – rsAsB] / [sA2 + sB2 – 2rsAsB] = (sB2 – sAB) / (sA2 + sB2 – 2sAB)

  7. Power of Diversification • As the number of assets (n) in the portfolio increases, the SD (total riskiness) falls • Assumption : • All assets have the same variance : si2 = s2 • All assets have the same covariance : sij = rs2 • Invest equally in each asset (i.e. 1/n)

  8. Power of Diversification (Cont.) • General formula for calculating the portfolio variance s2p = S wi2si2 + SS wiwjsij • Formula with assumptions imposed s2p = (1/n) s2 + ((n-1)/n) rs2 If n is large (1/n) is small and ((n-1)/n) is close to 1. Hence : s2prs2 Portfolio risk is ‘covariance risk’.

  9. Random Selection of Stocks Standard deviation Diversifiable / idiosyncratic risk C Market / non-diversifiable risk 0 1 2 ... 20 40 No. of shares in portfolio

  10. Example : 2 Risky Assets

  11. Example : Portfolio Risk and Return

  12. Example : Efficient Frontier 0, 1 0.5, 0.5 1, 0 0.75, 0.25

  13. Efficient and Inefficient Portfolios ERp A U x mp* = 10% x x L x mp** = 9% x x x P1 B x x x x x P1 x x x x x C sp** sp* sp

  14. Risk Reduction Through Diversification Y r = -0.5 r = -1 r = +1 B A r = 0.5 Z r = 0 C X

  15. Introducing Borrowing and Lending : Risk Free Asset • Stage 2 of the investment process : • You are now allowed to borrow and lend at the risk free rate r while still investing in any SINGLE ‘risky bundle’ on the efficient frontier. • For each SINGLE risky bundle, this gives a new set of risk return combination known as the ‘transformation line’. • Rather remarkably the risk-return combination you are faced with is a straight line (for each single risky bundle) - transformation line. • You can be anywhere you like on this line.

  16. Example : 1 ‘Bundle’ of Risky Assets + Risk Free Rate

  17. ‘Portfolio’ of Risky Assets and the Risk Free Asset • Expected return ERN = (1 – x)rf + xERp • Riskiness s2N = x2s2p or sN = xsp where x = proportion invested in the portfolio of risky assets ERp = expected return on the portfolio containing only risky assets sp = standard deviation of the portfolio of risky assets ERN = expected return of new portfolio (including the risk free asset) sN = standard deviation of new portfolio

  18. Example : New Portfolio With Risk Free Asset

  19. Example : Transformation Line 0.5 lending + 0.5 in 1 risky bundle No borrowing/ no lending -0.5 borrowing + 1.5 in 1 risky bundle All lending Standard deviation (Risk)

  20. Transformation Line Expected Return, N Borrowing/ leverage Z Lending X all wealth in risky asset L Q r all wealth in risk-free asset sX Standard Deviation, sN

  21. The CML – Best Transformation Line Transformation line 3 – best possible one Portfolio M ERp Transformation line 2 Transformation line 1 rf Portfolio A sp

  22. The Capital Market Line (CML) Expected return CML Market Portfolio Risk Premium / Equity Premium (ERm – rf) rf Std. dev., si 20

  23. The Security Market Line (SML) Expected return SML Market Portfolio Risk Premium / Equity Premium (ERi – rf) rf Beta, bi 0.5 1 1.2 The larger is bi, the larger is ERi

  24. Risk Adjusted Rate of Return Measures • Sharpe Ratio : SRi = (ERi – rf) / si • Treynor Ratio : TRi = (ERi – rf) / bi • Jensen’s alpha : (ERi – rf)t = ai + bi(ERm – rf)t + eit Objective : Maximise Sharpe ratio (or Treynor ratio, or Jensen’s alpha)

  25. Portfolio Choice IB Z’ ER Capital Market Line K IA Y M ERm ERm - r A Q r a L sm s

  26. Math Approach

  27. Solving Markowitz Using Lagrange Multipliers • Problem : min ½(Swiwjsij) Subject to SwiERi = k (constant) Swi = 1 • Lagrange multiplier l and m L = ½ Swiwjsij – l(SwiERi – k) – m(Swi – 1)

  28. Solving Markowitz Using Lagrange Multiplier (Cont.) • Differentiating L with respect to the weights (i.e. w1 and w2) and setting the equation equal to zero • For 2 variable case s12w1 + s12w2 – lk1 – m = 0 s21w1 + s22w2 – lk2 – m = 0 • The two equations can now be solved for the two unknowns l and m. • Together with the constraints we can now solve for the weights.

  29. The Two-Fund Theorem • Suppose we have two sets of weight : w1 and w2 (obtained from solving the Lagrangian), then aw1 + (1-a)w2 for -∞< a < ∞ are also solutions and map out the whole efficient frontier • Two fund theorem : If there are two efficient portfolios, then any other efficient portfolio can be constructed using those two.

  30. One Fund Theorem • With risk free lending and borrowing is introduced, the efficient set consists of a single line. • One fund theorem : There is a single fund M of risky assets, so that any efficient portfolio can be constructed as a combination of this fund and the risk free rate. Mean = arf + (1-a)m SD = asrf + (1-a)s

  31. References • Cuthbertson, K. and Nitzsche, D. (2004) ‘Quantitative Financial Economics’, Chapter 5 • Cuthbertson, K. and Nitzsche, D. (2001) ‘Investments : Spot and Derivatives Markets’, Chapters 10 and 18

  32. END OF LECTURE

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