Portfolio Management Primer • Primer I: Top-Down Portfolio Management • Capital vs. Asset allocation

Part VI: Active Portfolio

• Markowitz security selection model

• Primer II: Asset Pricing Models • CAPM (Theory & Practice)

Management

• Index & Multi-Factor Models

• Primer III: • Active Portfolio Management • Performance Measurement: Benchmarks & Rewards © Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Active Portfolio Management • Two examples so far • stocks-- Markowtiz security selection model (need inputs!) • bonds -- fixed-income portfolio management

• Relevant questions • why? • what? – market timing – security analysis » index model » multi-factor model

• how?

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Active Portfolio Management -- Why? • APM -- a contradiction in terms? • Nope! – market efficiency requires many investors to manage actively

• intuition – – – – –

mis-priced securities > deviations from passive strategies pay off > price pressures eliminate mis-pricing > active management does not pay off > securities become mis-priced again >…

• Theory » Grossman and Stiglitz

– key worry = control for risk © Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Active Portfolio Management -- Why? 2 • Evidence – other managers may beat the market » small but statistically significant » noise in security returns −> hard to disclaim – some portfolio managers are really good » hard to argue – anomalies » January effect, … » disappearance?

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

What is Active Portfolio Management? • Isn’t every strategy active? • 1. Security selection -- clearly – identify mis-priced securities

• 2. Asset allocation -- yep – different asset categories » require different forecasts – example » long-term bond return determinants » ≠ equity return determinants – international assets » things get worse © Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

1

What is Active Portfolio Management? 2

What is Active Portfolio Management? 3

• Isn’t every strategy active?

• Approach

• 3. capital allocation -- even that!

• definition – purely passive strategy » invest only in index funds » one fund per asset category (equity, bonds, bills) » proportions unchanged regardless of market conditions

– proportion invested in market portfolio

w* =

E [rm ] −r f 2 x0 .005 xAxσ 2M

• example

– requires to forecast E [rm ] and σ 2M

» 60% equity + 30% bonds + 10% bills » fixed for 5 years = entire investment horizon

– might also lead to market timing » market conditions change over time

• active management » requires control of risk

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

What is Active Portfolio Management? 4 • Objectives

Market Timing • Idea

• concentrate on portfolio construction – CAPM −> we can separate – construction of efficient portfolio – and allocation of funds » between risky asset and bills

• two components – security analysis » maximize Sharpe ratio (CAPM) – market timing » shift assets in and out of risky portfolio © Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Market Timing 2 • Reasons for differences • compounding » for all assets » importance for pension funds

• risk – mainly for equities ( Fig. p. 985 6th edition) » T-bills are mostly risk-free – irrelevant for market timing » exception: T-bill rate varies a little

• information » for market timing © Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

• market timers shift money » from money market (MM) to risky portfolio

• based on their forecasts of market return

• potential profits • huge • example ($1,000 reinvested from 1927 till 1978) » 30-day T -bills: $3,600 (r = 2.49%) » NYSE: $ 67,500 (r = 8.44%) » perfect market timing: $5,360,000,000 (r = 34.71%) • (Tables on p. 985, 6ht edition) © Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Market Timing 3 • Market timing as an option – much less risky than equities • standard deviation is misleading – perfect market timing – yields dominant payoffs in each state of the world (Fig. 27.1) » gives minimum return guarantee » + a non-negative random number

– market timing fees • market timers will charge for service » fees determined by option pricing © Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

2

Market Timing 4 • In practice

Market Timing 5 • Evaluating market timers • Basic idea

• clients – want managers to pick efficient portfolios » maximize Sharpe ratio – still need to pick the optimal proportion » to invest in the risk-free asset

• managers – need to update customers continuously » relative attractiveness of risky portolio changes – costly

• solution – let managers shift money in and out of MM funds » = solution used by most funds © Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

– Risk-return trade-off (Q1c, Assignment 2) – fund performance should improve with the market

• Intuition – as market improves, a good market timer shifts more money to market » caveat: true if short sales are ruled out

• Formally – Non-linear regressions (Fig. 24.5, BKM6) » Regress portfolio excess returns (ER) on market ER and ER^2

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Security Selection • Map

Security Selection 2 • A. Idea (Treynor-Black)

• A. idea

• 1. consider the entire set of securities » assume the entire set is there » index model (passive portfolio = market porfolio)

• B. portfolio construction • C. numerical example

• 2. focus on a small subset » as many as analysts can reasonably handle

• D. multi-factor models • E. use in practice » industry use » advantages vs. dangers

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

• 3. analyze – use index (single- or multi-) factor model » to estimate alpha, beta(s) and residual risk » of securities in subset – identify securities with positive expected “alpha” » assume securities outside the subset are correctly priced © Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Security Selection 3 • A. (continued) • 4. mix non-zero alpha securities » with passive portfolio (= market porfolio) – why? » want to maximize return » but need to control for risk » small subset −> too much risk if invest only in subset – how? » use the beta, alpha and residual risk estimated

• 5. optimal risky portfolio » = mix of active and passive portfolio © Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Security Selection 4 • B. Portfolio construction (NOT Exam Mat’l) • 1. assumptions (index model) – market portfolio M = efficient portfolio –

E[rm ] and σ 2M have been estimated » use them for passive portfolio » no need for market timing

– beta relationship

ri −r f =α i +β i ( rM −r f )+ei cov(ei ,e j )=cov(ei ,rM ) =0 © Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

3

Security Selection 5

Security Selection 6

• B. (continued)

• B. (continued)

• 2. research or estimate

• active portfolio

rk = r f + β k (rM − r f ) +ek +α k

» “A” comprises all the assets with non-zero alpha n

»

α k > 0 −> go long

»

α k < 0 −> go short

2 2 2 A = β Aσ M

∑α j / σ 2 (e j )

k=1

+σ (e A ) 2

n 2 = β Aσ 2M + ∑ w2kσ 2( ek ) k =1

n

» optimal weights: wk = n

β A = ∑ wk β k

k=1

σ α k / σ 2 (ek )

n

α A = ∑ wkα k

• 3. active portfolio » α k = 0 −> done (i.e., keep security in passive portf.)

∑ wk =1

k=1

cov(ei ,e j ) = 0

j=1

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Security Selection 7

Security Selection 8

• B. (continued)

• B. (continued)

• 4. mixing active (A) & passive (M) portfolios – portfolio A may lie above CML

• 5. formal construction – intuition

» interpretation (given analysis, M is not efficient after all)

» optimal combo of 2 risky assets & T-bills (Lecture 11)

» no need to know the “original” efficient frontier

E[ rP ]− r f

Max

– new frontier (BKM6 Fig. 27.2)

wA

» combine A and M » A and M not perfectly correlated

Max wA

– optimal risky portfolio

σP

s.t. wA + wM =1 (1 − wA ) E [R M ]+ wA E [R A ]

(1− w A ) 2 σ 2M + w 2Aσ 2A + 2 (1 − wA ) wA ρ A ,M σ Aσ M

» tangency point, given risk-free asset © Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Security Selection 9

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Security Selection 10

• B. (continued) • Optimal risky allocation

• 6. Optimal risky allocation

wo * wA= 1+ (1 − β A ) wo

Num wA = Den Num =( E [r A ]− r f )σ 2M − ( E[ rM ]− r f )cov(r A ,rM ) Den = ( E[ rA ] −r f )σ

2 + ( E [r M M

]− r f )σ

2 A

−( E[rM ]+ E[rA ] − 2r f ) cov(r A ,rM )

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

wo =

α A / σ 2 (e A ) ( E[ rM ]− r f )/ σ 2M

– intuition for wo » wo = ratio of reward-to-risk ratios for A and M » we mix for risk diversification reasons » the higher the reward for the extra risk taken » the more we invest in the (very) risky portfolio A © Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

4

Security Selection 11

Security Selection 12 • 7. Optimal security weights:

• Optimal risky allocation

wo 1+ (1 − β A ) wo

* wA=

wM =1− w*A

wk =

α k / σ 2 (ek ) n

∑α j / σ 2 (e j ) j=1

– intuition

» to “max” the composite portfolio’s Sharpe ratio 2 2  E[rM ]− rf   α A  α2  +   S 2P = S 2M + 2 A =   σ (eA )  σ M  σ (eA ) 

– intuition for w* » w* = adjustment for beta » the weight for A depends on diversif . opportunities » if β A =. 16= S2M SP  . 1556 . 1563 . 1154  20  20% 

S P=

2

» the appraisal ratio of each security

α / σ 2 (ek ) » we must have: wk = n k

  E [r ]− r α 2A M f 2 =  SM + 2 σ (e A )   σ M 

αA σ (e A )

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

=1. 1477x0.07+ ( −1.6212)x (−0.05)+1.4735x 0.03=20.56% n

β A = ∑ wk β k = 0 .9519 k =1

σ ( eA ) = 82.62 %

2 σ ( eA ) = 0. 6828

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

5

Security Selection 17

Security Selection 18

• optimal risky portfolio

• performance gain

– despite high alpha, small proportion in active portfolio

– Sharpe ratio

» large risk needs to be balanced out

S P = 0.2219= 0. 4711>

– small adjustment for beta » beta is close to 1

– M2 measure = 1.42% (large number, given only 3 securities)

α A / σ 2 (e A ) wo = = 0. 1506 ( E[ rM ]− r f )/ σ 2M wo * = 0. 1495=14.95 % wA= 1+ (1− β A )wo

8% = 0.4= S M 20%

» match risk (i.e., std-dev) of portfolio M » by mixing optimal risky portfolio and T-bills 2

» in proportions σ P

wM =1− w*A =85.05%

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

σ

2 M

2

and

1− σ 2P σM

respectively

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Security Selection 19

Security Selection 20 Data:

• D. Multi-factor models (NOT Exam Mat’l) • so far: index model • now: 2 -factor illustration of multi-factor extension • extension from index model is straightforward » the entire analysis is based on residual analysis » computations required, then proceed as before

E[rk ]− r f = βk 1 (E[r1 ]− r f )+ β k 2 ( E[r2 ] − r f ) + e f +α f 2 2 2 2 σ k = β k1σ 1 + β k 2σ 2 + 2 β k1 β k 2 cov(r1 ,r2 )+σ ( ek ) 2

cov(ri , r j ) = β i1β

2

2 2 j1σ 1 + β i2 β j2 σ 2 + ( β i1 β j 2 + β i 2 β j1) cov(r1 , r2 )

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

A portfolio management house approximates the return-generating process by a twofactor model and uses two-factor portfolios to construct its passive portfolio. The input table that is considered by the house analysts looks as follows: Micro Forecasts ----------------------------------------------------------------------------------------------------------------Asset Expected Return (%) Beta on M Beta on H Residual SD (%) ----------------------------------------------------------------------------------------------------------------Stock A 20 1.2 1.8 58 Stock B 18 1.4 1.1 71 Stock C 17 0.5 1.5 60 Stock D 12 1.0 0.2 55 ----------------------------------------------------------------------------------------------------------------Macro Forecasts ----------------------------------------------------------------------------------------------------------------Asset Expected Return (%) Standard Deviation (%) ----------------------------------------------------------------------------------------------------------------T-bills 8 0 Factor M portfolio 16 23 Factor H portfolio 10 18 ----------------------------------------------------------------------------------------------------------------

The correlation coefficient between the two-factor portfolio is 0.6.

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Security Selection 21

Security Selection 22 • optimal combo of 2 risky assets & T-bills (Lecture 9)

Max wM Questions:

Max wM

(a) What is the optimal passive portfolio? (b) By how much is the optimal passive portfolio superior to the single-factor passive portfolio, M, in terms of Sharpe’s measure? (c) What is the Sharpe measure of the optimal risky portfolio and what is the contribution of the active portfolio to that measure?

wM =

E[rP ]− r f σP

s.t. wH + wM =1

(1− wM ) E[RH ]+ wM E[ RM ] (1− wM )2 σ 2H + w 2Mσ 2M + 2(1− w M )wM ρ M ,H σ M σ H

2 Num Num =( E [rM ] −r f )σ H −( E [rH ] − r f ) cov(rM ,rH ) 2 Den Den= ( E[rM ]− r f )σ H + (E[ rH ]− rf )σ 2M

−( E[rH ] + E[rM ]− 2r f )cov(rM ,rH ) © Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

6

Security Selection 23

Security Selection 24 Answers:

wM =

(a) The optimal passive portfolio is obtained from equation (7.8) in Chapter 7 on Optimal Risky Portfolios – see Lecture 9.

Num Den

w M = [E(R M)σ H2 – E(RH )Cov(rH, r M )/{E(R M)σ H2 + E(RM)σ M2 – [E(R H)+E(RM)]Cov(rH, rM)}

Num =( E [rM ] −r f )σ 2H −( E [rH ] − r f ) cov(rM ,rH ) Den = ( E[ rM ]− r f )σ 2H + ( E[ rH ]− r f )σ 2M

−( E[rH ] + E[rM ] − 2r f ) cov(rM ,rH )

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Security Selection 25

where RM = 8%, RH = 2% and Cov(rH , rM) = ρσ Mσ H = 0.6 x 23 x 18 = 248.4. Thus,

wM = 8 x 18 2 – (2 x 248.4)/[8 x 18 2 + (2 x 232 ) – (8 + 2) 248.4] = 1.797,

and

wH = -0.797.

Because the weight on H is negative, if short sales are not allowed, portfolio H would have to be left out of the passive portfolio.

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Security Selection 26 • We now must

Answers:

• research or estimate (b)With short sales allowed,

rk = r f + β k (rM − r f ) +ek +α k

E(R passive) = 1.797 x 8 + (-0.797) x 2 = 12.78% σ

2

passive

2

2

= (1.797 x 23) + [(-0.797) x 18] + 2 x 1.797 x (-0.797) x 248.4 = 1202.54

σ passive = 34.68%. Sharpe’s measure in this case is given by: S passive = 12.78/34.68 = 0.3685,

• find the active portfolio » α k = 0 −> done (i.e., keep security in passive portf.) »

α k > 0 −> go long

»

α k < 0 −> go short

α / σ 2 (ek ) » optimal weights: wk = k n

and compared with the (simple) market’s Sharpe measure of

∑α j

S M = 8/23 = 0.3478.

j=1

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Security Selection 27

/ σ 2 (e

j)

n

∑ wk =1

k=1

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Security Selection 28

Answers: (c) The first step is to find the beta of the stocks relative to the optimized passive portfolio. For any stock i, the covariance with a portfolio is the sum of the covariances with the portfolio components, accounting for the weights of the components. Thus, βi = Cov(r i, rpassive )/ σ

2 passive

2

2

2

= (βiM wM σM + β iHw Hσ H )/ σ

passive

.

Now the alphas relative to the optimized portfolio can be computed: α i = E(ri) – r f - βi, passive x E(rpassive)

so that

α A = 20 – 8 – (0.5621 x 12.78) = 4.82% α B = 18 – 8 – (0.8705 x 12.78) = -1.12%

Therefore, 2

2

βA = [1.2 x 1.797 x 23 + 1.8 x (-0.797) x 18 ]/1202.54 = 0.5621

α C = 17 – 8 – (0.0731 x 12.78) = 8.07%

2 2 βB = [1.4 x 1.797 x 23 + 1.1 x (-0.797) x 18 ]/1202.54 = 0.8705

α D = 12 – 8 – (0.7476 x 12.78) = -5.55%

2

2

βC = [0.5 x 1.797 x 23 + 1.5 x (-0.797) x 18 ]/1202.54 = 0.0731 2

2

βD = [1.0 x 1.797 x 23 + 0.2 x (-0.797) x 18 ]/1202.54 = 0.7476

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

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7

Security Selection 29

Security Selection 30 From this point, the procedure is identical to that of the index model:

And the residual variances are now obtained from: 2 e

2 i

2

2

2

2

σ (i:passive) = σ – (β i:passive x σ passive), 2

2

where σ i = βM σ M + σ e (i). 2

2

2

2

σ e (A) = (1.3 x 23) + 58 – (0.5621 x 34.68) = 3878.01 σ e2 (B) = (1.8 x 23)2 + 712 – (0.8705 x 34.68)2 = 5843.59 2

2

2

2

2

2

2

2

σ e (C) = (0.7 x 23) + 60 – (0.0731 x 34.68) = 3852.78 σ e (D) = (1.0 x 23) + 55 – (0.7476 x 34.68) = 2881.80

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

wo 1+ (1 − β A ) wo

* wA=

α A / σ 2 (e A ) wo = ( E[ rM ]− r f )/ σ 2M

2

2

0.001243 -0.000192 0.002095 -0.001926 0.001220

2

(α / σ e )/( Σ α /σ e ) 1.0189 -0.1574 1.7172 -1.5787 1.0000

The active portfolio parameters are: α = 1.0189 x 4.82 + (-0.1574) (–1.12) + (1.7172 x 8.07) + (-1.5787)(–5.55) = 27.7% β = 1.0189 x 0.5621 + (-0.1574)(0.8705) + 1.7172 x 0.0731 + (-1.5787)(0.7476) = -0.619. 2 2 2 2 σ e = 1.0189 x 3878.01 + (-0.1574) x 5843.59 + 1.7172 x 3852.78 + (-1.5787)2 x 2881.80 = 22,714.03

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Security Selection 31 • Optimal risky allocation (index model)

α /σ e

Stock A B C D Total

Security Selection 31 The proportions in the overall risky portfolio can now be determined: 2

2

w 0 = (α /σ e )/[E(Rpassive )/ σ passive] = (27.71/22,714.03)/(12.78/1202.54) = 0.1148. w* = 0.1148/[1 + (1 + 0.6190) x 0.1148] = 0.0968. Sharpe’s measure for the optimal risky portfolio is:

– intuition for wo » wo = ratio of reward-to-risk ratios for A and P » we mix for risk diversification reasons » the higher the reward for the extra risk taken » the more we invest in the (very) risky portfolio A © Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

S 2 = S2 passive + (α/ σ e)2 = 0.36852 + [27.71 2 /22,714.03] = 0.1696 S = 0.4118, compared to Spassive = 0.3685.

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Security Selection 32 • E. Potential benefits (Treynor-Black) • in practice – not yet used often widely » hard to estimate alphas (bias correction needed) » correction requires constant monitoring & appraisal » shows alphas imprecise, second-guesses analysts » do you think analysts like that?

• yet, significant benefits » » » »

Part VI.B: Active Portfolio Management Evaluation

easy to implement allows for decentralized decisions can add significant return amenable to multi-factor analysis

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

8

Portfolio Performance Evaluation

Portfolio Return Measurement • One period vs. multiple periods

• Returns

• easy vs. unclear

• return measurement over several periods

» depends on number of periods » affected by intermediate investments/withdrawals

• Performance measures

• Time-weighted vs. dollar-weighted

• market timing • security analysis » Treynor, Sharpe, Jensen, appraisal ratio, M2

• average return vs. IRR • examples (Table 24.1)

» practical cases

• why use a time average? » performance measurement assigns responsibilities » cash ins and outs?

• Performance attribution • bogey; asset allocation; sector and security decisions © Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Portfolio Return Measurement 2

Portfolio Return Measurement 3 Question:

• Geometric vs. arithmetic averages • arithmetic – unbiased forecast of expected future performance » oriented towards the future

• geometric – constant rate » compounded, would yield same total return over period » downward bias relative to arithmetic » oriented towards the past

σ2 rG ≈ rA − 2

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

XYZ stock price and dividend histories are as follows: ------------------------------------------------------------------------------------------------------------Year Beginning of Year Price Dividend Paid at Year-End ------------------------------------------------------------------------------------------------------------1991 $100 $4 1992 $110 $4 1993 $ 90 $4 1994 $ 95 $4 ------------------------------------------------------------------------------------------------------------An investor buys three shares of XYZ at the beginning of 1991, buys another two shares at the beginning of 1992, sells one share at the beginning of 1993, and sells all four remaining shares at the beginning of 1994. (a) What are the arithmetic and geometric average time-weighted rates of return for the investor? (b) What is the dollar-weighted rate of return? (Hint: Carefully prepare a chart of cash flows for the four dates corresponding to the turns of the year for January 1, 1991 to January 1, 1994. Calculate the internal rate of return).

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Portfolio Return Measurement 4

Portfolio Return Measurement 5 (b)

Answer: (a) Time-weighted average returns are based on year-by-year rates of return. Year Return [(capital gains + dividend)/price)] -----------------------------------------------------------------------------1991-1992 [(110-100) + 4]/100 = 14% 1992-1993 [(90 – 110) + 4]/110 = -14.55% 1993-1994 [(95 – 90) + 4 ]/90 = 10% -----------------------------------------------------------------------------Arithmetic mean = 3.15% Geometric mean = 2.33%

Time Cash Flow Explanation ------------------------------------------------------------------------------------------------------------0 -300 Purchase of 3 shares at $100 each. 1 -208 Purchase of 2 shares at $110 less dividend income on 3 shares held 2 110 Dividends on 5 shares plus sale of one share at price of $90 each. 3 396 Dividends on 4 shares plus sale of 4 shares at price of $95 each. ------------------------------------------------------------------------------------------------------------$110 $396 ______________________________________|____________|_____ Date 1/1/91 1/1/92 1/1/93 1/1/94 | | ($300) ($208) Dollar-weighted return = Internal rate of return of cash-flow series = -0.1661%.

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

9

Market Timing Evaluation • Idea

• market timers shift money (MM risky portfolio) » based on their forecasts of market return

Market Timing Evaluation 2 • Overall quality • measure 1 – measure = P 1 + P2 - 1

• return from market timing » depends on # of times the timer is correct

» P1

= proportion of correct bull predictions

• two scenarios: bull vs. bear

= 1 if 100% correct

– must be correct in each scenario » example 1: always predict snow in Winter in Montreal

» P2

= proportion of correct bull predictions = 1 if 100% correct

right 95% of the time

– example: return maximization

but always wrong when no snow

» correct 100% of bulls, 0% of bears −>measure = 0 » correct 50% of bulls, 50% of bears ->measure = 0

» example 2: forward hedges

• overall quality vs. risk adjustment © Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Market Timing Evaluation 3 • Overall quality • measure 2

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Market Timing Evaluation 4 • Risk adjustment • problems » neither measure so far accounts for risk

– unclear » how large (P1 + P 2 - 1) must be » to ensure that we have seen “good performance”

» market timers constantly change portfolio risk profile

• solutions » time-varying dummy D (=1 for bull, 0 for bear)

– solution

rP − rf = a +b(rM − rf ) +c (rM − rf )D + eP

» statistical significance » application: hedging decisions (as time allows)

» Squared term

rP − rf = a + b( rM − rf ) + c( rM − rf ) D + eP

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Market Timing 5 • Bottom line -- evaluating market timers • Basic idea – Risk-return trade-off (Q1c, Assignment 2) – fund performance should improve with the market

• Intuition – as market improves, a good market timer shifts more money to market » caveat: true if short sales are ruled out

• Formally – Non-linear regressions (Fig. 24.5, BKM6) » regress portfolio excess returns (ER) on market ER and ER^2 or on ER and ER*timing dummy

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Evaluating Security Selection • Basic • idea and problems

• Traditional • response to problems » Sharpe, Treynor, Jensen, appraisal ratio

• time-changing beta and market timing

• In practice • performance attribution © Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

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Evaluating Security Selection 2 • Basic

Evaluating Security Selection 3 • Traditional

• idea

• idea

– compare returns with those of “similar” portfolios

– account for risk taken by manager

– depict percentiles (BKM 4-5-6, Fig. 24.1) » ex.: manager outperforms 90 out of 100 fund managers is in the 90th percentile » 5th, 95th percentiles; median, 25th and 75th_

– assume index model holds (and past performance matters) » and compute risk-adjusted excess returns

• problems – does extra performance cover fees

• problems

» difficulty to beat S&P 500 – estimation in practice » statistical significance?

– equities: allocations differ within groups – fixed income: durations vary © Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Evaluating Security Selection 4 • Traditional measures (continued) • Sharpe

rP − r f

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Evaluating Security Selection 5 • Traditional measures (continued) • Jensen

σP » appropriate for entire risky investment

α P =rP −[r f + β P (rM −r f ) ]

• appraisal ratio

αP σ (eP )

• Treynor

rP − r f

» benefit-to-cost ratio

βP

» appropriate for active portfolio (active PM)

» appropriate for one of many portfolios (Fig. 24.3) © Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Evaluating Security Selection 6 Question

Evaluating Security Selection 7 Answer:

Consider the two (excess return) index-model regression results for Stocks A and B. The risk-free rate over the period was 6%, and the market’s average return was 14%. i.

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

rA - rf = 1% + 1.2(r M - rf ) R-square = 0.576; residual std deviation , σ (eA) =10.3%; standard deviation of (rA -rf) = 26.1%.

ii. rB - r f = 2% + 0.8(rM - r f ) R-square = 0.436; residual std deviation , σ (eB ) =19.1%; standard deviation of (rB -r f) = 24.9%.

(a) To compute the Sharpe measure, note that for each portfolio, (rp – rf) can be computed from the right-hand side of the regression equation using the assumed parameters rM = 14% and r f = 6%. The standard deviation of each stock’s returns is given in the problem. The beta to use for the Treynor measure is the slope coefficient of the regression equation presented in the problem. Portfolio A

(a) Calculate the following statistics for each stock: i. ii. iii. iv.

Alpha. Appraisal ratio. Sharpe measure. Treynor measure.

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

(i) α is the intercept of the regression (ii) Appraisal ratio = α / σ(e) (iii) Sharpe measure = (rp – rf)/ σ (iv) Treynor measure = (rp – rf)/ β

1% 0.097 0.4061 8.833

Portfolio B 2% 0.1047 0.3373 10.5

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Evaluating Security Selection 8

(b) Which stock is the best choice under the following circumstances? i. This is the only risky asset to be held by the investor. ii. This stock will be mixed with the rest of the investor’s portfolio, currently composed solely of holdings in the market index fund. iii. This is one of many stocks that the investor is analyzing to form an actively managed stock portfolio.

Evaluating Security Selection 9

Answer: (a) (i) If this is the only risky asset, then Sharpe’s measure is the one to use. A’s is higher, so it is preferred. (ii) If the portfolio is mixed with the index fund, the contribution to the overall Sharpe measure is determined by the appraisal ratio. Therefore, B is preferred. (iii) If it is one of many portfolios, then Treynor’s measure counts, and B is preferred.

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Evaluating Security Selection 10 • Problems • does extra performance cover fees? » difficulty to beat S&P 500

• estimation in practice? » statistical significance? » time-varying beta? (Fig. 24.4) » solution: add a quadratic term in regression (Fig. 24.5)

• bottom line » still used » but not so much any more © Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Performance Attribution 2

Performance Attribution • Idea • hard to evaluate managers on risk-adjusted basis • important to allocate bonuses

• Split • excess returns • between contributions » broad asset allocation » industry choices within each market » security choices within each sector © Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Performance Attribution 3 Question 9 (20 points)

• Bogey (BKM6 Table 24.5) • base-line passive portfolio • assumed fixed for investment horizon

• Splits (BKM6 Tables 24.6 to 24.8) • broad asset » compare to bogey

• industry » given weights, compare with market weights

• security © Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

Consider the following information regarding the performance of a money manager in a recent month. The table represents the actual return of each sector of the manager’s portfolio in Column 1, the fraction of the portfolio allocated to each sector in Column 2, the benchmark or neutral sector allocations in Column 3, and the returns of sector indices in Column 4. Actual Return Actual Weight Benchmark Weight Index Return ----------------------------------------------------------------------------------------------------------------Equity 2% 0.70 0.60 2.5% (S&P 500) Bonds 1% 0.20 0.30 1.2% (SB Index)* Cash 0.5% 0.10 0.10 0.5% -----------------------------------------------------------------------------------------------------------------

* S&B Index = Salomon Brothers Index. (a) What was the manager’s return in the month? What was his or her overperformance or underperformance? (b) What was the contribution of security selection to relative performance? (c) What was the contribution of asset allocation to relative performance? Confirm that the sum of selection and allocation contributions equals his or her total “excess” return relative to the bogey.

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Performance Attribution 4

Performance Attribution 5

Answer:

(a) Asset Allocation:

(a) Bogey: 0.60 x 2.5% + 0.30 x 1.2% + 0.10 x 0.5% = 1.91% Actual: 0.70 x 2.0% + 0.20 x 1.0% + 0.10 x 0.5% = 1.65% Underperformance: 0.26%

Market

(a) Security Selection: Market

Differential Return Manager’s Portfolio Contribution Within Market Weight to Performance -----------------------------------------------------------------------------------------------------------Equity -0.5% 0.70 -0.35% Bonds -0.2% 0.20 -0.04% Cash 0 0.10 0% -----------------------------------------------------------------------------------------------------------Contribution of security selection -0.39%

Excess Weight: Index Return Contribution Manager - Benchmark minus Bogey to Performance -------------------------------------------------------------------------------------------------------------------Equity 0.10 0.59% 0.059% Bonds -0.10 -0.71% 0.071% Cash 0 -1.41% 0% -------------------------------------------------------------------------------------------------------------------Contribution of asset allocation 0.13% -------------------------------------------------------------------------------------------------------------------Summary:

Security selection = -0.39% Asset allocation = 0.13% Excess performance = -0.26%

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© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.

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