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1 Part 2 Microeconomic Analysis of Finance     金融のミクロ分析 Chapter 4 Household Finance        家計の金融 Naotsugu 1 Part 2 Microeconomic Analysis of Finance     金融のミクロ分析 Chapter 4 Household Finance        家計の金融 Naotsugu HAYASHI 林 直嗣 Professor of Economics 経済学教授 Faculty of Business Administration 経営学部 Hosei University 法政大学

2 1.Flow of Funds of Household  家計の資金循環 ①sources of funds= income (rewards to production 2 1.Flow of Funds of Household  家計の資金循環 ①sources of funds= income (rewards to production factors such as labor, capital and land) + increase in financial debt + sales of financial assets 資金源泉=所得(賃金、利子、地代等の要素報酬) +金融負債の増加+金融資産の売却 ②ways to spend funds = consumption + real investment + increase in financial assets + repaying of financial liabilities 資金使途=消費+実物投資+金融資産の増加+金融負債の返済 Since ①=②, a flow of funds equation for household is income−consumption=real investment+net increase in financial assets =real investment+financial investment = investment 家計の資金循環式は  所得−消費=実物投資+金融資産の増加−金融負債の増加      =実物投資+金融資産の純増加=実物投資+金融投資 = 投資

3 2.Definition of Savings 貯蓄の定義   savings = income – consumption = a residual 3 2.Definition of Savings 貯蓄の定義   savings = income – consumption = a residual of the income that was not consumed ⇒ investment = real investment + financial investment  貯蓄=所得−消費=所得のうち消費しなかった残余 ⇒投資=実物投資+金融投資 consumption/income + savings/income = average propensity to consume + average propensity to save = 1 消費/所得+貯蓄/所得=平均消費性向+平均貯蓄性向=1 additional consumption/additional income + additional savings/additional income = marginal propensity to consume + marginal propensity to save = 1 追加消費/追加所得+追加貯蓄/追加所得 =限界消費性向+限界貯蓄性向=1  

4 3.Savings and Portfolio Selection 貯蓄と資産選択   Portfolio Selection =a judgment of asset selection 4 3.Savings and Portfolio Selection 貯蓄と資産選択   Portfolio Selection =a judgment of asset selection whether I have savings in the form of cash, deposits, stocks and other financial assets or real assets  資産選択=貯蓄を現金、預金、株式等の金融資産で保有するか、      実物資産に投資するかという判断 Income – consumption = savings = real investment + financial investment 所得−消費=貯蓄=実物投資+金融資産の純増加(金融投資) If savings = real investment, then financial assets do not increase. 貯蓄=実物投資ならば、金融資産は増加しない Surplus unit … savings>real investment ⇒ net increase in financial asset Deficit unit … savings<real investment ⇒ net increase in financial liabilities   黒字主体…貯蓄>実物投資 ⇒ 金融資産の純増加 赤字主体…貯蓄<実物投資 ⇒ 金融負債の純増加

5 4 -1. Indifference Curve of Consumption and Savings (Future Consumption) 消費と貯蓄の無差別曲線 utility function 5 4 -1. Indifference Curve of Consumption and Savings (Future Consumption) 消費と貯蓄の無差別曲線 utility function U=U(C1, C2) … 2 -period model of present and future times 効用関数  U=U(C1, C2) …現在と将来の2期間モデル indifference curve =the combinations of present and future consumption goods that have the same level of utility 無差別曲線=現在消費C1と将来消費=貯蓄C2の 組合わせで、同じ効用水準をもたらす組み合わせを表す曲線 Indifference map = a group or map of indifference curves 無差別曲線群(indifference map)=無差別曲線の一群

6 4 -2. Indifference Curve of Consumption and Savings (Future Consumption) 消費と貯蓄の無差別曲線 Marginal Rate 6 4 -2. Indifference Curve of Consumption and Savings (Future Consumption) 消費と貯蓄の無差別曲線 Marginal Rate of Substitution; MRS = The ratio of increased future goods against decreased present goods to maintain the same level of utility MRS=−d. C2/d. C1 =(∂U/∂C1)/(∂U/∂C2) = marginal utility of present consumption / marginal utility of future consumption ⇔ a slope of the tangent line of indifference curve 限界代替率 =現在消費C1を1単位減らすときに、同じ効用水準に  とどまるために、増やさなければならない将来消費C2の割合 MRS=−d. C2/d. C1 =(∂U/∂C1)/(∂U/∂C2)  =現在消費の限界効用/将来消費の限界効用  ⇔ 無差別曲線の接線の勾配

7 5.Time Preference 時間選好 Time Preference = a tendency to prefer to consume goods 7 5.Time Preference 時間選好 Time Preference = a tendency to prefer to consume goods at present rather than in the future. It is because the quality of goods tends to deteriorate, a shortage of goods may happen and other uncertain matter may occur in the future 時間選好=同じ財でも将来より現在消費 することを好む傾向、消費を将来に回すと、  待忍の必要、品質低下、入手困難  など不確実なことが起こる Marginal Rate of Time Preference; MRTP = the marginal rate of substitution on the point C minus unity 限界時間選好率=原点からの 45度線と無差別曲線との交点のMRS−1 Marginal Rate of Time Discounting; MRTD = the marginal rate of substitution between present and future consumptions - 1 限界時差割引率=現在消費と将来消費との限界代替率−1

6 -1. Inter-temporal Budget Constraint : Consumption Possibility Set 異時点間の予算制約(消費可能集合) 8 ①When saving is 6 -1. Inter-temporal Budget Constraint : Consumption Possibility Set 異時点間の予算制約(消費可能集合) 8 ①When saving is impossible Present ConsumptionC1 ≦Present Income Y1 、 Future ConsumptionC2 ≦Future Income Y2 。Consumption Possibility Set = rectangle that is enclosed by the Origin O, present income Y1 , future income Y 2, and the point A.  貯蓄がない場合の予算制約 現在消費≦現在所得、将来消費≦将来所得   原点Oと現在所得Y1、将来所得Y2、及びA点で囲まれる四角形 ② When saving is possible Present ConsumptionC1 ≦Present Income Y1 、 Future ConsumptionC2 ≦(Present Income-Savings) +Future Income Y2 。 Consumption Possibility Set = trapezoid that is enclosed by the Origin O, present income Y1 + future income Y 2 (point B), and the point A  貯蓄がある場合の予算制約 現在消費≦現在所得、 将来消費≦(現在所得-貯蓄)+将来所得 原点Oと現在所得Y1、 将来所得Y2+現在所得Y1 (B点)、 及びA点で囲まれる台形

9 6 -2. Inter-temporal Budget Constraint : Consumption Possibility Set 異時点間の予算制約(消費可能集合) ③ When lending 9 6 -2. Inter-temporal Budget Constraint : Consumption Possibility Set 異時点間の予算制約(消費可能集合) ③ When lending is possible Present ConsumptionC1 ≦Present Income Y1 、 Future ConsumptionC2 ≦(Present Income-Savings)(1+Interest Rate r) +Future Income Y2 。Consumption Possibility Set = trapezoid that is enclosed by the Origin O, present income Y1, (1+Interest Rate r) present income Y1 + future income Y 2 (point C), and the point A  Market Interest Rate = Discounting Rate for Calculating Present Value 貸付が可能な場合の予算制約 現在消費C1 ≦現在所得Y1 、将来消費C2 ≦( 現在所得-貯蓄)(1+利子率 r) +将来所得 Y2  原点Oと現在所得Y1、将来所得Y2+現在所得Y1×(1+利子率r)、及びA点で 囲まれる台形 ∴市場利子率r=現在価値を求める割引率

10 6 -3. Inter-temporal Budget Constraint : Consumption Possibility Set 異時点間の予算制約(消費可能集合) ④ When lending 10 6 -3. Inter-temporal Budget Constraint : Consumption Possibility Set 異時点間の予算制約(消費可能集合) ④ When lending and borrowing are possible (1+Interest Rate r)( Present Income Y1− Present Consumption C1)+Future Income Y2=Future Consumption C2 ⇔  C1+C2/(1+r)=Y1+Y2/(1+r) Present Consumption + Discounted value of Present Consumption = Future Income + Discounted value of Future Income Consumption Possibility Set = triangle that is enclosed by the Origin O, the point Z, and the point C 金融が(貸付も借入も)可能な場合の予算制約  (1+r)(Y1−C1)+Y2=C2 ⇔ C1+C2/(1+r)=Y1+Y2/(1+r)  現在消費+将来消費の割引現在価値=現在所得+将来所得の割引現在価値  原点Oと現在所得Y1×(1−利子率r)、将来所得Y2+現在所得Y1×(1+利子率 r) で囲まれる三角形

7.Optimal Decision of Consumption and Savings 消費と貯蓄の最適決定 11 Utility Maximization under a budget constraint 7.Optimal Decision of Consumption and Savings 消費と貯蓄の最適決定 11 Utility Maximization under a budget constraint ⇒to select the optimal combination of present and future consumption goods ⇒choose a combination of goods at the point where the budget line comes into contact with the indifference curve ⇒ the slope of the indifference curve MRS = the slope of the budget line (1+r) = the marginal rate of time discount MRDT + 1 ∴MRDT = the market rate of interest 予算制約のもとで効用最大化 ⇒現在消費と将来消費との最適な組み合わせを選択 ⇒予算制約線と無差別曲線との接点で最適決定 ⇒ 無差別曲線の勾配MRS = 予算制約線の勾配(1+r) =(限界時差割引率 MRTD+1) ∴ MRTD=市場利子率

8.The Role of Money and Financial markets 貨幣と金融市場の役割 12 ① Function as a store 8.The Role of Money and Financial markets 貨幣と金融市場の役割 12 ① Function as a store of value ⇒to play a role in carrying over the value of present income into the future ⇒ to enlarge a consumption possibility set from the rectangle OY1 AY2 to the trapezoid OY1 AB ⇒ to enhance the utility level from A to E 貨幣の価値貯蔵機能⇒現在所得の価値を将来に持ち越す⇒消費可能集合を 四角形OY1 AY2から台形OY1 ABへ拡大⇒効用水準をAからEへ高める ②When lending is possible in the financial market ⇒to enlarge a consumption possibility set from the trapezoid OY1 AB to the trapezoid OY1 AC ⇒ to enhance the utility level from E to E' 金融市場で貸付が可能な場合⇒消費可能集合を台形OY1 ABから台形OY1 ACへ拡大⇒効用水準をEからE'へ高める ③ When lending is possible ⇒to enlarge a consumption possibility set from the trapezoid OY1 AC to the triangle OZC ⇒ to enhance the utility level from A to F 金融市場で借入も可能な場合⇒消費可能集合を台形OY1 ACから三角形台形 OZCへ拡大⇒効用水準をAからFへ高める

13 9. Interest Income and Capital Gains 利子所得と資本利得 Revenue of or return on financial 13 9. Interest Income and Capital Gains 利子所得と資本利得 Revenue of or return on financial assets = interest income (dividend income in case of stock) + capital gain Capital gain = sale price – purchase price Rate of return=interest rate+rate of capital gain  資産の保有・運用による 収益=利子所得(株式では配当所得)+資本利得   資本利得 =売却価格−取得価格 収益率=利子率+(売却価格−取得価格)/取得価格

10 -1. Standard Deviation as a Measure of Risk 危険の指標としての標準偏差 a rate of return 10 -1. Standard Deviation as a Measure of Risk 危険の指標としての標準偏差 a rate of return = x the sum of the rates of return = S = Σ i=0 nxi the mean of the rates of return = m = S / n the deviation of the rate of return = D = xi – m the sum of squared deviations = SSD = Σ i=0 n(xi - m) the variance = the mean of squared deviations = σ2 = SSD/ n the standard deviation = the root of the variance = σ = √σ2 = an average size of risk. 収益率=x   収益率の総和= S=Σ i=0 nxi   収益率の平均= m = S / n  各収益率の偏差= D=xi -m 各収益率の偏差平方和= SSD = Σ i=0 n(xi -m).  分散=σ2=SSD/n    分散の平方根=標準偏差=σ = √σ2 …リスクの平均的な大きさ 14

10 -2. Safe Asset and Risky Asset 安全資産と危険資産 15 Safe Asset = asset whose 10 -2. Safe Asset and Risky Asset 安全資産と危険資産 15 Safe Asset = asset whose earnings are certain (ex. deposits and fixedinterest-bearing securities) and whose standard deviation is zero Risky Asset = asset whose earnings are not certain (ex. stocks, mutual funds (investment trust)) and whose standard deviation is not zero 安全資産=収益が確定している資産(例:預金や確定利付証券)       標準偏差はゼロ 危険資産=収益が不確定な資産(例:株式、投資信託)       標準偏差はゼロでない

11. Marginal Utility and Risk Attitude   限界効用とリスク態度 16 Assets held increase by one unit 11. Marginal Utility and Risk Attitude   限界効用とリスク態度 16 Assets held increase by one unit ⇒ a variance or standard deviation tends to increase risk-averter = a consumer who experiences an additional decrease in incremental utility or a decrease in marginal utility when assets held increase risk-lover = a consumer who experiences an increase in marginal utility when assets held increase risk-neutral = a consumer who experiences a constant marginal utility when assets held increase 保有資産が1単位増加 ⇒ 分散ないし標準偏差が増加する傾向 危険回避者=保有資産が1単位増加するとき、限界効用が逓減する人 危険愛好者=保有資産が1単位増加するとき、限界効用が逓増する人 危険中立者=保有資産が1単位増加するとき、限界効用が不変の人

17 12. Mean-variance and Risk Attitude 平均・分散とリスク態度 Assets held increase by one unit ⇒ 17 12. Mean-variance and Risk Attitude 平均・分散とリスク態度 Assets held increase by one unit ⇒ a variance or standard deviation (risk) tends to increase risk-averter = a consumer who requires higher rate of return when risk increases by one unit risk-lover = a consumer who allows lower rate of return when risk increases risk-neutral = a consumer who do not care a constant rate of return when risk increases 保有資産が1単位増加 ⇒ 分散ないし標準偏差(危険)が増加する傾向 危険回避者=リスクが1単位増加するとき、平均収益も増えなければいけない人 危険愛好者=リスクが1単位増加するとき、平均収益は低くなっても良い人 危険中立者=リスクが1単位増加 するとき、平均収益が不変の人  

13. Expected Utility Theory 期待効用理論 18 a contract that has different conditions depending upon 13. Expected Utility Theory 期待効用理論 18 a contract that has different conditions depending upon uncertainties of the situation = contingent contract the goods whose conditions of transaction are different depending upon uncertainties of the situation = contingent goods  不確実な事態に応じて取引条件が違う契約= 条件付き契約  不確実な事態に応じて取引条件が違う財=条件付き財 Prize of lottery = inexpensive one x 1, expensive one x 2, winning probability of x 1 be p 1, winning probability of x 2 be p 2, (p1+p2= 1) utility obtained from the prize X is u=u(X) expected utility in the case of winning is v(X)=p 1 u(x 1)+p 2 u(x 2) von Neumann - Morgenstern theory of expected utility maximization Under uncertainty, people maximize an expected utility v(X) not utility u(X)  宝籤の賞金=低額賞金はx1、高額賞金はx2、当選する確率=低額がp1、高額がp2 、p1+p2=1、賞金xから得られる効用 u=u(X)  当選の場合の期待効用 v(X)=p1 u(x1)+p2 u(x2) ⇒ノイマン=モルゲンシュテルンの期待効用最大化説   不確実性の下では効用ではなく期待効用を最大化する

14 -1. Expected Utility and Risk Preference  期待効用とリスク選好 19 a consumer who gets larger 14 -1. Expected Utility and Risk Preference  期待効用とリスク選好 19 a consumer who gets larger utility u (X) obtained by a certain prize X than an expected utility v (X) in case of winning u(x1)…Point A, u(x2)…Point B, v(X)…Point V, u …Point U v(X)<u ⇒prefer point U than point V ⇒risk averter γ =insurance premium = premium that he may pay instead of not buying a lottery = negative risk premium u(x1)…A点、u(x2)…B点、v(X)…V点、u  …U点、 v(X)<u    ⇒V点よりU点を選好 ⇒危険回避者 γ=保険プレミアム =宝籤に参加しない代わりに  支払って良いプレミアム =負の危険プレミアム

14 -2. Expected Utility and Risk Preference  期待効用とリスク選好 v(X)>u    ⇒prefer point V than 14 -2. Expected Utility and Risk Preference  期待効用とリスク選好 v(X)>u    ⇒prefer point V than point U ⇒risk lover γ =risk premium = premium that he may pay if he can buy a lottery v(X)>u    ⇒U点よりV点を選好⇒危険愛好者 γ=危険プレミアム =宝籤に参加できるなら 支払って良いプレミアム 20

14 -3. Expected Utility and Risk Preference  期待効用とリスク選好 21 v(X)=u    ⇒indifferent between U 14 -3. Expected Utility and Risk Preference  期待効用とリスク選好 21 v(X)=u    ⇒indifferent between U and V ⇒ risk neutral v(X)=u    ⇒U点でもV点でも無差別 ⇒危険中立的 Arrow – Pratt’s degree of risk-aversion by using derivatives of utility function. Absolute risk aversion = |u (X)''/u '(X)| = |2 nd derivative of utility/1 st derivative of utility | Relative risk aversion = |Xu''(X)/u '(X)| = |X· 2 nd derivative of utility/1 st derivative of utility | アロー=プラットの危険回避度 絶対的危険回避度=|u”(x)/u’(x)|=|効用の2次微分/効用の1次微分| 相対的危険回避度=|xu”(x)/u’(x)|=|x・効用の2次微分/効用の1次微分|

15. Mean-variance Approach 平均・分散接近 22 the rate of return on risky assets be i 1 15. Mean-variance Approach 平均・分散接近 22 the rate of return on risky assets be i 1 in boom and i 2 in recession Probability of boom and recession be p 1 and p 2 Average rate of return u = p1 i1+p2 i2 Variance v2=p1 (i1−u)2+p2 ( i2−u)2 Average rate of return on risky assets A and B = uA, uB Holding ratio of them = a, b ( = 1 – b ) Average rate of return the two assets μ = auA+buB Variance of rates of return σ2=a2σA2+b2σB2+2 abρσAσB Where ρ = correlation coefficient, σA = standard deviation of A, σB = standard deviation of B  危険資産の収益率=好況時ではi1、不況時ではi2、  起こる確率=好況がp1、不況がp2  平均収益率u=p1 i1+p2 i2、  危険を表す分散v2=p1 (i1−u)2+p2 ( i2−u)2  危険資産AとBの平均収益率uA、uB、それぞれの保有比率a、b(= 1−a)  両者のポートフォリオの平均収益率μ=auA+buB 2 2 2   収益率の分散σ =a σA +b σB +2 abρσAσB ただし、ρ=相関係数、σA=Aの標準偏差、σB=Bの標準偏差

16 -1. Investment Opportunities and Effective Frontier  投資機会と有効フロンティア 23 Effective Frontier = a set 16 -1. Investment Opportunities and Effective Frontier  投資機会と有効フロンティア 23 Effective Frontier = a set of points that bring about a maximum rate of return with the same variance among feasible investment opportunities (1) the average return µ on risky assets R and Q on the vertical axis, its variance v 2 on the horizontal axis and the correlation coefficient = ρ the investment opportunity line of their portfolio is ① when ρ = 1 ⇒ a straight line that connects points R and Q ② When ρ = -1 ⇒ a polygonal line that connects points R, Q and S ③ When -1 <ρ <1 ⇒ a curve that connects points R and Q The upper part of the convex curve = Effective Frontier 有効フロンティア =実行可能な投資機会集合のうちで分散が同じなら最大の収益 率をもたらす点の集合 (1)危険資産RとQの収益率の平均uを縦軸に、 その分散v2を横軸にとった平面で、投資機会線は ①ρ=1の場合⇒RとQを結ぶ直線 ②ρ=-1の場合⇒Rと分散がゼロのS”とQを結ぶ折れ線 ③-1<ρ<1の場合⇒RとQを結ぶ曲線     ⇒この曲線の上方に凸の部分が有効フロンティア

16 -2. Investment Opportunities and Effective Frontier  投資機会と有効フロンティア 24 (2) In the portfolio consisting 16 -2. Investment Opportunities and Effective Frontier  投資機会と有効フロンティア 24 (2) In the portfolio consisting of two risky assets and one safe asset, Effective frontier = a tangent line that connects the point C and the curve between A and B The optimum combination of risky assets is determined by the tangent point M. (2) 2つの危険資産RとQ、および1つの安全資産Sとからなる  ポートフォリオでは、有効フロンティアは 点Sから曲線RQへの接線ST 危険資産の組合せはその接点Tで決まる

17 -1. Optimal Portfolio and Separation Theorem 最適ポートフォリオと分離定理 25 In the portfolio consisting of 17 -1. Optimal Portfolio and Separation Theorem 最適ポートフォリオと分離定理 25 In the portfolio consisting of two risky assets R and Q and one safe asset S, the effective frontier = a tangent line from the point S and to the curve between R and Q ⇒ the optimum combination of risky assets = the tangent point T That determines the optimal portfolio of risky assets  2つの危険資産RとQ、1つの安全資産Sとからなるポートフォリオでは、 有効フロンティア⇒点Sから曲線RQへの接線 ⇒その接点Tで危険資産だけの 最適ポートフォリオが決まる

17 -2. Optimal Portfolio and Separation Theorem 最適ポートフォリオと分離定理 26 The indifference curve of a 17 -2. Optimal Portfolio and Separation Theorem 最適ポートフォリオと分離定理 26 The indifference curve of a risk-averter is tangent to a point E and E is the optimal portfolio of the whole The optimal point T of risky assets is determined independently from the determination of the whole optimal point between safe assets and risky assets ⇔ Tobin's separation theorem  危険回避型の消費者の無差別曲線はその接線上の点Eで接する  ⇒E点が全体の最適ポートフォリオ ∴危険資産の最適点Tは安全資産と危険資産の全体の   最適点Eの決定とは独立  ⇔トービンの分離定理