Notes on Tobin s. Liquidity Preference as Behavior toward Risk
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1 otes on Tobin s Liquidity Preference s Behvior towrd Risk By Richrd McMinn Revised June 987 Revised subsequently
2 Tobin (Tobin 958 considers portfolio model in which there is one sfe nd one risky sset. The net rte of return is X on the sfe sset nd X on the risky sset. A portfolio consists of dollr mount invested in the sfe sset nd n mount invested in the risky sset where + = w nd w is the investor s initil welth. Let be rndom vrible denoting the return on the portfolio. Then ( ( = + x + + x ( ( = w + x + x x ( nd E μ ( ( = w + x + μ x Vr ( = Vr x = The mount determines both μ nd. The terms on which n investor cn obtin greter expected return t the expense of more risk is μ x μ = w+ + ( x (3 where w. (3 is the investor s trding line. The investor holds the mount = / of the risky sset. Let = λ w. Then μ x μ w = wx + λ w μ w μ x = x + λ w (4 This is clled the cpitl mrket line where λ is the stndrd devition per unit of μ x welth held in the risky sset nd is sometimes clled mrket price of risk.
3 The investor is ssumed to hve preferences which rnk ll (, μ pirs nd these preferences cn be represented by indifference curves. Then the individul mximizes expected utility subect to selecting pir (, μ on the trding line. There re two rtionles for supposing tht ( ( Eμ = f μ, (5 The first is tht ( μ, Let g(y; μ, be the density function of. Then ( ( ( = f ( μ, Eμ = μ y g y; μ, dy (6 nd the shpe of the indifference curve cn be inferred from the shpe of μ. To demonstrte this we perform the following trnsformtion. Let ( = Φ z μ + z μ z =φ( (7 i.e. φ=φ. Then
4 ( ( ( = μ( Φ( z ( = f ( μ, Eμ = Eμ Φ z h z;, dz (8 where z (, nd h is the stndrd norml density. Let Df nd Df denote the prtil derivtives of f with respect to μ nd respectively. Then Df = μ'h Df = μ'zh (9 Since f is constnt on the indifference curve or equivlently = df = Dfdμ + Dfd ( d d μ = Df D f = ( Φ ( ( ( Φ ( ( zu' z h z u' z h z ( Clim: if u > nd u < then dμ d >. Proof. Since the denomintor is clerly positive, it suffices to show tht u < implies ( ( ( zu' Φ z h z < ( ote tht { } (3 = lim zu'h + zu'h where h is symmetric, i.e. h(t = h(-t. Let z = -t. Then nd ( ( ( zu' Φ z h z dz = ( ( ( tu' Φ t h t ( dt { ( ( ( ( ( ( } { ( ( ( ( ( } = lim tu' Φ t h t dt + tu' Φ t h t dt = lim t u' Φ t u' Φ t h t < (4 (5 Since u' ( ( t u' ( ( t Φ > Φ.
5 Comprtive Sttistics Suppose tht u is n incresing concve function nd tht X (, μ. ow suppose μ chnges. Then we wnt to determine wht effect this will hve on the investor s optiml choice of. Recll tht ( z z ( ( Φ = μ + = w + x + μ x +z (6 nd the investor s problem is to select condition is to mximize u( Φ ( z h( z ( ( ( (.The first order u' Φ z μ x + z h z = (7 or equivlently μ x = u'zh u'h (8 ow let the function F:D,D be defined by ( μ = [ μ + ] F, u' x z h (9 Since DF <, it follows by the Implicit Function Theorem tht there exists differentible function f: such tht ( ( ote tht f' < > s DF > <. F μ,f μ = nd f = DF DF. { [ ]} D F = u' + u" μ x + z h [ ] = u'h + u" μ x + z h > ( if the second integrl on the RHS is non-negtive. To determine the sign let u'' u' denote the mesure of bsolute risk version. Assume tht ' nd let z be implicitly defined by Then μ x + z = (i.e. z μ x =. ( Φ ( z ( Φ ( z u'' ( Φ( z,z z u' ( or
6 ( ( ( ( ( ( u'' Φ z Φ z u' Φ z ( which in turn yields ( ( ( ( ( ( ( ( u'' Φ z μ x + z Φ z u' Φ z μ x + z (3 for z z. A similr rgument shows tht (3 holds for z < z. Hence, given ', u'' ( Φ( z ( μ x + z h ( Φ( z u' ( Φ( z ( μ x z + h (4 but the RHS is zero t the optiml. Hence, DF f' μ >. ote tht in Tobin s model (i.e. where x nd so the sfe sset is money f' ( μ > lso implies tht the demnd for money is decresing function of the expected return on the risky sset, which he clls the rte of interest. or equivlently ( ext consider chnge in the riskness of sset one. Let defined s G:D,D, be ( = ( Φ( ( μ + G, u' z x z h (5 Then s bove there exist function g such tht ( ( s DG > <. DG G,g = nd g' = < > DG ( DG = u'z+u''z μ x + z h ( = u'zh + u''z μ x + z h (6 Recll tht the first integrl on the RHS is negtive nd so DG < if the second integrl on the RHS is non-positive. Since u''z( μ x + z h = u'' z ( μ x + z h = u'' ( μ x + z ( μ x ( μ x + z h = u'' ( μ x + z h ( μ x u'' ( μ x +z h (7 It follows tht ' nd μ x suffice to show DG <.
7 Model with Mny Risky Assets Suppose there re risky ssets. Let (,..., = where,,..., = is the dollr mount held of risky sset nd is defined s before. Let x denote the rndom rte of return on sset. Then ( ( = + x + + x ( ( = w + x + x x (8 since w = Let E( x x Then nd = Ex Ex i i i ( ( μ = w + x + μ x (9 = Vr x = i= i i (3 In the specil cse = Vr ( x x E( x x ( E( ( x ( x ( ( ( ( + = + μ + μ = μ + μ = E x μ + E x μ x μ + E x μ = + + The set of points (3 for which μ is constnt is hyperplne. In the cse = { } the w ( + x +( μ ex =μ where e = (,..., nd (,..., shown μ = μ μ is
8 Let denote the vrince-covrince mtrix. Then T =. For = the { ( ( w+ x + μ ex =μ } where e = (,..., nd μ = ( μ,..., μ is shown Let denote the vrince-covrince mtrix. Then T =. For = = (3 ote is rel symmetric mtrix nd so there exists n orthogonl mtrix U such T tht U U is digonl mtrix whose digonl elements re the chrcteristic roots of. Since is rel symmetric mtrix, ll its chrcteristic roots re T n rel. Let U U = D( λ λ is the vector of chrcteristic roots, e. g. where Let D λ λ = ( = Uδ. Then we obtin the eqution λ (33 U U D( (34 = δ δ =δ λ δ = T T T T In sclr form we hve the fmilir eqution (i.e. for the liner opertor in We interpret the trnsformtion so the grph of λδ +λδ = (35 = Uδs trnsformtion of the coordinte system
9 T = (36 is not chnged. However the ltertion mkes the eqution so simple tht the nture of its grph my be determined by inspection. If λ, λ, nd then δ δ + = λ λ / / (37 If / λ > for, then (38 is the eqution of n ellipse. If λ =λ then (39 is circle. If λ nd λ re opposite in sign then (4 is hyperbol. n The chrcteristic roots λ of re ll positive if nd only if the qudrtic T form is positive definite nd the qudrtic form is positive definite if nd only if the leding principl minors of the mtrix of the form re ll positive, i.e. 3 K n >, >, 3 >,..., M O M > n L nn The dominnt re those which minimize subect to fixed vlue of μ. Hence, we hve the problem Minimize T Subect to w ( + x + ( μ + ex = μ
10 Or the LGrnge function ( T L ( δ, = δ w( + x +( μ ex μ (4 The first order conditions re ( ( ( ( DL= + δ μ x = DL= + δ μ x = DL= w + x + μ ex μ = δ (4 In mtrix nottion the first two conditions my be rewritten s = δ( μ ex (43 These conditions my s be expressed s + μ x = + μ x (44 which is the fmilir tngency condition. The LHS is the slope of the ellipse nd the T RHS is the slope of the constnt men line. Since f ( = is homogeneous of degree two the set of points which stisfy (45 or (46 is the liner function L. This result my lso be noted by observing tht t + t + = t + t + (47 Hence the investor holds the risky ssets in fixed proportions, i.e. = k. Thus we my form composite risky sset by letting x = IxI (48 where
11 x I = x (49 I nd = (5 I defines I. Then since = k we obtin x = x + k x + k + k I (5 Then ( ( = w + x + x x (5 I I nd it follows tht μ x μ = w+ + I ( x I (53 is the trding line. References Tobin, J. (958. "Liquidity Preference s Behvior Towrd Risk." Review of Economic Studies 5(:
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