1.6 Dynamics of Asset Prices*

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1 ESTOLA: THEORY OF MONEY 23 The greater the expectation rs2 e, the higher rate of return the long-term bond must offer to avoid the risk-free arbitrage. The shape of the yield curve thus reflects the risk premium investors require for the greater interest rate risk in long-term bonds. The more uncertain the future interest rates are, the higher the risk premium investors require. 1.6 Dynamics of Asset Prices* Demand of Different Assets In Section 10.4 we studied the equilibrium prices values) of bonds and shares of common stocks of corporations. At moment t, the value of such asset with term to maturity of n time units and discount rate r d 1/ t) is: P t) = N r d [1 1 + r d t) n Z 1 + r d t) n. 1.8) Formula 1.8) consists of the present value of the money flow N / t) the coupon payments of a bond or the dividends of a share), and the repayment of the principal value of a bond or the expected future selling price of a share; these both denoted by Z ). Now we can ask, what makes this equilibrium price to hold, that is, what causes the dynamics of asset prices? We model here the dynamics of asset prices similarly as we have done in other sections. An investor observes the prevailing price of a security, makes his estimate of the present value of future revenues of this asset, and buys one if he expects to gain from this or sells these assets in the opposite case if he has them. In the following we analyze the price dynamics of the share of a common stock of a corporation, and later we show that the model can be applied to other securities too. The net demand of investor i of the shares of common stock of a corporation during time unit t at moment t depends on quantity F is ): F is t) = N i P t), 1.9) where N i,, are the expectations of investor i of these quantities, and n i is the length of the investment horizon of the investor. We can interpret the net present value of future payments of the asset in 1.9) as the willingness-topay of investor i of the asset, and P t) is its price. Thus the decision-making of an investor is analogous to that of a consumer; buy an asset if you are willing to pay its price, and sell it in the opposite case if you have it. We

2 24 ESTOLA: PRINCIPLES OF QUANTITATIVE MICROECONOMICS can interpret quantity F is as the force by which investor i is acts upon the demand of the asset. The net demand D i t) unit/ t) of the shares of investor i is D i t) = f i F is t)), f if is ) > 0, f i 0) = ) The following results can be obtained from 1.10): D i t) = f if is ) ] > 0, N i D i t) = f P if is ) < 0, D i t) f = ig is ) > 0, D i t) = f Ni )n i t r if is ) di 1 + t) N ) i[1 1 + t) ni ], n i+1 rdi ) 2 D i t) = f Ni )ln1 + t) n if is ). i Increases in N i and thus increase the demand. Result D i t)/ P < 0 shows the stability of asset prices; an increase in an asset price decreases its demand, and vice versa. This explains the observed fluctuation in asset prices: once an asset price has increased enough, it starts to decrease and vice versa. Asset prices change according to their net demand, and when the majority of investors believe that an asset has lost its growth potential, they start selling it because they like to invest in assets with the greatest growth potential. An increase in an asset price decreases its internal rate of return, and once an asset price has increased enough with respect to its expected future value Z, it has lost its growth potential and investors start selling it. This explains the similarity between investors and card players who want to exchange bad cards to better ones. If N i <, then D i t)/ < 0 otherwise this result is ambiguous and then also D i t)/ n i < 0. The longer the investment horizon of investor i, the smaller his demand of these assets when N i <. We can model the demand of bonds, consols and T-bills of investor i as special cases of 1.10). For bonds, N i = N, = Z, n i = n for all investors in 1.9), for consols, N i = N, = 0, n i = n =, and for T-bills, N i = N =

3 ESTOLA: THEORY OF MONEY 25 0, = Z and n i = n = 1. These change F is as follows: F ib t) = N [1 1 + t) n F ic t) = N P t), F it t) = Z 1 + t P t), Z 1 + t) n P t), where subscripts B, C, T refer to Bond, Consol, and T-bill, respectively. The net demands of investor i of these assets are then: D ij t) = f ij F ij t)) f ijf ij ) > 0, f ij0) = 0, j = S,..., T. 1.11) In F ij, j = C, T, result D ij / < 0 holds certainly. Next we assume m investors in the market. In the equilibrium state in the market of the share of common stock of a corporation, the following holds: P t) = N i, i = 1,..., m. 1.12) The equilibrium states of different assets are obtained as special cases of 1.12). Adding the m equations in 1.12), and dividing by m, we get: P t) = 1 ) Ni. 1.13) m Thus in the equilibrium state, the asset price equals the arithmetic average of investors estimates of present values of net revenues from this asset Aggregate Investor Behavior According to 1.13), the force F acting upon the net demand of an asset is: F t) = 1 ) Ni P t). 1.14) m The net demand Dt) unit/ t) of the assets is then Dt) = ff t)), f F ) > 0, f0) = 0, 1.15) and the equation of motion for the asset price is P t) = gdt)), g D) > 0, g0) = )

4 26 ESTOLA: PRINCIPLES OF QUANTITATIVE MICROECONOMICS Taking the Taylor series approximations of functions f, g in 1.15) and 1.16) in the neighborhood of the points D = F = 0, and assuming the error-terms zero, we get P t) = g 0) f 0) F, F = HN i,,, n i ) P t), 1.17) where P t) / t) is the flow of the asset price, g 0) f 0) a positive constant with unit 1/ t, and the average of investors estimates of present values of revenues from one asset is denoted as: HN i,,, n i ) 1 m Ni In the Appendix of this chapter we approximate HN i,,, n i ) as: HN, Z, r d, n) a 0 m + a 1 m N + a 2 m Z + a 3 m r d + a 4 n. 1.18) m The units of a 0, a 1, a 3, a 4 are:, t, t and, respectively, and a 2 is dimensionless; a 1 > 0, a 2 > 0 and the signs of a 3, a 4 are ambiguous, see the appendix of this chapter. Substituting the approximation in 1.18) into Eq. 1.17), we get: P t) = g 0)f a0 0) m + a 1 m N + a 2 m Z + a 3 m r d + a ) 4 m n P t). 1.19) Denoting g 0)f 0) = 1/m A > 0 we can name the positive constant m A with unit t as the inertial mass of the asset price. Eq. 1.19) implies that the factors positively affecting the force F have an raising effect on the asset price, and vice versa, and the equilibrium price corresponds to: P t) = 0. To solve the Eq. 1.19) we assume x, x = N, Z, r d, n to be constants. Then, denoting g 0)f 0) = 1/m A the solution is P t) = a 0 + a 1 N + a 2 Z + a 3 r d + a 4 n m ). + C 0 e 1 m A t, 1.20) where C 0 ) is the constant of integration. The time path of P t) in 1.20) shows that with t, P t) will settle into its equilibrium state P : P = a 0 + a 1 N + a 2 Z + a 3 r d + a 4 n. m The higher the N, Z, the higher the equilibrium asset price P, and the effects of r d and n on P depend on whether the condition N < r d Z holds.

5 ESTOLA: THEORY OF MONEY 27 References: Friedman, M., The Role of Monetary Policy. American Economic Review, March. Jevons, W.S., Theory of Political Economy, 2nd ed. Original publication at 1879, Harmondsworth: Penguin. Mishkin, F. S., The Economics of Money, Banking, and Financial Markets. Sixth Edition. Addison Wesley. 1.7 Mathematical Appendix We approximate the quantity H i N i,,, n i ) N i 1.21) in the neighborhood of point x i0 = N i0, 0, 0, n i0 ) by taking the Taylor series approximation as: H i N i,,, n i ) = H i x i0 ) + N i x i0 )N i N i0 ) + x i0 ) 0 ) + x i0 ) 0 ) + n i x i0 )n i n i0 ) + ɛ i. 1.22) Assuming ɛ i = 0 i and adding the approximations of H i in 1.22), we get H i N i,,, n i ) a 0 + a 1 N + a 2 Z + a 3 r d + a 4 n, 1.23) where x = 1/m) m x i, x = N, Z, r d, n are the arithmetic averages of the quantities and: a 0 = H i x i0 ) x i0 )N i0 x i0 )0 x i0 )0 N i a 3 = x i0 )n i0 n i ), a 1 = x i0 ), a 4 = N i x i0 ), a 2 = n i x i0 ). x i0 ), The units of a 0, a 1, a 3, a 4 are:, t, t and, respectively, and a 2 is dimensionless. Now H i x i0 ) is positive at every N i,,, n i because every

6 28 ESTOLA: PRINCIPLES OF QUANTITATIVE MICROECONOMICS asset gives some revenues in the future. Thus a 0 > 0 let N i N i0, 0, 0, n i n i0 and ɛ i 0 i in 1.22)). In Section we showed that / N i > 0, / > 0, and /, / n i are ambiguous; thus a 1 > 0, a 2 > 0 and the signs of a 3, a 4 are ambiguous.