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Chapter 2 : Chapter 2 : Portfolio Theory and the Capital Portfolio Theory and the Capital Asset Pricing Model (CAPM)Asset Pricing Model (CAPM)

• Suppose you believe that investments in stock offer an expected rate of return of 12% while the expected rate of return on bonds is only 8%. Would you invest all your money in stock?

• Probably not, putting all of your eggs in one basket would violate the most basic notion of diversification.

• But what is the optimal combination of these two asset classes? And how will the opportunity to invest in other assets (e.g. real estate, foreign stocks etc.) affect your decision? Is there a best solution to your asset allocation problem?

• How should one measure the risk of an individual asset held as part of a diversified portfolio?

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• Asset Allocation Choice – a choice among broad investment class, rather than among the specific securities with each asset class.

• Risk-free assets: e.g. Treasury Bills• Risky assets: e.g. long term bonds, stocks, warrants• Let’s denote the investor’s portfolio of risk assets as

P, and the risk free asset as F.

• rp = actual rate of return of the risky portfolio P

• rf = rate of return of the risk free asset F

• E(rp) = expected rate of return of P

p = standard deviation of rp

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• The question is, how much, if anything, should be invested in a risky asset rather than investing in the risk free asset.

• The answer depends partially on the amount of expected extra return arises from investing in the risky asset.

• Risk premium = E(rp) – rf

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• Let E(rp) = 15% , p = 22% , rf = 7%

• Risk premium = E(rp) – rf = 8%

• Scenario 1 : Invest all funds into the risky assets, i.e. y = 1.0 (y is the proportion of investment budget to be allocated to the risky portfolio). We arrive at point P, where the expected return is 15%, and the standard deviation is 22%.

• Scenario 2 : put all fund into the risk-free asset.

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• Scenario 3 : allocate equal amounts of the funds to the risk-free and risky assets. i.e. y = 0.5. Arrive at point C where

E(rc) = 0.5 7% + 0.5 15% = 11%

E(rc) – rf =11% – 7% = 4%

c = 11%

• i.e. E(rc) – rf = y[E(rp) – rf ]

c = y p

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• In sum, both the risk premium and the standard deviation of the portfolio increase in proportion to the investment in the risky portfolio.

• The Capital Allocation Line (CAL) gives the risk-return combinations available by varying asset allocation.

Intercept = rf (7% in the present example)

The slope equals the increase in expected return that an investor can obtain per unit of additional risk.

• The slope is also known as the reward-to-variability ratio.

228

22715

Slope

p

fp rrE

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• The plot on the CAL to the right of P relates to the situation where the investor can borrow at the risk-free rate of rf = 7%

• e.g. suppose y = 1.25, then

E(rc) = 7 + 1.25 8 = 17

c = 1.25 22 = 27.5

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• But suppose the borrowing rate is rB = 9%. Then for y > 1.0 (the borrowing range), the reward-to-variability ratio will be

• Here the borrowing rate (rB) replaces the lending rate (rf), resulting in a “kink” on the CAL at Point P.

27.0226 pBp rrE

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The investment opportunity set with a risky asset The investment opportunity set with a risky asset and a risk-free assetand a risk-free asset

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The opportunity set with differential borrowing and The opportunity set with differential borrowing and lending rates lending rates

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• Sources of risk:

– Systematic risk – a risk that is attributable to market wide risk and is non-diversifiable.

– Nonsystematic risk – a risk pertinent to the specific firm and is diversifiable

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Portfolio risk as a function of the number of stocks in the Portfolio risk as a function of the number of stocks in the portfolioportfolio

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Portfolio risk decreases as diversification Portfolio risk decreases as diversification increasesincreases

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• Suppose a proportion denoted by WB is invested in the bond fund, and the remainder 1 – WB, denoted by Ws is invested in a stock fund.

• Rate of return : rp = WBrB + WSrS

• Expected rate of return:

E(rp) = WB E(rB) + WS E(rS) • Variance of rate of return of the portfolio:

where is the correlation coefficient between rB and rS.

To illustrate, let E(rB) = E(rS) = 10%;

B = S = 2% .

2 2 2 2 2

2 2 2 2

2

2

p B B S S B S BS

B B S S B S BS B S

W W W W

W W W W

BS

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E(rp) WB WS B S ρBS P

Case A 10% 1.0 0.0 2.0 2.0 1.0 2.0

Case B 10% 0.0 1.0 2.0 2.0 1.0 2.0

Case C 10% 0.5 0.5 2.0 2.0 1.0 2.0

Case D 10% 0.5 0.5 2.0 2.0 0.9 1.95

Case E 10% 0.5 0.5 2.0 2.0 0.5 1.73

Case F 10% 0.5 0.5 2.0 2.0 0 1.41

Case G 10% 0.5 0.5 2.0 2.0 0.5 1

Case H 10% 0.5 0.5 2.0 2.0 0.9 0.45

Case I 10% 0.5 0.5 2.0 2.0 1 0

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But what if E(rB) E(rS) and B S

To illustrate, let E(rB) = 10%,

E(rS) = 17% ,B = 12%, S = 25% and ρBS = 0

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Investment opportunity set for bond and stock fundsInvestment opportunity set for bond and stock funds

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• Investors prefer portfolios that lie to the northwest of the Investment opportunity set.

• Portfolio 1 is said to dominate portfolio 2 if all investors prefer 1 to 2. This will be case if it has higher mean return and lower variance:

i.e. E(r1) E(r2) and 1 2

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Investment opportunity sets for bonds and stocks Investment opportunity sets for bonds and stocks with various correlation coefficientswith various correlation coefficients

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• The smaller the value of ρBS, the better the effect of diversification

• If ρBS = – 1, then

and P2 = 0 when

22SSBBP WW

SB

SBW

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Investment opportunity sets for bonds and stocks Investment opportunity sets for bonds and stocks with various correlation coefficientswith various correlation coefficients

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• Now we can expand the asset allocation problem to include the risk free asset.

• Scenario : ρBS = 0.2

rf = 8%

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• Consider Portfolios A & B : (assume ρBS = 0.2)

1) Portfolio A (minimum variance portfolio):

WB = 0.8706 , WS = 0.1294

E(rA) = 10.91% A = 11.54%

The reward to variability ratio is :

2) Portfolio B:

WB = 0.65 , WS = 0.35

E(rB) = 12.45% B = 12.83%

The reward to variability ratio is :

25.0

54.11891.10

A

fAA

rrES

35.083.12

845.12 BS

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The opportunity set using bonds and stocks and two The opportunity set using bonds and stocks and two capital allocation linescapital allocation lines

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37.007.17

836.140 S

• The difference in the reward-to-variability ratio is SB – SA = 0.10, i.e. portfolio B provides an extra 0.1% of expected return for every percentage increase in risk.

• But why stop at Portfolio B? We should search for the CAL with the highest feasible rewarded-to-variability ratio. Therefore, the tangency portfolio O is the optimal risky portfolio to mix with the risk-free asset.

• For portfolio O,

WB = 0.3765 , WS = 0.6235

E(r0) = 14.36% 0 = 17.07%

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The optimal capital allocation line with bonds, stocks, and T-bills

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• We can extend the two-risky-assets portfolio to cover the case of many risky assets and a risk free asset.

The complete portfolio

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Single-factor Asset MarketSingle-factor Asset Market

• Let Ri = ri – rf = excess return on a security i.• Decompose Ri additively into three factors:

Ri = E(Ri) + iM + i

where M quantifies market factors;

i captures the impact of unanticipated firm specific events, independent of market factors

• Usually M is measured by the rate of return on a broad index, such as the S&P 500, HSI as a proxy for common macro factor.

Let M = RM = rm – rf : the excess return on the market index.

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• Risk analysis:

Var (Ri) = Var [E(Ri) + iRM + i]

= Var (iRM) + Var (i)

= i2 M

2 + i2

= systematic risk + firm specific risk

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1. The variance attributable to the uncertainty common to the entire market. This systematic risk depends on the volatility in RM (i.e., M

2 ) and the sensitivity of stock to the market’s fluctuation, measured by i

2. The variance attributable to firm specific risk factors, measured by i

2. This is the variance in the part of the stock’s return that is independent of market performance.

Therefore, the total variability of the rate of return of security i depends on 2 components :

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• The “beta” (i) coefficient is of paramount importance. It captures the responsiveness of individual stock to market events, as well as the trade-off between risk and return. The higher the value of beta, the greater the excess return of a security and the greater the systematic risk.

• An “average” security has a beta value of 1; an “aggressive” security has a beta greater than 1 (i.e., the security has above average return and above average risk); conversely, securities with betas lower than 1 are called “defensive” securities.

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The Capital Asset Pricing Model (CAPM)The Capital Asset Pricing Model (CAPM)

• It predicts the relationship between the risk and equilibrium expected returns on risky asset.

• We shall approach the CAPM in a simplified setting.• It is built on the premise that the risk premium of an

individual asset is proportional to its beta, which measures the security’s responsiveness to systematic risk. For example, if you double the systematic risk, you must also double its risk premium for investors still to be willing to hold the asset. So the ratio of risk premium to beta should be the same for all securities. i.e.,

12

2

1

1 fMff rrErrErrE

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• Therefore, for any security D,

In otherwords, the rate of return on any asset exceeds the risk free rate by a risk premium equals to the asset’s systematic risk measure (its beta) times the risk premium of the benchmark market portfolio.

• To move from a model cast in expectations to a realized-return framework, we rewrite the preceding theoretical model in the form of a regression.

fMDfD rrErrE

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where D and D are estimated using time series data. Taking expectation, we have

So the CAPM predicts that D = 0. D > (<)0 implies that the stock systematically returns more (less) than expected for any given level of market returns.

tDtftMDDtftD rrrr ,,,,,

.,,,, tftMDDtftD rrErrE

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• The conceptual model

is called the Security Market Line (SML), which is taken as a benchmark to assess the “fair” expected return on a risky asset.

• If D > 0, the expected return is greater than the fair return.

Equivalently, the stock is underpriced and is perceived to be a good buy.

fMDfD rrErrE

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The security market line and a positive-alpha stock

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SAS ExampleSAS Example

• From data file RETURNI.txt• time period : January 1978 – December 1986 (108

obs.)• contains monthly returns of the Gerber Corporation’s

common stock (GERBER) – a U.S. listed company that sells baby food and home care products.

• market return (R_M) : based on the value weighted transactions of all stocks listed on the American stock exchange.

• risk-free asset returns (R_F) : return on the 30-day U.S. Treasury Bills.

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datadata return1; return1; infile 'd:\teaching\ms4221\return1.txt';infile 'd:\teaching\ms4221\return1.txt'; input r_m r_f gerber @@;input r_m r_f gerber @@; retain date retain date '01dec77'd'01dec77'd;; date=intnx('month',date,date=intnx('month',date,11);); format date monyy.;format date monyy.;/* Creating of New Variables *//* Creating of New Variables */ r_gerber = gerber - r_f;r_gerber = gerber - r_f; r_mkt = r_m - r_f;r_mkt = r_m - r_f;/* Labeling Variables *//* Labeling Variables */ label r_m='Market Rate of Return'label r_m='Market Rate of Return' r_f='Risk-Free Rate of Return'r_f='Risk-Free Rate of Return' gerber='Rate of Return for Gerber Corporation'gerber='Rate of Return for Gerber Corporation' r_gerber='Risk Premium for Gerber Corporation'r_gerber='Risk Premium for Gerber Corporation' r_mkt='Risk Premium for Market';r_mkt='Risk Premium for Market';procproc printprint data=return1 (obs= data=return1 (obs=55) label;) label; var date r_gerber r_mkt r_m r_f gerber;var date r_gerber r_mkt r_m r_f gerber; title 'CAPM Analysis';title 'CAPM Analysis'; title2 'Returns and Risk Premiums';title2 'Returns and Risk Premiums';runrun;;

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CAPM Analysis Returns and Risk Premiums Rate of Risk Premium Risk Market Risk-Free Return for for Gerber Premium Rate of Rate of Gerber Obs date Corporation for Market Return Return Corporation 1 JAN78 -0.05287 -0.04987 -0.045 .00487 -0.048 2 FEB78 0.15506 0.00506 0.010 .00494 0.160 3 MAR78 -0.04126 0.04474 0.050 .00526 -0.036 4 APR78 -0.00091 0.05809 0.063 .00491 0.004 5 MAY78 0.04087 0.06187 0.067 .00513 0.046

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proc reg data=return1; model r_gerber = r_mkt / dw; slope: test r_mkt=1;run;

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The REG Procedure Model: MODEL1 Dependent Variable: r_gerber Risk Premium for Gerber Corporation Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 1 0.12676 0.12676 23.10 <.0001 Error 106 0.58166 0.00549 Corrected Total 107 0.70843 Root MSE 0.07408 R-Square 0.1789 Dependent Mean 0.01043 Adj R-Sq 0.1712 Coeff Var 710.15401 Parameter Estimates Parameter Standard Variable Label DF Estimate Error t Value Pr > |t| Intercept Intercept 1 0.00630 0.00718 0.88 0.3823 r_mkt Risk Premium for Market 1 0.53093 0.11046 4.81 <.0001

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The REG Procedure Model: MODEL1 Dependent Variable: r_gerber Risk Premium for Gerber Corporation Durbin-Watson D 2.250 Number of Observations 108 1st Order Autocorrelation -0.129 The REG Procedure Model: MODEL1 Test slope Results for Dependent Variable r_gerber Mean Source DF Square F Value Pr > F Numerator 1 0.09895 18.03 <.0001 Denominator 106 0.00549

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Interpretation of OutputInterpretation of Output

• R_GERBER = 0.006299 + 0.530929 (R_MKT)• R2 = 0.1789. It shows the proportion of the risk

premium of Gerber stock that is accounted for by the market’s risk premium. i.e. R2 measures the portion of the systematic (non-diversifiable) risk of the risk premium.

= 0.006299 with t-ratio = 0.88.

> (<) 0 indicates that the stock systematically returns more (less) than expected for any market return level. Note that H0 : = 0 cannot be rejected with the present data set.

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• = 0.530929. This value implies that if the market risk premium increases by 1 percent, the Gerber stock risk premium will increase by 0.530929%. So the stock moves with the market but its risk is less volatile than the market. Note that the hypothesis H0 : = 1 can be rejected, so the slope coefficient differs from 1.

• DW = 2.25. Suppose we want to test H0 : ρ = 0 vs. H1 : ρ < 0

at 0.05 level of significance.

DL = 1.65 and D = 1.69

As DW < 4 - Du, we cannot reject the null of no first order negative autocorrelation.

U

U

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Applying CAPM to additional stocksApplying CAPM to additional stocks

• Data file RETURN2.txt contains returns data on– Tandy Corporation (TANDY) – a computer and electronics

manufacturer based in Texas

– General Mills (GENMIL) – food distributor

– Consolidated Edison (CONED) – energy and telecommunication Co., mainly based in N.Y. state

– Weyerhanser (WEYER) – forest products Co., specializing in timber industry

– International Business Machine (IBM) -computer industry– Digital Equipment Corporation (DEC) - computer industry

(recently acquired by Compaq, then merged with Hewlett-Packard)

– Mobil Corporation (MOBIL) - petroleum, energy and fuels– Texaco Corporation (TEX)- Petroleum, energy and fuels

– Carolina Power and Light (CPL)- Electricity supplier in North and South Carolina.

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data return2;infile 'd:\teaching\ms4221\return2.txt'; input tandy genmil coned weyer ibm dec mobil tex cpl @@; retain date '01dec77'd; date=intnx('month',date,1); format date monyy.; label tandy='Rate of Return for Tandy Corporation' genmil='Rate of Return for General Mills' coned='Rate of Return for Con Edison' weyer='Rate of Return for Weyerhauser' ibm='Rate of Return for IBM' dec='Rate of Return for DEC' mobil='Rate of Return for Mobil Corporation' tex='Rate of Return for Texaco' cpl='Rate of Return for CPL'; data return3; merge return1 return2; by date; r_tandy = tandy - r_f; r_genmil = genmil - r_f; r_coned = coned - r_f; r_weyer = weyer - r_f; r_ibm=ibm - r_f; r_dec=dec - r_f; r_mobil = mobil - r_f; r_tex=tex - r_f; r_cpl=cpl - r_f; label r_tandy='Risk Premium for Tandy Corporation' r_genmil='Risk Premium for General Mills' coned='Risk Premium for Con Edison' weyer='Risk Premium for Weyerhauser' ibm='Risk Premium for IBM' dec='Risk Premium for DEC' mobil='Risk Premium for Mobil Corporation' tex='Risk Premium for Texaco' cpl='Risk Premium for Carolina Power & Light'; run;

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proc print data=return3 (obs=10) label; var date r_tandy; title2 'Risk Premiums for Tandy

Corporation';run;

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CAPM Analysis Risk Premiums for Tandy Corporation Risk Premium for Tandy Obs date Corporation 1 JAN78 -0.07987 2 FEB78 -0.00894 3 MAR78 0.11874 4 APR78 0.05009 5 MAY78 0.17087 6 JUN78 -0.01927 7 JUL78 0.18872 8 AUG78 0.21593 9 SEP78 -0.10645 10 OCT78 -0.21285

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proc reg data=return3 outest=capmest1; model r_gerber = r_mkt /dw; slope: test r_mkt=1;

model r_tandy = r_mkt / dw; slope: test r_mkt=1;

model r_genmil = r_mkt / dw; slope: test r_mkt=1;

model r_coned = r_mkt / dw; slope: test r_mkt=1;

model r_weyer = r_mkt / dw; slope: test r_mkt=1;

model r_ibm = r_mkt / dw ; slope: test r_mkt=1;

model r_dec = r_mkt / dw; slope: test r_mkt=1;

model r_mobil = r_mkt / dw; slope: test r_mkt=1;

model r_tex = r_mkt / dw; slope: test r_mkt=1;

model r_cpl = r_mkt / dw; slope: test r_mkt=1; run;

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CAPM Regression ResultsCAPM Regression Results

CompanyIntercept

(t-statistic)

Slope

(t-statistic)

F-statistic

Slope = 1.0 (p-value)R2

Gerber0.006299

(0.877)

0.530929

(4.806)

18.0316

(0.0001)

0.1789

Tandy0.013467

(1.286)

1.048230

(6.504)

0.0896

(0.7653)

0.2852

General Mills0.006604

(1.174)

0.138337

(1.598)

99.1320

(0.0001)

0.0235

Con Ed0.013044

(2.649)

0.102044

(1.347)

140.4491

(0.0001)

0.0168

Weyer – 0.003558

(– 0.574)

0.723119

(7.578)

8.4187

(0.0045)

0.3514

IBM 0.000095

(0.020)

0.395362

(5.298)

65.6575

(0.0001)

0.2094

DEC 0.005509

(0.724)

0.715704

(6.114)

5.8982

(0.0168)

0.2607

Mobil 0.004688

(0.752)

0.685563

(7.148)

10.7477

(0.0014)

0.3252

Texaco 0.000391

(0.061)

0.578628

(5.876)

18.3080

(0.0001)

0.2457

CP&L 0.005803

(1.145)

0.206764

(2.652)

103.5415

(0.0001)

0.0622

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data forecast; set capmest1; do f_r_mkt = -.02 to .07 by .01; pred=intercept + f_r_mkt * r_mkt; risk_fre=.03; return = pred + risk_fre; output; end; label _depvar_='Stock' f_r_mkt='Future Market Risk Premium‘ pred='Predicted Stock Risk Premium' return='Predicted Future Stock Return';run;

proc print data=forecast (obs=10) label; var _depvar_ f_r_mkt pred return; title2 'Point Estimates of'; title3 'Stock Risk Premiums and Returns';run;

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CAPM Analysis Point Estimates of Stock Risk Premiums and Returns Future Market Predicted Predicted Risk Stock Risk Future Stock Obs Stock Premium Premium Return 1 r_gerber -0.02 -0.004320 0.025680 2 r_gerber -0.01 0.000990 0.030990 3 r_gerber 0.00 0.006299 0.036299 4 r_gerber 0.01 0.011608 0.041608 5 r_gerber 0.02 0.016917 0.046917 6 r_gerber 0.03 0.022227 0.052227 7 r_gerber 0.04 0.027536 0.057536 8 r_gerber 0.05 0.032845 0.062845 9 r_gerber 0.06 0.038155 0.068155 10 r_gerber 0.07 0.043464 0.073464