Gini Coefficient. A supplement to Mahlerʼs Guide to Loss Distributions. Exam C. prepared by Howard C. Mahler, FCAS Copyright 2017 by Howard C. Mahler.

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1 Gini Coefficient A supplement to Mahlerʼs Guide to Loss Distributions Eam C prepared by Howard C. Mahler, FCAS Copyright 27 by Howard C. Mahler. Howard Mahler hmahler@mac.com

2 The Gini Coefficient, Gini Inde, or coefficient of concentration is a concept that comes up for eample in economics, when looking at the distribution of incomes. I will discuss the Gini coefficient and relate it to the Relative Mean Difference. The Gini coefficient is a measure of inequality. For eample if all of the individuals in a group have the same income, then the Gini coefficient is zero. As incomes of the individuals in a group became more and more unequal, the Gini coefficient would increase towards a value of. The Gini coefficient has found application in many different fields of study. Mean Difference: Gini Coefficient, HCM /2/7, Page Define the mean difference as the average absolute difference between two random draws from a distribution. Mean Difference = - y f() f(y) d dy, where the double integral is taken over the support of f. For eample, for a uniform distribution from to : Mean Difference = - y (/ ) (/ ) d dy = y (/) - y d dy + (/) y - d dy = =y = (/) 5 - y 2 / 2 + y 2 - y dy + (/) y 2 - y 2 / 2 dy = (/) (5 + /6-5) + (/)(/6) = /3. In a similar manner, in general for the continuous uniform distribution, the mean difference is: (width)/3. For a sample of size two from a uniform, the epected value of the minimum is the bottom of the interval plus (width)/3, while the epected value of the maimum is the top of the interval - (width)/3. Thus the epected absolute difference is (width)/3. This is discussed in order statistics, on the Syllabus of CAS Eam CAS S.

3 Gini Coefficient, HCM /2/7, Page 2 Eercise: Compute the mean difference for an Eponential Distribution. [Solution: Mean difference = - y e - / θ / θ d e - y / θ / θ dy = y (/θ 2 ) e - y / θ (y - ) e - / θ d dy + (/θ 2 ) e - y / θ ( - y) e - / θ d dy = = =y (/θ) e - y / θ {y (- e - y / θ ) + θe - y / θ + y e - y / θ - θ} dy + (/θ) e - y / θ {θe - y / θ + y e - y / θ - y e - y / θ } dy = y e - y / θ / θ + 2 e - 2y / θ - e - y / θ dy = θ + θ - θ = θ. Alternately, by symmetry the contributions from when > y and when y > must be equal. y Thus, the mean difference is: (2) (/θ 2 ) e - y / θ (y - ) e - / θ d dy = = (2/θ) e - y / θ {y (- e - y / θ ) + θe - y / θ + y e - y / θ - θ} dy = 2 y e - y / θ / θ + e - 2y / θ - e - y / θ dy = (2)(θ + θ/2 - θ) = θ. Comment: e - / θ d = -θ ( + θ) e -/θ. For a sample of size two from an Eponential Distribution, the epected value of the minimum is θ/2, while the epected value of the maimum is 3θ/2. Therefore, the epected value of the difference is θ.]

4 Gini Coefficient, HCM /2/7, Page 3 Mean Relative Difference: The mean relative difference of a distribution is defined as: mean difference. mean For the uniform distribution, the mean relative difference is: (width) / 3 (width) / 2 = 2/3. For the Eponential Distribution, the mean relative difference is: θ θ =. Eercise: Derive the form of the Mean Relative Difference for a Pareto Distribution. - (a- ) + θ Hint: (+ θ) a d = ( + θ) a - (a- )(a - 2) [Solution: For α >, = θ/(α-). f() = α θ α ( θ + ) α +. α θ Mean difference = - y α. θ + θ + y ( ) α + d α θ α ( ) α + dy By symmetry the contributions from when > y and when y > must be equal. Therefore, mean difference = 2 α 2 θ 2α ( - y) (θ + ) α + d (θ + y) α + dy. y= =y Now using the hint: (θ + ) α + d = =y yα + θ α (α - ) (θ + y) α. (θ + ) α + d = =y α (θ + y) α. Therefore, - y (θ + ) α + d = =y yα + θ α (α - ) (θ + y) α - y α (θ + y) α = α (α - ) (θ + y) α -. Thus, mean difference = 2 α θ2α α - (θ + y) 2 α dy = 2 α θ2α α - (2α - ) θ 2α - = 2 α θ (α - ) (2α - ). = θ/(α-). Thus, the mean relative difference is: 2 / (2α - ), α >.]

5 Gini Coefficient, HCM /2/7, Page 4 Lorenz Curve: Assume that the incomes in a country follow a distribution function F(). 2 Then F() is the percentage of people with incomes less than. The income earned by such people is: t f(t) dt = E[X ] - S() = S(t) dt. The percentage of total income earned by such people is: y f(y) dy = E[X ] - S(). Define G() = y f(y) dy = E[X ] - S(). 3 For eample, assume an Eponential Distribution. Then F() = - e -/θ. G() = E[X ] - S() = θ ( - e- / θ ) - e - / θ θ = - e -/θ - (/θ) e -/θ. Let t = F() = - e -/θ. Therefore, /θ = - ln( - t). 4 Then, G(t) = t - {-ln(-t)} (-t) = t + (-t) ln(-t). 2 Of course, the mathematics applies regardless of what is being modeled. The distribution of incomes is just the most common contet. 3 This is not standard notation. I have just used G to have some notation. 4 This is just the VaR formula for the Eponential Distribution.

6 Then we can graph G as a function of F: G(). Gini Coefficient, HCM /2/7, Page F() This curve is referred to as the Lorenz curve or the concentration curve. Since F() = = G() and F() = = G(), the Lorenz curve passes through the points (, ) and (, ). Usually one would also include in the graph the 45 reference line connecting (, ) and (, ), as shown below: % ofincome Lorenz Curve % ofpeople

7 Gini Coefficient, HCM /2/7, Page 6 G(t) = G[F())] = y f(y) dy. dg dt = dg d / df d = f() / f() = >. d 2 G dt 2 = d d / df d = f() >. Thus, in the above graph, as well as in general, the Lorenz curve is increasing and concave up. The Lorenz curve is below the 45 reference line, ecept at the endpoints when they are equal. The vertical distance between the Lorenz curve and the 45 comparison line is: F - G. Thus, this vertical distance is a maimum when: = df df - dg df. dg df =. =. =. Thus the vertical distance between the Lorenz curve and the 45 comparison line is a maimum at the mean income. Eercise: If incomes follow an Eponential Distribution, what is this maimum vertical distance between the Lorenz curve and the 45 comparison line? [Solution: The maimum occurs when = θ. F() = - e -/θ. From previously, G() = - e -/θ - (/θ) e -/θ. F - G = (/θ) e -/θ. At = θ, this is: e - =.3679.]

8 Eercise: Determine the form of the Lorenz Curve, if the distribution of incomes follows a Pareto Distribution, with α >. [Solution: F() = - θ θ + α. = Gini Coefficient, HCM /2/7, Page 7 θ θ. E[X ] = α α - θ α θ +. G() = E[X ] - S() = θ α - { - α - θ } - S() θ + = - θ / (α -) α - θ - (α-) θ + θ S(). Let t = F() = - θ α θ α. θ + θ + = S() = - t. Also, /θ = ( - t) -/α -. 5 Therefore, G(t) = - ( - t) (α-)/α - (α-){( - t) -/α - } ( - t) = t + α - tα - α (-t) -/α, t. Comment: G() = α - α =. G() = + α - α - =.] Here is graph comparing the Lorenz curves for Paretos with α = 2 and α = 5: % ofincome alpha = 5 alpha = % ofpeople 5 This is just the VaR formula for the Pareto Distribution.

9 The Pareto with α = 2 has a heavier righthand tail than the Pareto with α = 5. If incomes follow a Pareto with α = 2, then there are more etremely high incomes compared to the mean, than if incomes follow a Pareto with α = 5. In other words, if α = 2, then income is more concentrated in the high income individuals than if α = 5. 6 Gini Coefficient, HCM /2/7, Page 8 The Lorenz curve for α = 2 is below that for α = 5. In general, the lower curve corresponds to a higher concentration of income. In other words, a higher concentration of income corresponds to a smaller area under the Lorenz curve. Equivalently, a higher concentration of income corresponds to a larger area between the Lorenz curve and the 45 reference line. Gini Coefficient: This correspondence between areas on the graph of the Lorenz curve the concentration of income is the idea behind the Gini Coefficient. Let us label the areas in the graph of a Lorenz Curve, in this case for an Eponential Distribution: % ofincome A.2 B Gini Coefficient = % ofpeople Area A Area A + Area B. 6 An Eponential Distribution has a lighter righthand tail than either Pareto. Thus if income followed an Eponential, it would less concentrated than if it followed any Pareto.

10 However, Area A + Area B add up to a triangle with area /2. Area A Therefore, Gini Coefficient = = 2A = - 2B. Area A + Area B Gini Coefficient, HCM /2/7, Page 9 For the Eponentials Distribution, the Lorenz curve was: G(t) = t + (-t) ln(-t). Thus, Area B = area under Lorenz curve = t + (- t) ln(- t) dt = /2 + s ln(s) ds. Applying integration by parts, s = s ln(s) ds = (s 2 / 2) ln(s)] - (s 2 / 2) (/ s) ds = - /4 = -/4. s = Thus Area B = /2 - /4 = /4. Therefore, for the Eponential Distribution, the Gini Coefficient is: - (2)(/4) = /2. Recall that for the Eponential Distribution, the mean relative difference was. As will be shown subsequently, in general, Gini Coefficient = (mean relative difference)/2. Therefore, for the Uniform Distribution, the Gini Coefficient is: (/2)(2/3) = /3. Similarly, for the Pareto Distribution, the Gini Coefficient is: (/2){2 / (2α - )} = / (2α - ), α >. We note that the Uniform with the lightest righthand tail of the three has the smallest Gini coefficient, while the Pareto with the heaviest righthand tail of the three has the largest Gini coefficient. Among Pareto Distributions, the smaller alpha, the heavier the righthand tail, and the larger the Gini Coefficient. 7 The more concentrated the income is among the higher earners, the larger the Gini coefficient. 7 As alpha approaches one, the Gini coefficient approaches one.

11 Gini Coefficient, HCM /2/7, Page LogNormal Distribution: For the LogNormal Distribution: = ep[µ + σ 2 /2]. E[X ] = ep(µ + σ 2 ln() µ σ2 ln() µ /2) Φ σ + { - Φ σ } ln() µ σ2 = Φ σ + S(). Therefore, G() = E[X ] - S() ln() µ σ2 = Φ σ = Φ ln() - µ σ - σ. Let t = F() = Φ ln() - µ σ. Then the Lorenz Curve is: G(t) = Φ[ Φ - [t] - σ]. For eample, here a graph of the Lorenz curves for LogNormal Distributions with σ = and σ = 2: % ofincome sigma =.2 sigma = % ofpeople

12 Gini Coefficient, HCM /2/7, Page As derived subsequently, for a LogNormal Distribution, the Gini Coefficient is: 2Φ[σ/ 2 ] -. Here is a graph of the Gini Coefficient as a function of sigma: Gini Coef sigma As sigma increases, the LogNormal has a heavier tail, and the Gini Coefficient Increases towards. The mean relative distance is twice the Gini Coefficient: 4Φ[σ/ 2 ] - 2.

13 Gini Coefficient, HCM /2/7, Page 2 Derivation of the Gini Coefficient for the LogNormal Distribution: In order to compute the Gini Coefficient, we need to compute area B. % ofincome A.2 B % ofpeople B = G(t) dt = Φ[ Φ - [t] - σ] dt. Let y = Φ - [t]. Then t = Φ[y]. dt = φ[y] dy. B = Φ[y - σ] φ[y] dy. - Now B is some function of σ. B(σ) = Φ[y - σ] φ[y] dy. - B() = Φ[y] φ[y] dy = Φ[y] 2 y = / 2 ] - y = - = /2.

14 Gini Coefficient, HCM /2/7, Page 3 B(σ) = Φ[y - σ] φ[y] dy. Taking the derivative of B with respect to sigma: - Bʼ(σ) = - φ[y - σ] φ[y] dy = - 2π - - ep[-(y - σ) 2 / 2] ep[-y 2 / 2] dy = - 2π ep[-σ2 /2] ep[-(2y 2-2σy) / 2] dy - = - 2π ep[-σ2 /4] ep[-{( 2 y) 2-2( 2 y)(σ / 2) + (σ / 2) 2 } / 2] dy - = - 2π ep[-σ2 /4] ep[-{ 2 y - σ / 2} 2 / 2] dy. - Let = 2 y - σ / 2. dy = d / 2. Bʼ(σ) = - 2 π ep[-σ2 /4] ep[- 2 / 2] d = - 2π 2 π ep[-σ2 /4]. 8 - Now assume that B(σ) = c - Φ[σ/ 2 ], for some constant c. Then Bʼ(σ) = -φ[σ/ 2 ] / 2 = - 2π ep[-(σ/ 2 )2 /2] / 2 = - 2 π ep[-σ2 /4], matching above. Therefore, we have shown that B(σ) = c - Φ[σ/ 2 ]. However, B() = /2. /2 = c - /2. c =. B(σ) = - Φ[σ/ 2 ]. 9 Thus the Gini Coefficient is: - 2B = 2Φ[σ/ 2 ] -. 8 Where I have used the fact that the density of the Standard Normal integrates to one over its support from - to. 9 In general, Φ[a + by] φ[y] dy = Φ[a / + b2 ). - For a list of similar integrals, see

15 Gini Coefficient, HCM /2/7, Page 4 Proof of the Relationship Between the Gini Inde and the Mean Relative Difference: I will prove that: Gini Coefficient = (mean relative difference) / 2. As a first step, let us look at a graph of the Lorenz Curve with areas labeled: % ofincome..8.6 C.4 A.2 B % ofpeople A + B = /2 = C. B is the area on the Lorenz curve: G df. Area B is the area between the Lorenz curve and the horizontal ais. We can instead look at: C + A = area between the Lorenz curve and the vertical ais = F dg. Therefore, we have that: Area A = (/2) { F dg - G df}. F dg - G df = C + A - B = /2 + A - (/2 - A) = 2 A. Gini Coefficient = Area A Area A + Area B = 2A = F dg - G df. Based on Section 2.25 of Volume I of Kendallʼs Advanced Theory of Statistics, not on the syllabus.

16 Gini Coefficient, HCM /2/7, Page 5 Recall that G() = y f(y) dy. dg = f() d. Therefore, Gini Coefficient = F dg - G df = F(s) s f(s) ds - G(s) f(s) ds = s s f(t) dt f(s) ds - s t f(t) dt f(s) ds = s (s - t) f(t) dt f(s) ds. s (s - t) f(t) dt f(s) ds is the contribution to the mean distance from when s > t. By symmetry it is equal to the contribution to the mean distance from when t > s. s Therefore, 2 (s - t) f(t) dt f(s) ds = mean distance. Gini Coefficient = (mean difference) / 2 = (mean relative difference) / 2.

17 Gini Coefficient, HCM /2/7, Page 6 An Income Eample: Here is a Lorenz Curve for United States 24 Household Income: The Gini inde is calculated as twice the area between the Lorenz curve and the line of equality. In this case, the Gini inde is 48.%. See Figure 2 of of Generalized Linear Models for Insurance Rating, by Mark Goldburd, Anand Khare, and Dan Tevet, CAS monograph series number 5.

18 Gini Coefficient, HCM /2/7, Page 7 Gini Coefficient and Rating Plans: 2 The Gini Coefficient can also be used to measure the lift of an insurance rating plan by quantifying its ability to segment the population into the best and worst risks. Assume we have a rating plan. Ideally we would want the model to identify those insureds with higher epected pure premiums. The Lorenz curve for the rating plan is determined as follows:. Sort the dataset based on the model predicted loss cost On the -ais, plot the cumulative percentage of eposures. 3. On the y-ais, plot the cumulative percentage of losses. Draw a 45-degree line connecting (, ) and (, ), called the line of equality. Here is an eample: 4 This model identified 6% of eposures which contribute only 2% of the total losses. The Gini Coefficient is twice the area between the Lorenz curve and the line of equality, in this case 56.%. The higher the Gini Coefficient, the better the model is at identifying risk differences. 5 2 See Section of Generalized Linear Models for Insurance Rating, by Mark Goldburd, Anand Khare, and Dan Tevet, CAS monograph series number 5. 3 This should be done on a dataset not used to develop the rating plan. 4 See Figure 2 of Goldburd, Khare, and Tevet. 5 A Gini Coefficient does not quantify the profitability of a particular rating plan, but it does quantify the ability of the rating plan to differentiate the best and worst risks. Assuming that an insurer has pricing and/or underwriting fleibility, this will lead to increased profitability.

19 Gini Coefficient, HCM /2/7, Page 8 Problems:, (5 points) The distribution of incomes follows a Single Parameter Pareto Distribution, α >. a. (3 points) Determine the mean relative distance. b. (3 points) Determine the form of the Lorenz curve. c. (3 points) With the aid of a computer, draw and compare the Lorenz curves for α =.5 and α = 3. d. (3 points) Use the form of the Lorenz curve to compute the Gini coefficient. e. (3 point) If the Gini coefficient is.47, what percent of total income is earned by the top % of earners? 2. (5 points) For a Gamma Distribution with α = 2, determine the mean relative distance. Hint: Calculate the contribution to the mean difference from when < y. e - / θ d = - e -/θ θ - e -/θ θ 2. 2 e - / θ d = - 2 e -/θ θ - 2 e -/θ θ 2-2e -/θ θ 3.

20 Gini Coefficient, HCM /2/7, Page 9 Solutions to Problems:. a. f() = α θα α +, > θ. The contribution to the relative difference from when > y is: y ( - y) α θα α θα α + d y α + dy = α 2 θ 2α θ α -y α y α y α yα + dy = θ α 2 θ 2α α - - α y 2 α dy = θ 2α θ α α - (2α - ) θ 2α - = α (α -) (2α - ) θ. By symmetry this is equal to the contribution to the mean distance from when y >. α Therefore, the mean distance is: 2 (α -) (2α - ) θ. = α θ α -, α >. Therefore, the mean relative difference is: 2 2α -, α >. b. G() = θ y f(y) dy = θ y α θα y α + dy α θ α - = (α-) θ α y α dy = - θα α, > θ. θ Now let t = F() = - θα Then G(t) = - (-t) -/α, t. α, > θ. θ/ = (-t)/α.

21 Gini Coefficient, HCM /2/7, Page 2 c. For α =.5, G(t) = - (-t) /3, t. For α = 3, G(t) = - (-t) 2/3, t. Here is a graph of these two Lorenz curves: % ofincome alpha = 3 alpha = % ofpeople The Lorenz curve for α =.5 is below that for α = 3. The incomes are more concentrated for α =.5 than for α = 3. d. The Lorenz curve is: G(t) = - (-t) -/α, t. Integrating, the area under the Lorenz curve is: B = - /(2 - /α) = - α/(2α-) = (α-)/(2α-). Gini coefficient is: - 2B = - 2(α-)/(2α-) = 2α -, α >. Note that the Gini Coefficient = (mean relative difference) / 2 = (/2) 2 2α - = 2α -.

22 e..47 =. α =.564. = θ α / (α-) = θ. 2α - The 99th percentile is: θ ( -.99) -/.564 = 9. θ. The income earned by the top % is:.564 θ d = (.564/.564) θ.564 / (9θ).564 =.527 θ. 9θ Thus the percentage of total income earned by the top % is:.527 θ / (2.773θ) = 9.%. Comment: The mean relative distance and the Gini coefficient have the same form as for the two-parameter Pareto Distribution. The distribution of incomes in the United States has a Gini coefficient of about.47. For a sample of size two from a Single Parameter Pareto Distribution with α >, it turns out that: E[Min] = 2α θ 2α -. E[Ma] = Therefore, the mean difference is: 2α 2 θ (α - )(2α - ). 2α 2 θ (α - )(2α - ) - 2α θ 2α - = 2θ Since, = α θ α -, the mean relative distance is 2 2α -. Gini Coefficient, HCM /2/7, Page 2 α (α -) (2α - ).

23 Gini Coefficient, HCM /2/7, Page f() = α - e - / θ θ α Γ(α) = e -/θ / θ 2. The contribution to the relative difference from when < y is: (/θ 4 ) (y - ) e - / θ d y e - y / θ dy = (/θ 4 ) y e - / θ d - 2 e - / θ d y e- y / θ dy y y y = (/θ 4 ) {y(-ye - y / θ θ - e - y / θ θ 2 ) + y 2 e - y / θ θ + 2ye - y / θ θ 2 + 2e - y / θ θ 3 } y e - y / θ dy = (/θ 4 ) y 2 e - 2y / θ θ 2 + 2y e - 2y / θ θ 3 dy = (/θ 4 ) {θ 2 2(θ/2) 3 + 2θ 3 (θ/2) 2 } = θ 3/4. By symmetry this is equal to the contribution to the mean distance from when > y. Therefore, the mean distance is: θ 3/2. = αθ = 2 θ. Therefore, the mean relative difference is: θ 3 / 2 2θ = 3/4. Comment: The Gini Coefficient is half the mean relative difference or 3/8. One can show in general that for the Gamma the mean relative distance is 2-4 β(α+, α; /2). Then in turn it can be shown that for alpha integer, the mean relative distance is: 2α α / 2 2α-. For eample, for α = 4, the mean relative difference is: 8 4 / 2 7 = 7/28 = 35/64. The Gini Coefficient is half the mean relative difference, and is graphed below as a function of alpha: Gini Coef alpha