The Mean-Variance Model Revisited with a Cash Account ()

Chonghui Jiang, Yongkai Ma, Yunbi An

Odette School of Business, University of Windsor, Windsor, Canada.

School of Management and Economics,University of Electronic Science and Technology of China,Chengdu, China.

**DOI: **10.4236/jmf.2012.21006
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Odette School of Business, University of Windsor, Windsor, Canada.

School of Management and Economics,University of Electronic Science and Technology of China,Chengdu, China.

Fund managers usually set aside a certain amount of cash to pay for possible redemptions, and it is believed that this will affect overall fund performance. This paper examines the properties of efficient portfolios in the mean-variance framework in the presence of a cash account. We show that investors will retain a portion of their funds in cash, as long as the required return is lower than the expected return on the portfolio corresponding to the point of intersection between the traditional efficient frontier and the straight line that passes through the minimum-variance portfolio and the origin in the mean-variance plane (intersection portfolio). In addition, the portion of funds allocated to risky assets is invested in the intersection portfolio, and this investment is more efficient than the corresponding traditional efficient portfolio. Using a simulation, we illustrate that 6% to 9% of total funds are retained in the cash account if a no-short- selling constraint is imposed. Based on real data, our out-of-sample empirical results confirm the theoretical findings.

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C. Jiang, Y. Ma and Y. An, "The Mean-Variance Model Revisited with a Cash Account," *Journal of Mathematical Finance*, Vol. 2 No. 1, 2012, pp. 43-53. doi: 10.4236/jmf.2012.21006.

1. Introduction

Since Markowitz’s (1952) ground-breaking publication [1], mean-variance analysis has become an important portfolio management approach both in academics and in practice. This approach is not only the basis of asset pricing theories [2-5], but also the foundation of various extended portfolio selection models [6-8] and portfolio performance evaluations [9]. It implies that investors allocate funds among risky assets to achieve mean-variance efficiency. Such a mean-variance efficient portfolio is simply a linear convex combination of the minimum-variance portfolio and the intersection portfolio—the portfolio corresponding to the point of intersection of the efficient frontier and the straight line that passes through the minimumvariance portfolio and the origin in the mean-variance plane. One crucial assumption behind this conclusion is that funds are fully invested in all risky assets.

In practice, however, investors typically choose to allocate a part rather than all of their funds in risky assets, keeping a small portion in cash, even though the opportunity set includes only risky assets. This might be caused by regulations or restrictions on market transactions, but it is primarily due to risk management and other considerations. Fund managers might adjust the position in risky assets as per market conditions to control the overall portfolio risk; they may also set aside a certain amount of cash to pay for possible redemptions by individual investors. The traditional mean-variance model eliminates the possibility of a cash account, thereby failing to accommodate this general practice.

This paper examines the mean-variance portfolio selection problem with a cash account and explores the composition and properties of efficient portfolios under this modified model. More specifically, we derive the new efficient frontier and the conditions under which fund managers will retain a certain amount of cash to achieve high portfolio efficiency. Our focus is also on the relation between new and traditional efficient portfolios. These issues are practically relevant, and thereby are of particular interest to practitioners as well as academics.

We show that investors will retain a part of their funds in cash, as long as the required return is lower than the expected return on the intersection portfolio on the traditional efficient frontier. In this case, the investment decision is to allocate funds between the intersection portfolio and cash. Using a simulation, we illustrate that 6% to 9% of total funds are retained in the cash account if a no-short-selling constraint is imposed. Importantly, we demonstrate that the efficient portfolios of risky assets under our model are more efficient than traditional efficient portfolios. This is because these new efficient portfolios have a substantially lower risk, while still achieving returns as high as those from traditional efficient portfolios. This is not only in line with the pseudo efficiency of traditional portfolios documented in [10], but also provides another explanation for the reason why, in practice, investors keep a certain amount of cash. Using real data, we find that the portfolios determined by our model outperform the traditional mean-variance portfolio in terms of out-of-sample risk and Sharpe ratio measures.

The remainder of this paper is organized as follows. Section 2 briefly reviews the traditional mean-variance model, which serves as a benchmark in our analysis. Section 3 describes our modified model and analyzes the properties of efficient portfolios in comparison with traditional efficient portfolios. Section 4 provides an empirical analysis, while Section 5 concludes the paper.

2. Traditional Mean-Variance Model

Suppose there are n different risky assets available in the market with a column return vector r. A portfolio is a vector q = (q_{1}, q_{2}, ···, q_{n})^{T}, where q_{i} is the proportion of the portfolio invested in asset i and the superscript T represents the transpose operation. Thus, the portfolio return is. A traditional mean-variance optimizer solves the following optimization problem:

(1)

s.t.,

where stands for the covariance matrix of risky asset returns, and matrix is non-singular. represents the required expected return on the portfolio, while 1 is an n-column vector with all elements being equal to one. The second constraint implies that all funds are invested in risky assets and no cash is retained.

Using Lagrange multipliers for the two constraints, we can obtain the solution to problem (1) as follows:

, (2)

where is the minimum-variance portfolio in the traditional mean-variance model, and

is the point of intersection of the efficient frontier and the straight line that passes through and the origin in the mean-variance plane [11], called the intersection portfolio. In the above expressions, ,

. Further, the expected return on portfolio

is, and the expected return on

is, where. Note that, , and, while can be negative but is typically positive.

Equation (2) suggests that efficient portfolios are a linear convex combination of and. This is the so-called two-fund separation theorem, where the two funds refer to and [12,13]. All the portfolios that correspond to different required returns constitute the portfolio frontier, whereas those above the minimum-variance portfolio constitute the efficient frontier. The variance of a portfolio as a function of is given by

. (3)

The traditional efficient frontier is illustrated in Figure 1.

3. Mean-Variance Model with a Cash Account

3.1. The Modified Model

The traditional mean-variance model does not reflect the fact that investors are able to keep a portion of their funds in the cash account. To accommodate this fact, the second constraint in model (1) should be relaxed as

Figure 1. The traditional mean-variance efficient frontier.

. If investors are able to borrow cash at a zero cost, then the constraint no longer exists. Accordingly, the mean-variance model becomes

(4)

s.t..

The cash account in this paper is defined as a banking account that pays no interest. Thus, cash provides a return of zero with no risk. Note that the cash account is assumed to be available to any investors, even though the investment opportunity set includes only n risky assets. This is in contrast with the risk-free asset considered in the traditional portfolio selection problem, as the riskfree asset is not available if it is not included in the opportunity set. Suppose the solution to model (4) is, then the proportion of total funds invested in risky assets is. If, it means that investors allocate only a part of their total funds to risky assets, and of their total funds are retained in cash. In the case that, investors allocate all funds to risky assets, and portfolio coincides with portfolio. On the other hand, if, then investors borrow of their total funds and invest them, together with the original funds, in risky assets. Thus, the efficient portfolios determined by model (1) are typically no longer efficient according to model (4), indicating that the presence of the second constraint in model (1) leads to a loss in efficiency [14].

3.2. The Solution and Properties

Using the Lagrange multiplier approach, we obtain the first-order optimality condition for problem (4) as follows

. (5)

Solving for from Equation (5) gives

. (6)

Plugging Equation (6) into the constraints in Equation (4) yields

.

Therefore, the efficient portfolio for the given required return in our model is

. (7)

Consequently, the proportion of total funds invested in risky assets is calculated as

. (8)

The following discussion considers the case in which. The cases of and will be investigated in the last subsection.

Proposition 1. If the required return, then investors place only a part of their total funds in risky assets; if the required return, then investors invest their total funds in risky assets; if the required return, then investors borrow certain funds and invest them, together with their original amount, in risky assets.

Proof. It follows immediately from Equation (8).

Proposition 1 suggests that instead of investing all their funds in risky assets, investors will keep a part of their total funds in cash, as long as the required return is lower than the expected return on. This partially explains the general practice observed in the investment funds industry.

Next, we analyze how the portion of funds invested in risky assets is allocated among these risky assets in our model. According to Equations (7) and (8), the relative weights allocated in each of these assets are given

. (9)

Interestingly, the relative weights in the risky assets are the same as the weights of portfolio, regardless of the required return on the portfolio. Equations (8) and (9) yield

. (10)

The intuition behind Equation (10) is that if, then investors only invest (percentage) of their total funds in portfolio and the rest is retained in cash. On the other hand, if, then investors borrow (percent) of their total funds and invest all these funds including the original amount in portfolio. This demonstrates that the efficient portfolios of risky assets under the modified mean-variance model are directly determined by the percentage (or times) of total funds in portfolio, while the rest is kept in cash (or borrowed). In other words, all funds will be allocated between portfolio and cash. This is the twofund separation theorem in our model, where the two funds are and the cash account.

It is believed that setting aside a certain amount of cash by fund managers may affect portfolio efficiency. However, our result indicates that this is not necessarily true. Portfolio efficiency depends on whether the portion of funds in risky assets is allocated in accordance with the relative weights of portfolio, and has nothing to do with whether or not certain funds are retained in the form of cash.

According to the new two-fund separation theorem, the portfolio selection process can be described as follows. First, for a given expected return, investors decide the proportion of their total funds invested in risky assets, the rest to be deposited in the cash account or borrowed. Second, investors allocate these funds among these risky assets according to the weights of portfolio. Therefore, all investors will hold the same risky portfolio regardless of risk aversion. Given that the market portfolio is the linear convex combination of portfolios of individual investors when the market is in equilibrium, portfolio is the market portfolio in the modified model.

For any given required return, the variance of the portfolio determined by Equation (7) is

. (11)

is the efficient frontier generated by the risky assets in the presence of a cash account. The minimumvariance portfolio is the one in which all funds are retained in cash with a return of zero.

3.3. The New Efficient Frontier versus the Traditional Efficient Frontier

To compare the efficient portfolios determined by model (4) with the efficient portfolios in the traditional model, we investigate their relative risks and Sharpe ratios for the same required return. From Equations (3) and (11), we note that

(12)

Given that and, is always nonnegative, demonstrating the loss in efficiency of the traditional efficient portfolios relative to the new efficient portfolios. Namely, efficient portfolios determined by model (4) can achieve even lower risk than traditional efficient portfolios for any given required return that is not equal to. In addition, the bigger the difference between and, the greater the loss in efficiency of traditional efficient portfolios.

Assume without loss of generality that the risk-free rate is zero, then the Sharpe ratio of efficient portfolios determined by model (4) is given as

.

On the other hand, the Sharpe ratio of traditional efficient portfolios is

.

Apparently, is a constant, and is always greater than as long as the required return is not equal to.

Proposition 2. Portfolio is the only portfolio on the traditional efficient frontier that also lies on the new efficient frontier determined by model (4), and all other traditional efficient portfolios are dominated by the corresponding new efficient portfolios for the given expected return. The traditional efficient frontier is innertangent to the efficient frontier determined by model (4) at point.

Given that both efficient frontiers are parabolas and that is always non-negative, the traditional frontier must stay inside of the efficient frontier determined by model (4) with the only point of intersection. Thus, Proposition 2 follows immediately. Figure 2 displays the relation between these two efficient frontiers.

3.4. No Zero-Cost Borrowing

It is practically impossible to borrow funds at a zero cost.

Figure 2. The traditional mean-variance efficient frontier and the efficient frontier determined by model (4).

If investors are not allowed to borrow, then the proportion of total funds in risky assets cannot exceed one, i.e.,. To accommodate this restriction, model (1) becomes

, (13)

s.t.,

.

From Proposition 1, we notice that investors place a portion of their total funds in risky assets when, while they invest all their funds in risky assets when. In both cases, the no-borrowing restriction is satisfied, and models (13) and (4) imply the same portfolio decision.

However, when, the solution to model (4) violates the no-borrowing restriction. In this case, the decision implied by model (13) is consistent with the traditional mean-variance decision. To illustrate this point, we consider the following model:

, (14)

s.t.,

.

In this model the proportion of total funds in risky assets is parameterized as (). The efficient frontier determined by model (14) is solved as

. (15)

Equation (15) demonstrates that when or, decreases with in the domain of

. To achieve the minimum variance, must be maximized, or. As a result, the model reduces to the traditional mean-variance problem when, and yields the same decision as that in the traditional model.

Proposition 3. If borrowing is not allowed, then the efficient frontier is characterized by the following expression:

(16)

The solid line in Figure 3 displays the efficient frontier under this circumstance. Apparently, the portfolios determined by model (13) still perform at least as well as traditional efficient portfolios in terms of risk reduction.

3.5. b ≤ 0

When, the net funds in risky assets are negative according to Equation (8), indicating short-sales of risky assets. More specifically, investors will short sell portfolio and keep all the money in cash, regardless of whether or not borrowing is allowed. Moreover, the higher the required return, the more assets will be short-sold. In this case, the traditional efficient frontier is inner-tangent to the lower part of the portfolio frontier determined by the modified model.

In particular, when, the portfolios of risky assets are self-financing, and the traditional efficient frontier reduces to

.

Therefore, the traditional efficient frontier of risky assets is located to the right of the efficient frontier determined by the modified model with the distance being.

4. A Numerical Analysis

Since the mean-variance model with a no-short-selling

Figure 3. The traditional mean-variance efficient frontier and the efficient frontier determined by model (4) if borrowing is not allowed.

constraint cannot be solved analytically, our previous theoretical analysis does not impose the no-short-selling constraint. However, short-selling is difficult to implement in practice. This section begins by using a simulation to illustrate the performance of efficient portfolios determined by our model with the no-short-selling constraint relative to the traditional portfolios. Then, we demonstrate our theoretical findings using real data. To this end, we compare the following two groups of portfolios: the traditional minimum-variance portfolio (MVP) versus the efficient portfolio in the modified model with the same return as that on the MVP (EP0), as well as the equally-weighted portfolio (EWP) versus the efficient portfolios in both the traditional (EP1) and modified models (EP2) that generate the same return as does the EWP.

4.1. Data

The data used for the numerical analysis include the average value-weighted monthly returns on 10 industry portfolios in the US for the sample period from January 1976 to October 20101. These industry portfolios are constructed from all AMEX, NYSE and NASDAQ stocks by using their four-digit SIC codes from Compustat. Whenever Compustat SIC codes are not available, the CRSP SIC codes are taken. The data is obtained from Kenneth R. French’s personal website.

Table 1 reports the summary statistics of these portfolio returns. Apparently, NoDur and Enrgy yielded the highest returns, while HiTec and Durbl exhibited the highest risk over the past 30 years. Utils seems to be the safest industry among all 10 industries with the lowest return. In addition, the correlation coefficients between any pair of industry returns range from 0.26 to 0.84.

Conflicts of Interest

The authors declare no conflicts of interest.

[1] | H. Markowitz, “Portfolio Selection,” Journal of Finance, Vol. 7, No.1, 1952, pp. 77-91. doi:10.2307/2975974 |

[2] | F. Black, “Capital Market Equilibrium with Restricted Bo- rrowing,” Journal of Business, Vol. 45, No. 3, 1972, pp. 444-455. doi:10.1086/295472 |

[3] | W. F. Sharpe, “Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risks,” Journal of Finance, Vol. 19, No. 3, 1964, pp. 425-442. doi:10.2307/2977928 |

[4] | J. Lintner, “The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolio and Capital Budgets,” Review of Economics and Statistics, Vol.47, No. 1, 1965, pp. 13-37. doi:10.2307/1926735 |

[5] | J. Mossin, “Equilibrium in a Capital Asset Market,” Eco- nometrica, Vol. 34, No. 4, 1966, pp. 768-783. doi:10.2307/1910098 |

[6] | M. J. Brennan and Y. H. Xia “Dy-namic Asset Allocation under Inflation,” Journal of Finance, Vol. 57, No. 3, 2002, pp. 1201-1238. |

[7] | C. Jiang, Y. Ma and Y. An, “An Analysis of Portfolio Selection with Background Risk,” Journal of Banking and Finance, Vol. 34, No. 12, 2010, pp. 3055-3060. doi:10.1016/j.jbankfin.2010.07.013 |

[8] | B. H. Solnik, “Infla-tion and Optimal Portfolio Choices,” Journal of Financial & Quantitative Analysis, Vol. 13, No. 5, 1978, pp. 903-925. doi:10.2307/2330634 |

[9] | W. F. Sharpe, “Mutual Fund Performance,” Journal of Bu- siness, Vol. 39, No. 1, 1966, pp. 119-138. doi:10.1086/295080 |

[10] | X. Cui, D. Li and J. Yan, “Classical Mean Variance Model Revisited: Pseudo Efficiency,” 2009. http://ssrn.com/abstract=1507453. |

[11] | R. Roll, “A Mean/Variance Analysis of Tracking Error,” Journal of Portfolio Management, Vol. 18, No. 4, 1992, pp. 13-22. doi:10.3905/jpm.1992.701922 |

[12] | R. Merton, “An Analytical Derivation of the Efficient Portfolio Frontier,” Journal of Financial & Quantitative Analysis, Vol. 7, No. 4, 1972, pp. 1851-1872. doi:10.2307/2329621 |

[13] | J. Tobin, “Liquidity Preference as Behaviour towards Risk,” Review of Economic Studies, Vol. 25, No. 2, 1958, pp. 65-86. doi:10.2307/2296205 |

[14] | G. J. Alexander and A. M. Baptista, “Active Portfolio Management with Benchmarking: A Frontier Based on Alpha,” Journal of Banking and Finance, Vol. 34, No. 9, 2010, pp. 2185-2197. doi:10.1016/j.jbankfin.2010.02.005 |

[15] | M. J. Best and R. R. Grauer, “On the Sensitivity of Mean-Variance-Efficient Portfolios to Changes in Asset Means: Some Analytical and Computational Results,” Review of Financial Studies, Vol. 4, No. 2, 1991, pp. 315-342. doi:10.1093/rfs/4.2.315 |

[16] | F. Black and R. Litterman, “Global Portfolio Optimization,” Financial Analysts Journal, Vol. 48, No. 5, 1992, pp. 28-43. doi:10.2469/faj.v48.n5.28 |

[17] | O. Ledoit and M. Wolf, “Improved Estimation of the Covariance Matrix of Stock Returns with an Application to Portfolio Selection,” Journal of Empirical Finance, Vol. 10, No. 5, 2003, pp.603-621. doi:10.1016/S0927-5398(03)00007-0 |

[18] | O. Ledoit and M. Wolf, “Honey, I Shrunk the Sample Co- variance Matrix,” Journal of Portfolio Management, Vol. 30, No. 4, 2004, pp. 110-119. |

[19] | V. DeMiguel, L. Garlappi and R. Uppal, “Optimal versus Naive Diversification: How Inefficient Is the 1/N Portfolio Strategy?” Review of Financial Studies, Vol. 22, No. 5, 2009, pp. 1915-1953. doi:10.1093/rfs/hhm075 |

[20] | R. Duchin and H. Levy, “Markowitz versus the Talmudic Portfolio Diversification Strategies,” Journal of Portfolio Management, Vol. 35, No. 2, 2009, pp. 71-74. doi:10.3905/JPM.2009.35.2.071 |

[21] | J. Fletcher, “Risk Reduction and Mean-Variance Analysis: An Empirical Investigation,” Journal of Business Finance & Accounting, Vol. 36, No. 7-8, 2009, pp. 951-971. doi:10.1111/j.1468-5957.2009.02143.x |

[22] | M. Kritzman, S. Page and D. Turkington, “In Defense of Optimization: The fallacy of 1/N,” Financial Analyst Journal, Vol. 66, No. 2, 2010, pp.31-39. doi:10.2469/faj.v66.n2.6 |

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