Financial Market Analysis (FMAx) Module 6

Transcription

1 Financial Market Analysis (FMAx) Module 6 Asset Allocation and iversification This training material is the property of the International Monetary Fund (IMF) and is intended for use in IMF Institute for Capacity evelopment (IC) courses. Any reuse requires the permission of the IC.

2 Preamble: Why should you care about portfolio allocation? You might be an investor. Your institution might be an investor. As a policymaker, you ll be interested in investor behavior. Your country may be interested in what drives international investment.

3 Preamble: In this module we will 1. Review statistical concepts related to return and risk.. mphasize the importance of correlation. 3. xplore how to choose an optimal portfolio. Of assets Of n > assets 4. o the same for an international portfolio.

4 An Introduction - Concepts - Portfolio Theory - Harry Markowitz (195) Investments are compared in terms of trade-off between Risk (variance) Return (expected reward) An investor who cares only about risk and return, will always prefer Highest mean return for given amount of risk; and Lowest risk given the mean return.

5 An Introduction - Concepts - Insights from H. Markowitz: iversification does not rely on returns being uncorrelated, but rather on having them be imperfectly correlated. Risk reduction from diversification is limited by the extent to which returns are correlated.

6 An Introduction - Review of Statistics - xpected Returns: Over all possible scenarios s: where p s = probability of s, r s = return if s occurs S ( r) = s= 1 p s r s Using historical data: 1 ( r) = r = [( 1+ r )( 1+ r )( 1+ r )...( 1+ )] T 1 g 1 3 r T ( r) = r a = r 1 + r + r T r T

7 An Introduction - Review of Statistics - Risk: (Variance or Standard eviation of Returns) Over all possible scenarios s: Using historical data: Var S ( ) r = = p ( r ( r) ) s= 1 s s Var ( r) = r g = ( ( )) ( ( )) r r + r r + ( r ( r) ) 1... T T

8 An Introduction - Review of Statistics - Correlation: (egree of co movement between two variables.) Correlation coefficient ranges from: + 1 (perfect movement together) 0 (variables are independent) - 1 (perfect movement apart) Cov Cov ρ AB ( r, r ) = p r ( r ) A ( r, r ) A B B Cov = = ( r, r ) A A B S s= 1 T t= 1 B s ( )( r ( r )) ( r ( r )) r ( r ) A, t A, s A T A ( ) B, t B, s B B

9 Introducing Two Risky Assets efine Portfolio P containing two risky assets (portfolios): Bonds () and quity (), Weights w and w xpected rate of return: ( r ) w ( r ) w ( r ) = + P Correlation coefficient ρ : ρ = Cov ( r, r )

10 Introducing Two Risky Assets Risk of P: Question: What is the largest possible value for ρ? When ρ =1 (perfect correlation): ( ) ( ) [ ] P P P w w w w r r P Var ρ + + = = = ( ) ( ) P P w w w w w w w w P Var + = + = + + = =

11 Introducing Two Risky Assets Unless correlation between and is perfect: < w + P w Note that: Hedge asset: negative correlation with other assets (ρ < 0) xpected return of P is unaffected by the correlations We often call the difference between p and weighted average of individual s the gains from diversification.

12 Introducing Two Risky Assets In the extreme case of a perfect hedge (ρ = -1): p = ( w w ) p = abs( w w ) Can construct a zero risk portfolio: = w w = 0, w + w = 1 p w =, w = = 1 w + +

13 Graphical Representation - Two-Assets Portfolio - xample: (r ) = %, (r ) = 6%, = 5%, = 10%, ρ = 0. Return and Risk: r ( ) = 0.0w w p = (0.05) w + (0.10) w ww p = w 0.01w 0.00ww

14 Graphical Representation - Two-Assets Portfolio - Minimizing Risk: Min = Min w + w + w w ρ p st.. : w + w = 1, or w = 1 w Solving the above for w, w w min ( ) = + ρ w ( ) = 1 w ( ) min ρ min

15 Graphical Representation - Two-Assets Portfolio - (r p ) ρ = -1.0 ρ = 0.0 Portfolio 100% in ρ = 1.0 Portfolio 100% in

16 Graphical Representation - Two-Assets Portfolio - Benefits from iversification: Come from imperfect correlation between returns. The smaller ρ, the greater the benefits from diversification. If ρ = 1, no risk reduction is possible. Adding extra assets with lower correlation with the existing ones decreases total risk of the portfolio. iversification can eliminate some, but not all risk.

17 Comparing ifferent Portfolios - Two-Assets - The Risk-Free Asset: IF it is possible to borrow/lend at the risk-free rate r f, THN the portfolio selection problem is to maximize the excess return over the risk-free rate, for a given amount of risk: Max ( ( ) ) + ( ( ) ) r ( p) r w r f rf w r rf = w + w + wwρ st..: w + w = 1 p

18 Comparing ifferent Portfolios - Two-Assets - Our Numerical xample: Assume r f = 1.5% efine A as the minimum-variance portfolio earlier obtained (w = 0.84, w = 0.16) Capital allocation line (CAL): combinations of A and the risk-free asset. Slope of CAL = reward-to-variability, or Sharpe ratio: S A r ( A) rf = = = 4.78 A 0.4

19 Comparing ifferent Portfolios - Two-Assets - Consider an alternative portfolio B: W = 0.65, W = 0.35 Is it better than portfolio A? S B r ( B) rf = = = 3.4 B The optimal portfolio will be such that the reward-to-variability ratio is maximized (depends on r f ): r ( ) N p rf Max S p = s t wi = w i p i= 1.. 1

20 Comparing ifferent Portfolios - Two-Assets - (r p ) CAL B CAL A B r f A P

21 Selecting the Best Portfolio - Two-Assets - The Optimization Problem: Max ( ( ) ) + ( ( ) ) r ( p) r w r f rf w r rf = w + w + wwρ st..: w + w = 1 p After some algebra, and using w = 1 - w : w ( ) = ( r ( ) rf ) ( r ( ) rf ) cov( r, r) ( ) + ( ) ( + ) S max r rf r rf r rf r rf r r ( ) ( ) ( ) ( ) cov(, )

22 Selecting the Best Portfolio - Two-Assets - xpressing the Numerator in Matrix Notation: (1) ( ) cov( r, r ) r rf ( ) max ( ) cov(, ) r r ws r r f ws max ( ) = r ( ) rf r ( ) rf r ( ) rf r ( ) rf cov( r, r) ( ) + ( ) ( + ) Let s call the numerator of (1) a column vector z, which will be: () z cov( r, r ) r ( ) r 1 f z = ( ) cov( r, r) r r f z z 1 = 1 ( f ) S R R

23 Selecting the Best Portfolio - Two-Assets - You can verify that the denominator of (1) is equal to the sum of the z s. Therefore, the solution to the weights of the optimal portfolio P* is: (3) z1 ws max ( ) z w ( ) = z + z S max 1

24 Selecting the Best Portfolio - Two-Assets - xample: w ( ) = 0.04, w ( ) = 0.96 Smax Smax r ( ) = 5.8%, = 9.6%, S = p p p This is the tangency portfolio P*; no other portfolio achieves a higher reward-to-variability (Sharpe ratio) with respect to the risk-free rate.

25 Selecting the Best Portfolio - Two-Assets - Questions: If the risk-free rate declines (monetary loosening), what happens to the composition of the optimal portfolio (w and w )? its mean return? its risk? its Sharpe ratio?

26 Selecting the Best Portfolio - Two-Assets - Answers: If the risk-free rate declines (monetary loosening), then Optimal portfolio P* shifts toward the Southwest: w and w Both mean return and risk decline As does its Sharpe ratio

27 Recap of Optimal Portfolio - Two-Assets - Main Ideas: Investor chooses the combination of, that maximizes reward-to-variability relative to the risk-free rate. This reward to variability = Sharpe ratio = slope of the CAL. Two methods Algebraic (matrix solution) Numerical (using Solver) z z 1 = 1 ( f ) S R R z1 w ( ) z = ws ( ) z + z a Sa max max 1 Recall: benefits to diversification depend on (imperfect) correlation between and.

28 Recap of Optimal Portfolio - Two-Assets - Questions: If the correlation between and increases, what happens to the composition of the optimal portfolio (w and w )? its mean return? its risk?

29 Recap of Optimal Portfolio - Two-Assets - Answers: If the correlation between and increases, then: the optimal portfolio P* shifts to the Northeast: w and w (in this case, it means even shorting ) Both mean return and risk increase Sharpe ratio increases slightly (from to 0.454)

30 Completing the Investor ecision - Two-Assets - What next? We have (1) the tangency portfolio P* and () the riskless asset. But each investor must now decide how much to invest in each. Will depend on personal preference (risk aversion or tolerance). This can be expressed by a utility function: U = where A = degree of risk aversion; if A = 0, risk-neutral ( r ) 0.005A C C

31 Completing the Investor ecision - Two-Assets - Individual Investor Choice: Maximize utility subject to the tangency portfolio and the riskless asset, to obtain the proportions to be invested in each (w P* and w rf ). The resulting portfolio will be called C. MaxU = ( r ) 0.005A C ( ) ( ) C = w r + r 1 w 0.005Aw P P f P P P In our example, assuming risk aversion A = 600, solve for w P* : w P = = %

32 Completing the Investor ecision - Two-Assets - (r p ) U CAL C P* r f w P* =78% w P* =100% P 3

33 Completing the Investor ecision - Two-Assets - Separation Property: Technical information guides the decision to choose the optimal portfolio of risky assets P*: Mean returns, relative to risk-free rate Volatilities Correlation The choice of ultimate investor position (how much in P*, how much in risk-free) depends on individual preferences (A). The two decisions are separate.

34 Generalizing to n Assets - Part 1 - Maximize reward-to-variability, subject to all weights summing to 1. The solution, w i*, is the same matrix as before, now generalized to N assets: z = S 1 [ R rf ] ( NN, ) ( N,1) ( N,1) N: # risky assets R: The column vector: expected returns r f : risk-free rate w: The column vector: portfolio shares S: NxN variance-covariance matrix * wi = z i z

35 Generalizing to n Assets - Part 1 - Finding the optimal portfolio, step-by-step: 1. Construct an xcess Return (R) matrix for the N assets. T: # observations N: # risky assets R ra,1 ra rb,1 rb rn,1 rn ra, ra rb, rb rn, r N = r r r r r r AT, A BT, B N, T N (T rows, N columns)

36 Generalizing to n Assets - Part 1 - Finding the optimal portfolio, step-by-step:. Multiply R by its transpose, divide by the number of time observations, to obtain the Variance-Covariance Matrix (S). ra Cov( ra, rb ) Cov( ra, r ) N T Cov( ra, rb ) rb Cov( rb, rn ) S R R / T = = ( NT, ) ( TN, ) ( NN, ) Cov( ra, rn ) Cov( rb, rn ) rn (N rows, N columns)

37 Generalizing to n Assets - Part 1 - Finding the optimal portfolio, step-by-step: 3. Find the inverse of S and multiply it by the difference between the mean returns and the risk-free rate (R - r f ). This gives us the z vector. 4. The optimal portfolio weights w i* will be equal to the z vector divided by the sum of z s. z = S 1 [ R rf ] ( NN, ) ( N,1) ( N,1) * wi = z i z

38 Generalizing to n Assets - Part - With the optimal portfolio obtained (P*), compute the (1) expected or mean return, () variance, and (3) standard deviation. xpected or Mean Return: Variance: Standard eviation T r ( P* ) = W P* R (1,1) (1, N ) ( N,1) T P* = W P* S W P* (1,1) (1, N) ( NN, ) ( N,1) P = * P* (1,1) (1,1)

39 Building the ntire Frontier - n Assets - You have computed the optimal portfolio P*, which holds for a certain level of the risk-free rate. Vary r f and compute a second optimal portfolio P**. Can choose any arbitrary r f The frontier can be generated as a series of linear combinations of P* and P**.

40 Building the ntire Frontier - n Assets - The mean return and standard deviation of each combination of P* and P** can then be computed: Mean Return: r ( ) = αr ( ) + (1 α) r ( ) P*, P** P* P** Standard eviation: = α + (1 α) + α(1 α) Cov( P, P ) * ** P P* P** Covariance between two market portfolios: * ** T Cov( P, P ) = W P* S W P** (1,1) (1, N) ( NN, ) ( N,1)

41 Building the ntire Frontier - n Assets - (r) Aggressive investor Conservative investor r f1 r f Individual Assets Min variance P

42 International iversification - Part 1 - The main principles related to diversification within the domestic market continue to apply when diversifying internationally: Including additional (international) assets that are imperfectly correlated with domestic assets will produce gains (risk reduction). But not all risk can be eliminated. Currency fluctuations introduce an additional element (Part ).

43

44 International iversification - Part - Investing across international borders introduces additional risk-return elements coming from currency fluctuations. There are two investor reactions to this 1. o nothing.. Hedge their foreign currency exposure (in different ways).

45 International iversification - Part - xample: A Japanese investor wants to buy U.S. stock [Apple, Inc.]. Must buy dollars today at the spot exchange rate (S 0, /$ ) in order to buy the shares at today s price (P A, 0 ). Today s (Nov 30, 015) cost (in ) is therefore: P A,0 x S 0 P A,0 = $118.88, S 0 = 13.6 /$ Cash flow in one year: if unhedged, can sell the shares and exchange US$ for at the spot rates: P A,1 x S 1

46 International iversification - Part - xample: A Japanese investor wants to buy U.S. stock [Apple, Inc.]. Suppose Apple s stock price rises by 5% in one year: P A,1 = $ x ( ) = $14.8 Regarding the exchange rate, suppose there are two possible scenarios (6% depreciation or appreciation): S 1,d = 13.6 x ( ) = /$ S 1,a = 13.6 x (1-0.06) = /$

47 International iversification - Part - Unhedged Return: P S P S ru PA,0 S0 r r + r r r r A,1 1 A,0 0 = ( ) U A FX A FX FX : return on US stock in $ : % change in the spot exchange rate (1$= S Yen) = S S S r Ud, = 11.3% r Ua, =?

48 International iversification - Part - Full currency hedging: ach investment in a foreign stock is fully hedged by a forward position; the investor agrees today to sell at the 1-year forward rate. F 1,0 = 11.3 /$ Cash flow (per share) one year from now: CF 1,d = ( x 11.3) + ( ) x = 15,188 CF 1,a = (118.8 x 11.3) + ( ) x = 15,101

49 International iversification - Part - Hedged Return: r r H H Value in Japanese Yen of the American Stock at time 1 PA,0 F1,0 + ( PA,1 PA,0) S1 PA,0 S0 = PA,0 S0 F S 1,0 1 ru = S0 Initial Investment Forward return Initial Investment r Hd, r Ha, ( ) + ( ) ( ) = = 3.7% =?

50 International iversification - Part - Two Strategy Comparison: As expected, much less variation under hedging (although exchange rate risk is not eliminated entirely). Hedging misses out on potentially large returns. Risk vs Reward yet again. Alternative hedging strategies: using Markowitz approach, incorporating ρ s,p

51 Wrap-Up Portfolio allocation main messages: iversification benefits come from imperfect correlation (ρ<1) Var(portfolio) < Weighted average of Var(individual assets) If r f is available, then portfolio choice maximizes excess return (over r f ) relative to the additional risk: P* In other words, maximizes the Sharpe ratio or Reward-to-variability (slope of CAL) Finally, investor chooses a combination of P* and r f according to risk attitude Separation Property

52 Wrap-Up Portfolio allocation problem, in a nutshell: Slope = ( r ( ) r) P P f CAL By choosing another portfolio (B vs A), the CAL becomes steeper. Return P* The Sharpe Ratio of the portfolio increases. r f B A This continues until the CAL is tangent to the investment opportunity set. The optimal portfolio P* maximizes the Sharpe Ratio