otes on Tobin s Liquidity Preference s Behvior towrd Risk By Richrd McMinn Revised June 987 Revised subsequently
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.
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
( ( ( = 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.
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
( ( ( ( ( ( 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 <.
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
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
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 > 3 3 33 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
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
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(: 65-86.