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Conditional Weighted Value + Growth Portfolio

(a.k.a MCP)

Midas Asset Management

Under the instruction of Prof. Campbell Harvey Feb 2005

Assignment 1 for GAA

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Goal

Optimize weights between value and growth trading styles periodically (monthly) on basis of conditional information available at the end of last period, so that the total returns and/or risk adjusted returns of our dynamic trading rule beat those of the benchmark portfolios and/or other selected benchmarks.

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Part 1: Methodology

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Security Universe

We select the top 5,000 U.S. stocks in market capitalization as the universe.

S&P 500: universe size too small

Russell 2000: only small- to mid cap.

We select 01/1983 to 08/1996 (163 months) as in sample, and 09/1996 to 11/2004 (99 months) as out of sample.

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Value and Growth Portfolio (a)

Value portfolio sorting variableBook(t-1)/Price(t-1)

Growth portfolio sorting variableEarnings growth per price dollar

[E(t-1)-E(t-13)]/[│E(t-13) │*P(t-1)]

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Value and Growth Portfolio (b)

• For each period, long F(1) stocks and short F(10) stocks in our universe.

• Within the two groups (N,N), equally value weighted.

• Portfolio return for each period: Rv or Rg=1/N*[Ra-Rz] Ra=sum of return of top F(1) Rz=sum of return of bottom F(10)

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Risk Adjusted Returns

Selected risk factor model: CAPM Risk adjusted return for Ra and Rz, for

Ra’(t)=Ra(t)-Rf(t)-β(a)*[Rm(t)-Rf(t)]

Rz’(t)=Rz(t)-Rf(t)-β(z)*[Rm(t)-Rf(t)]Here a, z represent a stock.

So we have risk adjusted return for each of the constructed portfolio (value portfolio and growth portfolio) and each period.

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Conditional Weighted Trading Rule (1)

For each period, assign w(v,t) to the value portfolio and w(g,t) to the growth portfolio.

w(v,t)+w(g,t)=1 Total trading rule return (TTRR)

TTRR(t)=w(v,t)*Rv(t)+w(g,t)*Rg(t)

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Conditional Weighted Trading Rule (2)

Alternatively, we use two sets of weights, one for 1 (value will out-perform growth), one for 0. And then we use in-the-sample R(v,t) and R(g,t) data, and optimizer to maximize positive excess return (over benchmark trading rule) and minimize negative excess return.

Suppose two sets of weights are {w(v,1),w(g,1)}, w(v,1)>=w(g,1), w(v,1)+w(g,1)=1{w(v,0),w(g,0)}, w(v,0)<=w(g,0), w(v,0)+w(g,0)=1

Then, if F(t,ω(t))=1,

TTRR(t)=w(v,1)*R(v,t)+w(g,1)*R(g,t)if F(t,ω(t))=0,

TTRR(t)=w(v,0)*R(v,t)+w(g,0)*R(g,t)

F(t,ω(t)) stands for the logistic predictive regression model. ω(t) stands for information set available at time t (at the end of t-1)

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Objective Function to Solve for Weights

Objective function for Optimizer (solve for optimal conditional weights)

Maximize Midas Conditional Portfolio (MCP) holding period return over the whole in-the-sample period.

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Trading costs

Use two thresholds to minimize between-portfolio turnover

Need to model within-portfolio turnover

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Logistic Predictive Regression Model

F(t,ω(t)) stands for the logistic predictive regression model. ω(t) stands for information set available at time t (at the end of t-1, lagged predictors).

F(t,ω(t)) takes on a probability between 0 and 1 given the predictors of period t-1.

F(t,ω(t)) conditions MCP.

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Map it out: the big picture of the steps

Total Trading Rule Return = if(C4=1,w(v,1)*G4+w(g,1)*H4, w(v,0)*G4+w(g,0)*H4)

Regression Sorting and portfolio construction Trading/Benchmarking

Periods Step : TTRR(t)Value Growth

Dec-04 F(t,ω(t)): 0/1 Predictor 1 (t-1) Predictor n (t-1) Rb(v,t) Rb(g,t) Provide conditional info TTRR(t)

Out-of SampleTest F(t)

Dec-94

Feedback: change sorting variables, weights?

Feedback: change predictors, model?

Step: F(t,ω(t)) Step: Rb(v/g,t)

Out of sample

test

In the sample

data

Challenge: predictorsChallenge: Sorting variables

Challenge: WeightsTransaction costs

each row is sorted long short return for that period for value or growth.

Optimizer to find out w(v,1), w(g,1) w(v,0), w(g,0)

Generate 0/1

Forecast 0/1

Starting Point

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Part 2: Results

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Selected Predictors in Logistic Regression Model

o 3ContMonValBetter (categorical variable): “1” means value portfolio outperforms growth portfolio in previous three consecutive months.

o 3ContMonGroBetter (categorical variable): “1” means growth portfolio outperforms value

portfolio in previous three consecutive months. o ValLessGrow: value portfolio return minus growth portfolio return o TenLess3MUpDn (categorical variable): Term structure. o BaaLessAaaUpDn (categorical variable): Credit spread o PELessMA: P/E minus 12 months moving average o Spread10YLessFedUpDn (categorical variable): “1” means10 year bond is higher than fed

fund rate.

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Coefficients

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Model Statistics

All-in-sample regression First, we tried to run the regression using all data points as in-sample data. The Cox & Snell R Square is 4.9% and the Naqelkerke R Square is 6.6%.

Step -2 Log

likelihood Cox & Snell R Square

Nagelkerke R Square

1 351.037(a) .049 .066

. The overall correct percentage is 61.2% as shown in the following classification table.

Classification Tablea

88 48 64.7

54 73 57.5

61.2

Observed0

1

ValueBetter

Overall Percentage

Step 10 1

ValueBetter PercentageCorrect

Predicted

The cut value is .500a.

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Conditioning & Weight Optimization

rg results rg result>=threshold H? rg result<threshold L? W(v) to use Turnover0.57876 1 0 1.226843107 00.59554 1 0 1.226843107 00.66772 1 0 1.226843107 00.6069 1 0 1.226843107 0

Analysis and presentation Target cell Changing cellsObjective: max conditional return threshold H 0.55

3.903361619 threshold L 0.45w(v) when rg>=threshold 1.226843107w(v) when rg<threshold -0.672121772

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Performance of MCP

In the sampleOut of sample

Value portfolio Growth portfolio Midas Conditional Market T-bill

Annualized return2.6% 6.7% 10.5% 15.9% 6.3%1.2% 2.0% 26.2% 8.9% 3.6%

Volatility17.5% 16.3% 30.0% 14.1% 0.6%27.1% 24.8% 37.5% 17.0% 0.5%

skewness0.268947156 -0.172674905 -0.452138825 -0.947330467 0.121644780.318139691 -0.876256154 0.181344591 -0.44332867 -0.26165161

Correlation0.176735719 -0.058663088 0.063230298 1 0.0209204040.233018437 -0.139552486 -0.046958957 1 0.04701875

Beta0.219764601 -0.067990626 0.134849794 1 0.0008656830.371168031 -0.20299067 -0.103476132 1 0.001496022

Alpha-5.78% 1.04% 2.97% 0.00% -0.01%-4.39% -0.51% 23.09% 0.00% -0.01%

Sharpe Ratio-0.209427986 0.023896531 0.142303888 0.683493189 0-0.08992208 -0.063678409 0.601034212 0.307982992 0

Midas conditional portfolio turnover times20 8.15 average months per turn over7 14.14285714 one turnover average month

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Performance of MCP (1)

Annualized Return

Annualised Return

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

30.0%

Value portfolio Growth portfolio Midas Conditional Market T-bill

Per

cen

tage

In the sample

Out of sample

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Performance of MCP (2)

Volatility

Volatility

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

30.0%

35.0%

40.0%

Value portfolio Growth portfolio Midas Conditional Market T-bill

In the sample

Out of sample

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Performance of MCP (4)

Skewness

Skewness

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

Value portfolio Growth portfolio Midas Conditional Market T-bill

In the sample

Out of sample

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Performance of MCP (4)

Correlation

Correlation

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Value portfolio Growth portfolio Midas Conditional Market T-bill

In the sample

Out of sample

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Performance of MCP (5)

Beta

Beta

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Value portfolio Growth portfolio Midas Conditional Market T-bill

In the sample

Out of sample

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Performance of MCP (6)

Sharpe Ratio

Sharpe Ratio

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Value portfolio Growth portfolio Midas Conditional Market T-bill

In the sample

Out of sample

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Performance of MCP (7)Alpha (at least, in the way we calculated it. Yes, we are still wondering, is this real?)

Alpha

-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

Value portfolio Growth portfolio Midas Conditional Market T-bill

In the sample

Out of sample

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The concern of transaction costs

Partially addressedTurnover

0

2

4

6

8

10

12

14

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average months per turn over

In the sample

Out of sample

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Part 3: Future Research

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Suggested Future Research

Midas is an intriguing figure. Interesting research topics arise around him.

For example, Women like gold; but do they like to be turned into gold??