Naotsugu HAYASHI Professor of Economics Faculty of Business Administration Hosei University

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Flow of Funds of Household sources of funds income (rewards to production factors such as labor, capital and land) + increase in financial debt + sales of financial assets ways to spend funds = consumption + real investment + increase in financial assets + repaying of financial liabilities Since , a flow of funds equation for household isincomeconsumptionreal investmentnet increase in financial assets real investmentfinancial investment = investment = *

Definition of Savings savings income consumption = a residual of the income that was not consumed investment = real investment + financial investment

consumption/income + savings/income = average propensity to consume + average propensity to save = 1

additional consumption/additional income + additional savings/additional income = marginal propensity to consume + marginal propensity to save = 1 *

3Savings and Portfolio Selection 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 Income consumption = savings = real investment + financial investment If savings = real investment, then financial assets do not increase.

Surplus unit savingsreal investment net increase in financial asset Deficit unit savingsreal investment net increase in financial liabilities *

-1 Indifference Curve of Consumption and Savings (Future Consumption) utility function UU(C,C) 2-period model of present and future times UU(C,C)

indifference curve the combinations of present and future consumption goodsthat have the same level of utilityCC

Indifference map = a group or map of indifference curves indifference map*

-2 Indifference Curve of Consumption and Savings (Future Consumption) Marginal Rate of Substitution; MRS = The ratio of increased future goods against decreased present goods to maintain the same level of utility MRSdCdC (UC)(UC)= marginal utility of present consumption / marginal utility of future consumption a slope of the tangent line of indifference curve CCMRSdCdC (UC)(UC)*

Time Preference 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 Marginal Rate of Time Preference; MRTP = the marginal rate of substitution on the point C minus unity 45MRSMarginal Rate of Time Discounting; MRTD = the marginal rate of substitutionbetween present and future consumptions 1*

-1 Inter-temporal Budget Constraint : Consumption Possibility Set When saving is impossible Present ConsumptionPresent Income Y Future Consumption Future Income Y Consumption Possibility Set = rectangle that is enclosed by the Origin O, present income Y , future income Y2, and the point A. OYYA When saving is possible Present ConsumptionPresent Income Y Future Consumption (Present Income-Savings) +Future Income Y Consumption Possibility Set = trapezoid that is enclosed by the Origin O, present income Y, present income Y + future income Y2 (point B), and the point A ()OYYY (B)A*

-2 Inter-temporal Budget Constraint : Consumption Possibility Set When lending is possible Present ConsumptionPresent Income Y Future Consumption (Present Income-Savings)(1+Interest Rate r) +Future Income Y Consumption Possibility Set = trapezoid that is enclosed by the Origin O, present income Y, (1+Interest Rate r) present income Y + future income Y2 (point C), and the point A Market Interest Rate = Discounting Rate for Calculating Present Value Y (-)(1+ r) + Y OYYY(1r)A r

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-3 Inter-temporal Budget Constraint : Consumption Possibility Set When lending and borrowing are possible (1Interest Rate r)( Present Income Y Present Consumption C)Future Income YFuture Consumption C CC(1r)YY(1r) 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 (1r)(YC)YC CC(1r)YY(1r) OY(1r)YY(1r)

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Optimal Decision of Consumption and Savings Utility Maximization under a budget constraintto select the optimal combination of present and future consumption goodschoose 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 + 1MRDT = the market rate of interest MRS (1r)( MRTD1) MRTD=*

The Role of Money and Financial markets 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 OYAY to the trapezoid OYAB to enhance the utility level from A to E OYAYOYABAEWhen lending is possible in the financial market to enlarge a consumption possibility set from the trapezoid OYAB to the trapezoid OYAC to enhance the utility level from E to E' OYABOYACEE' When lending is possible to enlarge a consumption possibility set from the trapezoid OYAC to the triangle OZC to enhance the utility level from A to F OYACOZCAF*

Interest Income and Capital Gains 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 returninterest raterate of capital gain

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10-1. Standard Deviation as a Measure of Risk a rate of return = x the sum of the rates of return = S = 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 = i=0n(xi m) the variance = the mean of squared deviations = 2 = SSDn the standard deviation = the root of the variance = = 2 = an average size of risk. S= i=0nxi m = S / n Dxi m SSD = i=0n(xi m).2SSDn = 2 *

10-2. Safe Asset and Risky Asset

Safe Asset = asset whose earnings are certain (ex. deposits and fixed-interest-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

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11. Marginal Utility and Risk AttitudeAssets held increase by one unit a variance or standard deviation tends to increaserisk-averter = a consumer who experiences an additional decrease in incremental utility or a decrease in marginal utility when assets held increaserisk-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

1 111*

12. Mean-variance and Risk Attitude Assets held increase by one unit a variance or standard deviation (risk) tends to increaserisk-averter = a consumer who requires higher rate of return when risk increases by one unitrisk-lover = a consumer who allows lower rate of return when risk increasesrisk-neutral = a consumer who do not care a constant rate of return when risk increases1 111*

13. Expected Utility Theory a contract that has different conditions depending upon uncertainties of the situation = contingent contractthe goods whose conditions of transaction are different depending upon uncertainties of the situation = contingent goods= Prize of lottery = inexpensive one x1, expensive one x2,winning probability of x1 be p1, winning probability of x2 be p2, (pp1)utility obtained from the prize X is uu(X)expected utility in the case of winning is v(X)p1u(x1)p2u(x2)von Neumann - Morgenstern theory of expected utility maximizationUnder uncertainty, people maximize an expected utility v(X) not utility u(X)xxpp ppx uu(X) v(X)pu(x)pu(x) *

14-1. Expected Utility and Risk Preference 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(x)Point A, u(x)Point B, v(X)Point V, u Point U v(X)u prefer point U than point Vrisk averter insurance premium = premium that he may pay instead of not buying a lottery = negative risk premium

u(x)Au(x)Bv(X)Vu U v(X)u VU*

14-2. Expected Utility and Risk Preference v(X)u prefer point V than point U risk lover risk premium premium that he may pay if he can buy a lottery

v(X)u UV

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14-3. Expected Utility and Risk Preference v(X)u indifferent between U and V risk neutralv(X)u UV

Arrow Pratts 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)| = |X2nd derivative of utility/1st derivative of utility |u(x)/u(x)xu(x)/u(x)x*

15. Mean-variance Approachthe rate of return on risky assets be i1 in boom and i2 in recession Probability of boom and recession be p1 and p2Average rate of return u = pipiVariance vp (iu)p ( iu)Average rate of return on risky assets A and B = u, uHolding ratio of them = a, b ( = 1 b )Average rate of return the two assets = aubuVariance of rates of return ab2abWhere = correlation coefficient, = standard deviation of A, = standard deviation of Biippupipivp (iu)p ( iu)ABuuab(1a) aubuab2ab AB*

16-1. Investment Opportunities and Effective Frontier 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 on risky assets R and Q on the vertical axis, its variance v2 on the horizontal axis and the correlation coefficient = the investment opportunity line of their portfolio is when = 1 a straight line that connects points R and Q When = -1 a polygonal line that connects points R, Q and S When -1

16-2. Investment Opportunities and Effective Frontier (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 BThe optimum combination of risky assets is determined by the tangent point M.

(2) 2RQS SRQST T

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17-1. Optimal Portfolio and Separation Theorem 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 TThat determines the optimal portfolio of risky assets

2RQSSRQ T*

17-2. Optimal Portfolio and Separation Theorem 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

EE T E*