第Ⅰ講貨幣と金融

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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
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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
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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
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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
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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
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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＝−dC２／dC１
＝(∂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＝−dC２／dC１＝(∂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
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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）
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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 Y2, 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, present income Y１ + future income Y2 (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
６-2． Inter-temporal Budget Constraint : Consumption Possibility9
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 Y2 (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
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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
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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
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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 OY１AY２ to the trapezoid OY１AB ⇒ 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OY１AY２から台形OY１AB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 OY１AB to the
trapezoid OY１AC ⇒ 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OY１AB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 OY１AC 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OY１ACから三角形台形
OZCへ拡大⇒効用水準をAからFへ高める
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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 = x
the sum of the rates of return = S = Σ i=0nxi
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=0n(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=0nxi
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=0n(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
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10-2. Safe Asset and Risky Asset
安全資産と危険資産
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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
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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
12. Mean-variance and Risk Attitude 平均・分散とリスク態度
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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
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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 x1, expensive one x2,
winning probability of x1 be p1, winning probability of x2 be p2, (p１＋p２＝1)
utility obtained from the prize X is u＝u(X)
expected utility in the case of winning is v(X)＝p1u(x1)＋p2u(x2)
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)＝p１u(x１)＋p２u(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
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a consumer who gets larger utility u (X) obtained by a certain prize X than an
^
expected utility v (X) in case of winning
(x)
^ A, u(x２)…Point B, v(X)…Point V, u …Point U
(x)
u(x１)…Point
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点、
u(x１)…A点、u(x２)…B点、v(X)…V点、u(x)
^ ⇒V点よりU点を選好
v(X)＜u (x)
⇒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
^ ⇒prefer point V than point U ⇒risk lover
v(X)＞u (x)
γ ＝risk premium
＝ premium that he may pay if he can buy a lottery
^ ⇒U点よりV点を選好⇒危険愛好者
v(X)＞u(x)
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
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14-3. Expected Utility and Risk Preference
期待効用とリスク選好
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^ ⇒indifferent between U and V
v(X)＝u(x)
⇒ risk neutral
^ ⇒U点でもV点でも無差別
v(X)＝u(x)
⇒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)|
= |2nd derivative of utility/1st derivative of utility |
Relative risk aversion = |Xu''(X)/u '(X)|
= |X·2nd derivative of utility/1st 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｜
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15. Mean-variance Approach 平均・分散接近
the rate of return on risky assets be i1 in boom and i2 in recession
Probability of boom and recession be p1 and p2
Average rate of return u = p１i１＋p２i２
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＋2abρσＡσＢ
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＝p１i１＋p２i２、
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
収益率の分散σ２＝a２σＡ２＋b２σＢ２＋2abρσＡσＢ
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
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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 v2 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
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(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
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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
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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

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