Influence of Real Interest Rate Volatilities on Long-term Asset Allocation

Transcription

1 200 2 Ó Ó 4 4 Dec., 200 OR Transactions Vol.4 No.4 Influence of Real Interest Rate Volatilities on Long-term Asset Allocation Xie Yao Liang Zhi An 2 Abstract For one-period investors, fixed income securities without default are risk-free asset, because the return of these securities can be determined at the beginning of investment period. However, considering long-term investment, investors are able to adjust their portfolio since fixed income securities would have risk of interest rate volatilities from reinvestment. So fixed income securities are no longer risk-free.this paper discusses long-term asset allocation under a frame of special habit formation utility function. Under some assumption for simplicities, we derive the influence of real interest rate volatilities on weight of risky asset allocation, and provide theoretical basis and algorithm for calculating real optimal long-term asset allocation. Keywords Operations research, long-term asset allocation, real interest rate volatility, habit formation utility function, portfolio Subject Classification GB/T ) 0.74 ± ¹ µ² ¼ ½»¾º 2 Ö ÍÛ«Â Ì Ë Ü ÍÛ«Â ÍÛ ½ Ã Ä Æ É Ú Å ÍÛ«Â º Å Ü «Ì Ø Ï Ò Ñß± Æ Ø ¹ È Ê Å ÁÆ ÞÐ Àµ Ê Æ Ð Ù¾² Ó Æ Ê Å Ò Ñß± ³ ³ GB/T ) 0.74 Î Å Å * The research is supported by the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of ChinaNO708040), National Natural Science Foundation of China under Project and Leading Academic Discipline Program, the 0th five year plan of 2 Project for Shanghai University of Finance and Economics.. School of Finance, Shanghai University of Finance and Economics, Shanghai , China; Ç ¼ Ô»ÆÔĐ Ç Department of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai , China; Ç ¼ ÔÝ Ô Ç Õ Corresponding author

2 20 Xie Yao, Liang Zhi An 4 Introduction Mean-variance analysis developed by Harry Markowitz in [] is a classic model of portfolio analysis. In this model variance of portfolio return is regarded as measure of risks and it makes assumption that investors trade off between expected return and variance to get the optimal portfolio. Many researches have shown that in single period investment decisions, mean-variance analysis also belongs to the frame of maximizing investors utilities, in which utility functions are set on wealth at terminal period. Investors maximize their terminal period utility in restrict of investment budget in order to derive an optimal rate of asset allocation. The same method can also be used in multiple-period portfolio problems. We assume that investors make a decision at every beginning of periods and their aim is to maximize the consumption utilities during the whole investment period. Like most of the economic analysis, we use time-added utility function to maximize investors expected utility during the whole period. So we are able to analyze the optimal consumption and portfolio conditions. With Epstein-Zin utility function [2], John Campbell and Luis M. Viceira have shown that volatility of real interest rate affects long-term asst allocation [3]. Epstein-Zin utility function is time-separable function but time-nonseparable function is not considered. We discuss long-term asset allocation under one type of time-nonseparable utility function. We also derive the influence of real interest rate on long-term asset allocation. In Section 2 of this paper, we first introduce habit formation utility function, and develop asset allocation model under one type of special habit formation utility function. In Section 3, we mainly analyze Euler equation satisfied by the optimal portfolio. Then we get Euler equation of consumption and equation of risk premium. In Section 4, we develop linear intertemporal budget constrain and show that covariance of risky asset returns with consumption can be expressed as variance with optimal portfolio return if some assumption is made. In Section 5 the influence of real interest rate on optimal portfolio return is discussed. In Section 6, we compare long-term investment with single term, and show that the risk-free interest is risk free for short-term investors. However it is risky for long-term investors. Because of volatility of real interest rate, the risk-free interest such as yield to maturity of treasury bond, is risky. As a risk aversion investor, he or she holds risky asset not just for risky premium but for avoiding consumption volatility that comes from real interest rate. So, volatility of real interest rate must be considered in long-term asset allocation. 2 Model with one type of habit formation utility function Habit formation utility function is one type of time-nonseparable function. Constantinides [4] and Sundaresan [5] have discussed the importance of these functions. In these models, uc t, X t ) denotes the consumers utility at time t, where C t and X t denotes level of con-

3 4 Parallel-batch Scheduling on Unrelated Machines to Minimize the Sum Objectives 2 sumption and the so-called habit at time t, respectively. In this paper, we consider a special type of habit formation utility function, which is proportion power utility function developed by Abel [6]. Set uc t, X t ) = Ct/Xt), which is a power function and where X t influences utility at time as a habit. In the models of this paper, we set X t = Ct k k 0). So uc t, X t ) = Ct/Ck t ), which shows that the consumption level of previous period C t influences current consumption utility. As a parameter, k controls influence degree of previous period on current utility. When k is set to zero, this utility function becomes general power function. In all models of this paper, current time is set to be t, so that the objective function of investor is max U t = E t [ j=0 δ j C t+j/x t+j ) ]. 2.) As mentioned above, X t can be expressed by one-term lagged consumption level, set X t = C k t k 0), 2.2) which shows that consumption habit affects current utility. Parameter k measures degree of the influence and expresses degree of time-nonseparability. In order to emphasize influence of real interest rate on asset allocation, we simplify the problem. Assume that consumption stream only comes from financial assets and other incomes such as labor wage are not considered. So, investors intertemporal budget constraint is W t+ = + R p,t+ )W t C t ), 2.3) where W t denotes the asset level at time t, C t denotes the consumption level at time t, and R p,t+ denotes the portfolio return in the period from t to t+. Assume that investors make a decision at every beginning of investment period, the return of each period would realize at the end of period. R i,t+ denotes the risky asset return from time t to t +. Because it is risky, R i,t+ is a random variable at time t, the real value of which can be derived at time. R f,t+ denotes the return of risk-free rate from time t to t +. Although the return will realize at time t +, the value of it has been known at beginning of period because of its risk-free. Now, long-term asset allocation problem is expressed by 2.) 2.3). In this paper, we use method of log linearization, so we set lower case letter to express natural log term, such as, c t lnc t, r t lnr t+ ). There is an important equation that will be used frequently. Let ξ be a normal variable, that is lne[ξ] = E[lnξ] + V ar[ln ξ]. 2.4) 2 3 Consumption Euler equation and risky premium First-order conditions of optimal problem 2.)-2.3) is E t [ u t / C t ] = δe t [ + R t+ ) u t+ / C t+ ]. 3.)

4 22 Xie Yao, Liang Zhi An 4 Note that utility function of 2.) is time-nonseparable. Current marginal utility is affected by pervious level of consumption, and current consumption level will influence marginal utility of future time. Marginal utility at time t is u t / C t = C k ) t C t Substituting 3.2) into 3.) we get Euler equation δkc k ) t C t+ C t+/c t ). 3.2) = δe t [ + R t+ )C t /C t ) k ) C t+ /C t ) ], 3.3) where R t+ is return of asset or portfolio, however, it is also can be risk-free return. For the sake of simplification, we assume that consumption and return of asset follow normal distribution. We derive linear expression of 3.3) by 2.4) E t [ c t+ ] = lnδ + E t[r p,t+ ] + 2 V ar t[r p,t+ c t+ ] + k ) c t. 3.4) From equation 3.4), we recognize that expected consumption increase composed by four parts: time preference the first term of the right side of the equation), expected return of optimal portfolio the second term), uncertainty of return of portfolio and consumption increase the third term), and current consumption increase the last term). The influence of current consumption on expected consumption increase is a special characteristic of timenonseparable utility function. Substituting R t+ in 3.3) as risk-free rate R f,t+, we get r f,t+ = lnδ 2 2 V ar t c t+ ) + E t [ c t+ ] k ) c t. 3.5) Substituting the return rate in 3.3) as risky asset return R i,t+ and then risk premium can be induced from 3.3) and 3.5) E t [r i,t+ ] r f,t+ + 2 V ar t[r i,t+ ] = Cov t [r i,t+, c t+ ]. 3.6) From 3.6) we get that risk premium is determined by covariance of return and consumption increase and absolute risk averse parameter. Now, we consider that there are n available risk assets i =,, n, n ), then equation 3.6) can be transformed into vector form E t [r t+ ] r f,t+ ι + σ2 t 2 = σ ct, 3.7) where,r t+ [r,t+,, r n,t+ ] ; ι [,...,] ; σ 2 t [ V ar t r,t+,,v ar t r n,t+ ) ], and σ ct [ Cov t r,t+, c t+ ),..., Cov t r n,t+, c t+ ) ]. In order to derive the optimal portfolio, we need to calculate the right side of 3.7). That is to say, we should substitute c t+ from Cov t [r i,t+, c t+ ].

5 4 Parallel-batch Scheduling on Unrelated Machines to Minimize the Sum Objectives 23 4 Linearization of intertemporal budget 4. A lineal approximation of intertemporal budget Under assumption that consumption stream only comes from financial assets, investor s wealth in next period equals current reinvestment weal multiplied by total return of this investment period. So, investor s intertemporal budget can be expressed by 2.3). By taking the log form of 2.3), we can transform log wealth increase to log portfolio return and log consumption-wealth rate w t+ = k + r p,t+ + ) c t w t ), 4.) ρ where ρ = expc w), k = lnρ) + [ ρ)ln ρ/ρ, and c w denotes mean value of c t w t. We deduce log ratio of consumption and wealth by linear constraint. Note that the identical transformation w t+ c t+ + c t w t ) c t+ w t+ ). 4.2) From 4.) and 4.2), we get difference equation of log consumption and wealth. Under assumption lim j c t+j w t+j ) = 0, we solve the equation by forward iteration. When the ratio of consumption and wealth is a stationary process, the assumption is justified. Solving difference equation to get k + r p,t+ + ) c t w t ) = c t+ + c t w t ) c t+ w t+ ) ρ c t w t = r p,t+j c t+j ) + ρk ρ. 4.3) 4.3) shows that the current ratio of consumption-wealth is determined by future return of portfolio and consumption increase. Take conditional expectation of 4.3) to get [ ] c t w t = E t r p,t+j c t+j ) + ρk ρ. 4.4) Equation 4.4) shows that if future return of portfolio is expected to increase, current ratio of consumption to wealth would increase. 4.2 Express Cov t r i,t+, c t+ ) to covariance of r i,t+ with function of r p,t+ From 3.4), we show that the expected consumption variance composes by four parts under the assumption of this paper. For the sake of simplicity and to derive real influence of interest

6 24 Xie Yao, Liang Zhi An 4 rate on long-term asset allocation, we assume that consumption level and portfolio return have identical variance. Then the second term in 3.4) lnδ + 2 V ar t[r p,t+ c t+ ] is constant, which is expressed by µ. Under frame of habit formation utility, we have to consider the influence of passed consumption level on today s consumption utility which also affects future consumption utility, which is just the difference from time-separable function. This difference reflected in 3.4) is expected consumption increase composed by the four parts mentioned ) above. Compared to time-separable function, there is an additional term k c. So, 3.4) can be transformed by E t [ c t+ ] = µ + E t[r p,t+ ] + k ) c t 4.5). 4.5) is a differece equation, which can be solved under some assumption. Then the expected consumption can be expressed by return of portfolio. First, we set k >. generally it is greater than zero, but it needn t greater than one. If we set k >, previous consumption level C t affects current marginal utility u t / C t more greatly. On the other hand, we generally assume that investors are risk averse, and ) they have more risk aversion that is greater, is closer to. So, set φ k, and we would believe φ >. We again assume lim m φ E m t [r p,t+ ] = 0. Only if the optimal portfolio return rate process {r p,t+ } is a stable process, this assumption is justified. The difference equation 4.5) can be transformed as c t = φ E t[ c t+ ] φ E t[r p,t+ ] µ φ. 4.6) From forward iteration and assumption lim m φ m E t [r p,t+ ] = 0, we can derive a relationship between expected consumption increase and return of portfolio E t [ c t+ ] = m= φ m E t[r p,t+ ] µ φ. 4.7) Note that 4.7) can be substituted into 4.3) to express log ratio of consumption and wealth by expected future portfolio return [ ] [ c t w t = E t r p,t+ + E t j=i m= ] φ m r p,t+j+m ρµ φ ) ρ) + ρk ρ. 4.8) Now we discuss what composes next period consumption strike c t+ E t [c t+ ]. From 4.8) we can express difference between consumption strike and wealth strike by expected future portfolio return c t+ E t [c t+ ] {w t+ E t [w t+ ]} = E t+ E t ) r p,t++j )+ E t+ E t ) m= φ m r p,t++j+m ). 4.9)

7 4 Parallel-batch Scheduling on Unrelated Machines to Minimize the Sum Objectives 25 Note that the approximation of intertemporal budget 4.) shows that increase of wealth comes from portfolio return in next period. So, wealth strike totally comes from strike of portfolio return. To understand this, we take expectation of 4.) to get Minusing 4.0) from 4.) we get E t [w t+ w t ] = k + E t [r p,t+ ] + ) c t w t ). 4.0) ρ w t+ E t [w t+ ] = r p,t+ E t [r p,t+ ]. 4.) Substituting 4.) into 4.9) to get ) c t+ E t c t+ ) =r p,t+ E t [r p,t+ ] + E t+ E t ) r p,t++j + E t+ E t ) m= φ m r p,t++j+m ). 4.2) From 4.2) we show that conditional covariance of individual risk asset return r i,t+ and consumption increase c t+ at time t equals covariance of r i,t+ and r p,t+ added covariance of expected future portfolio return change, which is Cov t [r i,t+, c t+ ] =Cov t [r i,t+, r p,t+ ] + Cov t [r i,t+, E t+ E t ) r p,t++j + Cov t [r i,t+, E t+ E t ) m= φ m r p,t++j+m. 4.3) When consumption and portfolio are optimal, premium of risk asset premium can be expressed by 3.6). We have showed that covariance of risk asset and consumption increase comes from right side of 4.3). So equation of risk premium can be transformed to E t [r i,t+ ] r f,t+ + 2 V ar t[r i,t+ ] = Cov t [r i,t+, r p,t+ ] ] + Cov t [r i,t+, E t+ E t ) r p,t++j + Cov t [r i,t+, E t+ E t ) m= ) ] φ m r p,t++j+m. 4.4) 5 Influence of volatility on long-term asset allocation In order to show what rules asset allocation needs to follow and what factors it is affected, ) we need firstly discuss the change of expected portfolio return E t+ E t ) r p,t++j is affected by what factors. Return r p,t+ is log return of portfolio, which can not be expressed

8 26 Xie Yao, Liang Zhi An 4 by linear function of individual asset return. But there is a linear approximation, which is more accurate when investment period is shorter r p,t+ = r f,t+ + α te t [r t+ r f,t+ ι] + 2 α t 2 α tσ t α t, 5.) where vector α t = α,t,,α n,t ) denotes the weight on the n risk assets at time. Σ t denotes variance and covariance of risk assets at time t. Take expectation on 5.), we get E t [r p,t+ ] = r f,t+ + α te t [r t+ r f,t+ ι] + 2 α tσ 2 t 2 α tσ t α t. 5.2) Equation 5.2) shows that expected return of portfolio varied with time and this variation comes from three parts: the First comes from variation of risk-free asset, the second from variation of risk premiums, and the last from variation of variance and covariance of risk assets. If we set variance and covariance of risk assets is a constant and risk premium is also a constant, there is only one factor- risk-free return- change the expected portfolio return. Short term interest rate, such as zero coupon bonds yield, is an important factor that investors must consider. Under the previous assumption, we have Cov t [r i,t+, E t+ E t ) r p,t++j ρ + Cov t [r j i,t+, E t+ E t ) φ m r p,t++j+m m= =Cov t [r i,t+, E t+ E t ) r f,t++j + Cov t [r i,t+, E t+ E t ) Substituting 5.3) to 4.5), we have m= φ m r f,t++j+m. 5.3) E t [r t+ ] r f,t+ ι + σ2 t 2 = Σ tα t + σ f,t. 5.4) From 5.) covariance of risk asset return and portfolio return can be expressed by Cov t r t+, r p,t+ ) = Σ t α t. 5.5) σ f,t in 5.4) is the vector of risk asset return and expected risk-free return [ σ f,t Cov t r,t+, E t+ E t ) r f,t++j,, ) Cov t [r n,t+, E t+ E t ) r f,t++j +

9 4 Parallel-batch Scheduling on Unrelated Machines to Minimize the Sum Objectives 27 [ Cov t r,t+, E t+ E t ) m= Cov t [r n,t+, E t+ E t ) m= φ m r f,t++j+m,, φ m r f,t++j+m ). 5.6) Equation 5.4) shows that when investor maximizes the whole utility of investment period to make optimal on consumption and portfolio, the risk premium must satisfy 5.4). So solving α t from 5.4), we get investor s weight on risk assets from time t to time t+. From 5.4), there is α t = Σ t E t [r t+ r f,t+ ι] + σ2 t 2 ) + Σ t σ f,t. 5.7) 6 Comparison between long-term asset allocation and short-term investment When considering short-term investment, making utility function on wealth at end period is the same as on consumption level. Because problem of short-term investment has assumption that investors consume all of the wealth at the end of investment period, and doesn t reinvest the wealth. So the investor only cares of the wealth level. Let the utility function on wealth of the end period be power function W investor s objective function is t+, so [ W t+ max E ] t, 6.) where W t+ = + R p,t+ )W t. We take the linear approximation of the objective function and use equation 5.) to get optimal portfolio α t = Σ t E t [r t+ ] r f,t+ ι + σ2 t 2 ). 6.2) Comparing 5.7) and 6.2), we can see that the first term of right side of 5.7) is the weight on risk assets of short-term investor, which is determined by risk premium, variance-covariance matrix and absolute risk averse. The second term of right side of 5.7) is the influence of interest volatility on investment that long-term investor must consider. That is long-term investor invest on risk assets because they want to hedge the influence of interest rate on their consumption stream. That is the difference between long-term and short-term investment. For extreme investor of risk averse, the absolute risk averse +. Under this condition, the short-term investor s weight on risk asset equals zero,α t. That is to say extreme investor of risk averse doesn t invest on risk assets and they will invest all of their wealth on risk-free asset. But for long-term investors, when +, from 5.7) and expectation of σ f,t, we

10 28 Xie Yao, Liang Zhi An 4 know that the weight on risk asset is [ α t =Σ t Cov ) ] t r,t+, E t+ E t ) r f,t++j,, Cov t [r ) ]) n,t+, E t+ E t ) r f,t++j. Extreme investor of risk averse would invest on risk asset because they want to avoid influence of real interest rate volatility on consumption stream. This paper is deferent from that of John. Campbell and Luis M. Viceira 200). That paper is under frame of Epstein- Zin utility function. This paper uses time-nonseparable utility function- habit formation utility. So, the conclusion of long-term risk asset allocation 5.7) - influence of real interest rate on asset allocation is more complex than their conclusion. Because we have considered the influence of previous consumption on current utility, there is one more term in σ f,t [ ρ Cov j ) ] t r,t+, E t+ E t ) φ m r f,t++j+m,, Cov t [r n,t+, E t+ E t ) m= m= ) ]) φ m r f,t++j+m. There are many models that discuss real interest rate volatility. We can get one of them to calculate the volatility of real interest rate in order to get optimal portfolio weight. The parameter of random models of interest rate can be estimated by real data of risk-free securities, such as treasury bonds. This paper is a basis of empirical analysis on long-term asset allocation. References [] Harry Markowitz. Portfolio Selection[J]. The Journal of Finance, 952, 7: [2] Larry G., Epstein and Stanley E. Zin. Risk Aversion and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework[J]. Econometrica, 989, 57: [3] John Y. Campbell and Luis M. Viceira. Strategic Asset Allocation: Portfolio Choice for Long- Term Investors[M]. New York: Oxford University Press, 200. [4] George M. Constantinides. Habit Formation: A Resolution of the Equity Premium Puzzle[J]. The Journal of Political Economy, 990, 98: [5] Suresh M. Sundaresan. Intertemporally Dependent Preferences and the Volatility of Consumption and Wealth[R]. Review of Financial Studies, 989, 2: [6] Abel A. Asset Prices under Habit Formation and Catching Up with the Joneses[J]. American Economic Review 80, Papers and Proceedings, 990, [7] John Y. Campbell, Andrew W. Lo, and A. Craig Mackinlay. The Econometrics of Financial Markets[M]. Princeton University Press, 997.