1. Copyright K. Cuthbertson and D. Nitzsche 1 LECTURE : PORTFOLIO THEORY AND RISK Note that only a selection of these slides will be dealt with in detail, in the lecture All other slides are there to guide you towards the key points in Cuthbertson/Nitzsche “Investments” and in the end of chapter questions Revise your elementary stats before the lecture 11/9/2001

2. Copyright K. Cuthbertson and D. Nitzsche 2 Basic Ideas Efficient Frontier Transformation Line, Capital Market Line and the Market Portfolio Practical Issues in Portfolio Allocation Self-Study Slides TOPICS

3. Copyright K. Cuthbertson and D. Nitzsche 3 Investments:Spot and Derivative Markets, K.Cuthbertson and D.Nitzsche CHAPTER 10: Section 10.1: Overview Section 10.2: Portfolio Theory Note: Chapter 18 also contains much useful material for those who wish to learn more ! READING

4. Copyright K. Cuthbertson and D. Nitzsche 4 Basic Ideas

5. Copyright K. Cuthbertson and D. Nitzsche 5 PORTFOLIO THEORY Portfolio theory works out the ‘best combination’ of stocks to hold in your portfolio of risky assets. You like return but dislike ‘risk’ We assume the investor is trying to ‘mix’ or combine stocks to get the best return relative to the overall riskiness of the chosen portfolio. As we shall see ‘Best’ has a very specific meaning.

6. Copyright K. Cuthbertson and D. Nitzsche 6 PORTFOLIO THEORY Question 1 What proportions of your own $100 should you put in two different stocks (e.g. ‘weights’ = 25%, 75% which implies $25, $75) Different ‘weights’ give rise to different ‘risk-return’ combinations and this is the ‘efficient frontier’ Question 2 We now allow you to borrow or lend (from the bank), How does this alter your choice of ‘weights’ and the amount you actually choose to borrow or lend? Latter depends on your ‘love of risk’

7. Copyright K. Cuthbertson and D. Nitzsche 7 Statistics: Some Definitions  Expected Return of Portfolio E(R P ) = w 1 ER 1 + w 2 ER 2 Variance of Portfolio   P = w 2 1   1 + w 2 2   2 + 2 w 1 w 2  12   P = w 2 1   1 + w 2 2   2 + 2 w 1 w 2 (   1  2 ) Also, ‘proportions’ are: w 1 + w 2 = 1. Note  12 =   1  2 - from statistics

8. Copyright K. Cuthbertson and D. Nitzsche 8 Some Intuition: Domestic Assets Risk of a single asset is the variance (SD =  1 ) of its return ( eg. Man.Utd share) Risk of a portfolio of shares depends crucially on covariance (correlation) between the returns. (Eg. Man Utd and Arsenal)

9. Copyright K. Cuthbertson and D. Nitzsche 9 Random selection of shares Standard Deviation 2040 Diversifiable / Idiosyncratic Risk Market / Non-Diversifiable Risk No. of shares in portfolio 12... Note: 100%=risk when holding only one asset 100% Increasing the size (=n) of the portfolio (each asset has ‘weight’ w i = 1/n)

10. Copyright K. Cuthbertson and D. Nitzsche 10 Some Intuition: International Diversification  US resident invests $100 in UK Stock index (FTSE100) Suppose whenever FTSE100 goes up by 1% the sterling exchange rate always goes down by 1% - perfect negative correlation between the two returns Then the US resident has zero US dollar risk Hence negative correlations (strictly any  < +1) reduces risk (True, she also has zero expected USD return but seeing as she is holding zero risk, that seems OK. ‘It’s the 1st rule of finance, stupid!’)

11. Copyright K. Cuthbertson and D. Nitzsche 11 Random Selection: International Portfolio Standard Deviation 2040 Domestic Only International No. of shares in portfolio12... 100% Note: 100%=risk when holding only one asset

12. Copyright K. Cuthbertson and D. Nitzsche 12 Efficient Frontier

13. Copyright K. Cuthbertson and D. Nitzsche 13 Can we do better than “random selection” ? Assumptions You like return and dislike portfolio risk (variance/ SD). Assume everyone has the same view of future returns ER i and correlations  12,  12. 2-Stage Decision Process STAGE 1 Use only “own wealth” of $100 and work out the risk- return combinations which are open to you by distributing this $100 in different combinations (proportions, w i ) in the available stocks. This gives the “efficient frontier” Consider ‘Return’ together with ‘Risk’

14. Copyright K. Cuthbertson and D. Nitzsche 14 Efficient Frontier: Diversification Expected Return RISK,  A w i = (50%, 50%) w i = (25%,75%).. B C Own wealth of $100 split between 2 assets in proportions w i. As you alter the proportions you move around ABC Individual variances and correlation coefficients are held constant in this graph

15. Copyright K. Cuthbertson and D. Nitzsche 15

16. Copyright K. Cuthbertson and D. Nitzsche 16 Transformation Line the Capital Market Line CML and the Market Porfolio

17. Copyright K. Cuthbertson and D. Nitzsche 17 Borrowing and Lending, ‘safe rate’= r STAGE 2 You are now allowed to borrow and lend at 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 combinations =“transformation line, TL” ~ which is a ‘straight line’ Each risky asset bundle has its ‘own’ TL You can move along this TL by altering your borrowing/lending

18. Copyright K. Cuthbertson and D. Nitzsche 18 + Transformation Line(s) TL ER ER m r This is TL to point M =‘CML’ M.. mm  Point M corresponds to fixed w i (e.g. 50%, 50%) Point Z corresponds to fixed w i (e.g. 25%, 75%) Z Everyone would choose the ‘highest’ TL = point M and proportions 50-50. A.. B TL = Combination of ANY SINGLE ‘risky bundle’ and the safe asset This is TL to point Z r

19. Copyright K. Cuthbertson and D. Nitzsche 19 CML: Some Properties NO BORROWING OR LENDING (ONLY USE OWN $100) You are then at point M LEND SOME OF $100 (e.g lend $90 at r and $10 in risky bundle) You are then at point like A BORROW (say $50 ) and put all $150 in risky assets You are then at point like B Surprisingly the proportions at A and B are the same as at M (I.e. 50%,50%) - but the $ amounts are NOT the same! (Tricky !)

20. Copyright K. Cuthbertson and D. Nitzsche 20. CML and Market Portfolio (M) ER ER m r CML M mm  ER m - r w i - optm proportions at M mm A B M/s-B less risk averse than M/s-A w i maximises “reward to risk ratio” - “Sharpe Ratio”

21. Copyright K. Cuthbertson and D. Nitzsche 21 Market Portfolio = Passive Investment Strategy Optimal w i maximises “reward to risk ratio” - “Sharpe Ratio”. At the time you choose your optimal proportions you expect to obtain a ‘reward to risk ratio’ of S = ( ER m - r ) /  m Note that both M/s-A and M/s-B have the same Sharpe ratio Of course the ‘out-turn’ for the Sharpe ratio could be very different to what you envisaged (because your forecasts turned out to be poor). Ball park estimate for Sharpe ratio for S&P500 (annual) = 0.4 [= (12-4)/20]

22. Copyright K. Cuthbertson and D. Nitzsche 22 Practical Issues in Portfolio Allocation

23. Copyright K. Cuthbertson and D. Nitzsche 23 ‘Active’ versus ‘Passive’ Strategy Sharpe Ratio for any portfolio-k S k = ( ER k - r ) /  k Active portfolio managers must try and beat the Sharpe ratio of the ‘passive’ investment strategy (I.e. holding the market portfolio, month in-month-out ). ER k = average of ‘out-turn’ values for monthly portfolio returns (net of transactions costs) over say 3 years, for any portfolio-k and any ‘strategy’ (e.g. trying to pick winners)   = sample SD of these monthly returns (over 3 years) Compare investment strategies: The investor with the highest value of S k is the ‘winner’

24. Copyright K. Cuthbertson and D. Nitzsche 24 Practical Issues 1) Suppose all investors do not have the same views about expected returns and covariances. ~ we can still use our methodology to work out optimal proportions/weights for for each individual investor. 2) The optimal weights will change as forecasts of returns and correlations change - the ‘passive’ portfolio needs ‘some rebalancing’ - ‘Tracking Error’ 3)The method can be easily adopted to include transactions costs of buying and selling, and investing “new” flows of money. 4) Lots of weights might be negative, which implies short-selling, possibly on a large scale. If this is ‘impractical’ you can re- calculate, where all the weights are forced to be positive.

25. Copyright K. Cuthbertson and D. Nitzsche 25 No-Short Sales Allowed  P (=SD) ‘Unconstrained’ Efficient Frontier - allows short sales ER P Efficient Frontier - no short sales 1) Always lies ‘within’ or ‘on’ frontier which allows short sales 2) Deviates more at ‘high’ levels of expected return and  P (ie. All ‘weights’ > 0 )

26. Copyright K. Cuthbertson and D. Nitzsche 26 5) The optimal weights depend on estimate/forecasts of expected returns and covariances. If these forecasts are incorrect, the actual risk-return outcome may be very different from that envisaged when you started out Put another way a small change in expected returns can radically alter the optimal weights - ie. Extreme sensitivity to the” inputs”. The optimal weights are relatively insensitive to errors in forecasts of correlations and variances - hence some investors choose weights to min. SD only. Practical Issues

27. Copyright K. Cuthbertson and D. Nitzsche 27 Forecast Errors, (ER,  P )  Error in ‘proportions’ A MIN VARIANCE PORTFOLIO,Z Confidence band around Z may be relatively small - because it does not use ‘poor’ forecasts of ER i C It is possible that (90%,10%) lies within a 95% CONFIDENCE BAND  P (=SD) x x x x X x x z x x M = mathematical optimum = (50%, 50%) say Each ‘cross’ represents a different set of ‘weights’ w i ER P x x x x 90% S&P500 + 10% Europe. Optimal for US investor ?. x x x x x

28. Copyright K. Cuthbertson and D. Nitzsche 28 6) To overcome this “sensitivity problem” try: a) Choose the weights to minimise portfolio variance - the weights are then independent of the “badly measured” expected returns. (Note:does not imply a zero expected return - see fig). b) Choose “new proportions” which do not deviate from existing proportions by more than 2%. c) Choose “new proportions” which do not deviate from “index tracking proportions” (eg. S&P500) by more than 2%. d) Do not allow any short sales of risky assets ( All w i >0). e) Limit the analysis to investment in say 5 sectors, so sensitivity analysis can be easily conducted (A sophisticated version of which is Monte Carlo Simulation). Practical Issues

29. Copyright K. Cuthbertson and D. Nitzsche 29 Tries to take advantage of “lower” (own) return correlations compared to solely domestic investments. -this can arise because of different timing of business cycles. (eg. US is booming, Japan is in recession) Diversification benefits can also arise because of exchange rate correlations. e.g.Suppose whenever FTSE100 goes up by 1% the sterling exchange rate goes down by 1% (perfect negative correlation). Then a US based investor faces no risk in dollar terms from his UK investments. Above extreme case is unlikely in practice so the issue of currency hedging arises (via forwards, futures and options). International Diversification

30. Copyright K. Cuthbertson and D. Nitzsche 30 ‘ HOME BIAS’ PROBLEM It appears that investors, invest too much in the home country relative to the results given by “optimal” portfolio weights BUT - actual weights may not be statistically different from the optimal weights, given that the latter are subject to (large ? ) estimation error. - actual weights might reflect “ a long view” of returns, including the fact that purchases of goods (when investments are cashed in) are largely made the “home currency”. International Diversification

31. Copyright K. Cuthbertson and D. Nitzsche 31 INVESTMENT COMMITTEES usually make STRATEGIC ASSET ALLOCATION decisions based on a long term view of risk and return (including political risk). This gives them their ‘baseline’ asset allocation between countries. (e.g. no more than 10% portfolio in S.America over next 3 years) - conventional portfolio theory largely ignores political/default risk but could in principle incorporate this in forecast of expected returns, variances etc - but usually done on an ad-hoc basis. The ‘international portfolio’ may then be ‘fine tuned’ using portfolio theory, but the weights will be heavily constrained (to not move far from those set by the Investment committee). International Diversification

32. Copyright K. Cuthbertson and D. Nitzsche 32 Within a particular country, either portfolio theory will be used to guide proportions in each industrial sector, or they will try just ‘track’ the respective domestic indices (e.g. the S&P500, FTSE 100). There is some evidence that INVESTMENT COMMITTEES are moving towards choosing industrial sector weights, subject to limits on the resulting country proportions. This is to ‘gain’ from the disparate business cycles between industries (e.g. world car industry has different cycle to world chemicals) This is because ‘country indices’ are beginning to have ‘high correlations’ (e.g. US and UK aggregate business cycles are now more highly correlated. International Diversification

33. Copyright K. Cuthbertson and D. Nitzsche 33 TACTICAL ASSET ALLOCATION Use part of funds for market timing’ the business cycle’ (e.g. switch 10% of speculative funds out of US and into SE Asia ) -might use a macro-economic model for forecasts -does not easily ‘fit’ into portfolio theory because usually little or no formal estimate of risk is made International Diversification

34. Copyright K. Cuthbertson and D. Nitzsche 34 LECTURE ENDS HERE

35. Copyright K. Cuthbertson and D. Nitzsche 35 SELF STUDY SLIDES The following slides provide a simple numerical example to construct the efficient frontier,the capital market line and the market portfolio These slides will NOT be covered in the lectures

36. Copyright K. Cuthbertson and D. Nitzsche 36 STATISTICS REVISION: Some Definitions  Expected Return of Portfolio E(R P ) = w 1 ER 1 + w 2 ER 2 Variance of Portfolio   P = w 2 1   1 + w 2 2   2 + 2 w 1 w 2  12   P = w 2 1   1 + w 2 2   2 + 2 w 1 w 2 (   1  2 ) Also, ‘proportions’ are: w 1 + w 2 = 1. Note  12 =   1  2 - from statistics The above are used to derive the EFFICIENT FRONTIER by (arbitrarily) altering the w’s

37. Copyright K. Cuthbertson and D. Nitzsche 37 STAGE 1: 2 Risky Assets: Real world data (statistician)

38. Copyright K. Cuthbertson and D. Nitzsche 38 STAGE 1: Construct Efficient Frontier Now plot values of ER p and  p and construct the Efficient Frontier Choose different w’s and calculate ER p and  p combinations)

39. Copyright K. Cuthbertson and D. Nitzsche 39

40. Copyright K. Cuthbertson and D. Nitzsche 40 Efficient Frontier with ‘n’ - Risky Assets ER Z A B C Excel solver changes the weights to minimise risk (SD) for any arbitrarily chosen level expected return, ER z So, Z moves to the left  P (=SD) x x x x x X x x End of Excel minimisation w z = 25%,75%, say x x Z Each ‘cross’ represents a different set of ‘weights’ w i ER P ‘Start’ Excel (50%,50%, say)  z ‘Finish Excel’ X You require EXCEL ‘SOLVER’ to ‘draw’ the EFFICIENT FRONTIER (=A-B) ER z

41. Copyright K. Cuthbertson and D. Nitzsche 41 STAGE 2: Transformation Line We have ‘constructed’ the efficient frontier Now introduce a “safe asset” What does the risk-return “trade-off” look like when we allow borrowing or lending at the safe rate and we combine this with any ‘single bundle’ of risky assets? ‘New Portfolio’=1-safe asset + 1 “bundle of risky assets” Answer = Straight Line relationship between ER and 

42. Copyright K. Cuthbertson and D. Nitzsche 42 STAGE 2: Transformation Line What is a ‘Risky Asset bundle’ ?: Keep (arbitrary) fixed weights in risky assets eg. 20% in asset-1, 80% in asset-2 So, if you have W 0 = $100 you will hold $20 in asset-1 and $80 in asset-2 Assume this gives rise to a fixed “bundle” of risky assets” called “q” with ER q =22.5% and s q = 24.8% Now combine ‘fixed risky bundle’ with the safe asset by borrowing/lending different $ amounts of safe asset

43. Copyright K. Cuthbertson and D. Nitzsche 43 Construct ‘One’ Transformation Line FORMULAE FOR EXPECTED RETURN AND SD OF ‘NEW’ PORTFOLIO N= “new” portfolio of: ‘safe + risky ‘bundle’  q 2 = variance of the risky ‘bundle’ x = proportion held in ‘risky asset’ (1-x) = proportion held in safe asset(with  = 0 ) Expected Return: E(R N ) = (1- x). r + x ER q THEN: Variance (SD) of NEW PORTFOLIO of “ 1-safe + 1 risky asset”  N 2 = x 2  q 2 or  N = x  q Data

44. Copyright K. Cuthbertson and D. Nitzsche 44 “New Portfolio (N) : “Arbitrarily alter ‘x’ to give different Expected Return ER N and risk combinations  N This gives a straight line = Transformation Line

45. Copyright K. Cuthbertson and D. Nitzsche 45 Variance of ( 1-safe + 1 risky asset BUNDLE) Note: Borrowing: When proportion (1-x)= - 0.5 is in the safe asset, this implies x = 1.5 held in risky asset Suppose ‘own’ initial wealth W 0 =$100 Hence above implies borrowing 50% of “own wealth” (=$50) to add to your initial $100 and putting all $150 into the bundle of risky assets (in the fixed proportions 20%, 80%, I.e $30 and $120 in each risky asset) - this is referred to as ‘leverage’ and involves a higher expected return but also higher risk (SD). ‘Its the first law of finance again! Now plot the combinations ER and  in the previous slide

46. Copyright K. Cuthbertson and D. Nitzsche 46 Borrow -0.5, put all 1.5 in risky bundle 0.5 lending + 0.5 in risky bundle All lending No Borrowing/ No lending 24.87 22.5 Note: At “no borrow/lend” position, ER and  of “new” portfolio equals that for the risky asset alone (not surprisingly)

47. Copyright K. Cuthbertson and D. Nitzsche 47 Transformation Lines Safe asset plus ANY ONE ‘arbitrary’ risky bundle, gives a specific transformation line (which is straight line) between r and the s.d of the risky bundle Every single, risky bundle has its own transformation line Which transformation line is “best”? “THE HIGHEST ACHIEVABLE” = Capital Market Line

48. Copyright K. Cuthbertson and D. Nitzsche 48  q =24.87  k = 10 r = L L’L’ q and k are both ‘points’ on the efficient frontier. So q might represent(20%,80%) in risky assets and k might represent (70%,30%). Each “fixed weight” risky bundle has its own transformation line

49. Copyright K. Cuthbertson and D. Nitzsche 49 A C D B CML L’ “B” is highest attainable transformation line, while still remaining on the efficient frontier. ‘B’ represents the optimal weights (50%,50%) for the risky bundle. L

50. Copyright K. Cuthbertson and D. Nitzsche 50 Point-B is therefore a rather special portfolio and hence is known as the “Market Portfolio” (as indicated by the subsript ‘m’ in the next slide) IF everyone has the same expectations about returns, standard deviation and correlations then: Everyone chooses point-B (which here gives 50%, 50% held in each risky asset) Market Portfolio

51. Copyright K. Cuthbertson and D. Nitzsche 51 + CML and Market Portfolio (M) ER ER m r CML M mm  ER m - r w i - optm proportions at M mm A B M/s-B less risk averse than M/s-A w i maximises “reward to risk ratio” - “Sharpe Ratio”

52. Copyright K. Cuthbertson and D. Nitzsche 52 How Much Should an Individual Borrow or Lend? ~while still maintaining the 50:50 proportions in the 2-RISKY assets ? This depends on the individual’s “preferences” for risk versus return M/s-A is VERY “risk averse” (=dislike risk) implies uses e.g. $90 of her $100 “own wealth” to invest in the safe asset and puts only V= $10 in the risky “bundle” thus holding $5 in each risky asset (5/10 = 50%) M/s-B is LESS “risk averse” (=not too worried about risk) She borrows say $60 and invests the whole V= $160 in the risky bundle thus holding $80 in each risky asset (80/160 =50%) Hence both A and B invest the same PROPORTIONS in the risky assets but DIFFERENT $-amounts. The latter implies A and B hold different DOLLAR risk (For the ‘experts’: $-RISK = V x  m )

53. Copyright K. Cuthbertson and D. Nitzsche 53 END OF SLIDES