Capital Market Line Based on Efficient Frontier of Portfolio with Borrowing and Lending Rate

Capital Asset Pricing Model (CAPM) is a general equilibrium model. It not only allows improved understanding of market behavior, but also practical benefits. However, there exists a risk-free asset in the assumption of the CAPM. Investors are able to borrow and lend freely at the rate may not be a valid representation of the working of the marketplace. Therefore, in this paper, it studies that the efficient frontier of portfolio in different borrowing and lending rate. This paper solves the highly difficult problem by matrix operation method. It first denotes the efficient frontier of Markowitz model with the matrix expression of portfolio. Then it denotes the capital market line (CML) with the matrix expression too. It is easy to calculate by using Excel function. The aim of this study is to develop the mean- variance analysis theory with regard to market portfolio and provide algorithmic tools for calculating the efficient market portfolio. Then explain that the portfolio frontier is hyperbola in mean-standard deviation space. It constructs CML in order to get more returns than that of efficient frontier if risk-free securities are included in the portfolio. A proposed step for CML on efficient frontier of portfolio with borrowing and leading rate is presented. Under these tools, it is easy calculation SML and CML by using Excel function. An example show that proposed method is correct and effective, and can improve the capability of the mean-variance portfolio efficiency frontier model.


Introduction
Capital market theory represented a major step forward in how investors should think about the investment process. In discussion the Markowitz portfolio is its average covariance with all other assets in the portfolio. The CAPM extends Capital market theory in a way that allows investors to evaluate the risk-return trade-off for both diversified portfolios and individual security. The CPAM redefines the relevant measure of risk from total volatility to just the no diversifiable portion of the total volatility (i.e., systematic risk). This risk measure is call beta coefficient, and it calculates the level of a security's systematic risk compared to that of the market portfolio. One of the first assumptions of the CAPM was that investor could borrow and lend any amount of money at the risk free rate.
CAPM uses the Security Market Line (SML), which is a trade-off between expected return and security's risk (beta risk) relative to market portfolio. Each investor will allocate to the market portfolio and the risk free asset in according to own risk tolerance. Capital Market Line (CML) represents the allocation of capital between risk free securities and risky securities for all investors combined. An investor is only willing to accept higher risk if the return rises proportionally. The optimal portfolio for an investor is the point where the new CML in tangent to the old efficient frontier when only risky securities were graphed.
CAPM discovered by Shap (1964), Lintner (1965), Treynor (1965) and Mossin (1966) is a general equilibrium model for portfolio analysis. It derived from two fund separation theory. Some important researches in CAPM are Mas-celell et al. (1995), Constantinides and malliarias (1995) , Elliott and Kopp (1999) , Merton (1992) , Duffie (1992), and Cochrance (2001). Markowitz (1952) firstly offers the Portfolio theory. This theory is a model of modern financial economics. In ordering to introduce Markowitz's Portfolio theory, Jarrow (1988) first proposed a general equilibrium mode, and then proposed the concept of mean variance efficient frontier. Markowitz's Portfolio theory has somewhat cumbersome expressions. Levy and Sarmat (1970) Mangram (2013) found that diversification cannot eliminate all risk, i.e. it cannot eliminate systematic risk but unsystematic risk can be eliminated to a large extent by diversification. Mitra and Khanna (2014) present a simplified perspective of Markowitz's contributions to modern portfolio theory. It is to see the effect of duration of historical data on the risk and return of the portfolio and to see the applicability of risk-reward logic. Chen et al. (2010) introduce the theory and the application of computer program of modern portfolio theory and introduce the mean-variance spanning test which follows directly from the portfolio optimization problem.
Since the assumption of the CAPM that there exists a risk-free asset and investors are able to borrow and lend freely at the rate may not be a valid representation of the working of the marketplace. Therefore, discuss portfolio issues under different borrow and lend rate in theory and financial practice become significance. So it is necessary to discus the portfolio problem in different borrowing and leading rate.
The following statement described the structure of this study. Section 2 deals with Markowitz efficient frontier portfolio. In this section it proposed a procedure for mean-variance efficient portfolio. It will examine the portfolio frontier is a hyperbola in mean-standard deviation space, and finding the minimum variance portfolio of risky assets. Different borrowing and lending rates' capital market line is discussed in section 3. Section 4 gives an illustration. It help in understanding of this procedure, a demonstrative case is given to show the key stages involving the use of the introduced concepts. Section 5 is discussion. Section is conclusion.

Markowitz Efficient Frontier Portfolio
The Markowitz portfolios (1959) is that one that gives the highest expected return of all feasible portfolios with the same risk. It is called the mean-variance efficient portfolio. The Markowitz model describes a set of rigorous statistical procedures used to select the optimal project portfolio for wealth maximizing /risk-average investors. This model is described under the framework of risk-return tradeoff graph. Investors are assumed to seek our maximizing their return while minimizing the risk involved.

Markowitz Portfolio Efficiency Frontier Model
This paper use the following notations n: number of available assets It takes the derivate of it with respect to Solving and rearranging from Equation (4) yields.
Equation (9) can be written as Solving the Equation (1) yields.  (15) Equation (15) can be written as: It explains that the portfolio frontier is a hyperbola in mean-standard deviation space.

Global Minimum Variance Portfolio Point
Find the minimum variance portfolio of risky assets in this section. The minimum variance portfolio of risky assets denotes as g W . Differentiating Equation (16) with respect to p r and setting it equation to zero yields. Figure 1 denotes the minimum variance portfolio hyperbola and efficiency Frontier

Tangent Line of Portfolio Efficiency Frontier Model.
The CML leads all investors to invest in the in the sample risky asset portfolio T W (see Figure 2). It uses the tangent concept to find the weight of tangent point of the portfolio of assets consisting entirely for risk.
Since Equation (18) Assume that  In this section, it finds Capital market line and Security market line. SML shows that the trade-off between risk and expected return as a straight line intersecting the vertical axis (i.e., zero-risk point) at the risk free rate. CML efficient investment portfolios were those that provided that highest return for chosen level of risk, or conversely, the lowest risk for a chosen level of return. The risk-return relationship show in Equation (24) holds for every combination of the risk-free asset with any collection of risk assets. The CML (Equation (24)) offers a precise way of calculating the return that investors can expect for  The availability of this zero-beta portfolio will not affect the CML, but it will allow construction of a linear SML. The combination of this zero-beta portfolio and the market portfolio will be a linear relationship in return and risk, because the covariance between the zero-beta portfolio and the market portfolio is similar to what it was with the risk-free assets.    b CML is slightly kinked as the borrowing rate is higher than the risk-free lending rate and small part of the concave efficient frontier is a segment of the b CML .

The Steps of this Proposed Method
Portfolio optimization involves a mathematical procedure called quadratic programming problem (OPP). There considered two objectives: to maximize return and minimize risk. The OPP can be solved using constrained optimization techniques involving calculus or by computational algorithm applicable to non-linear programming problem. This paper use matrix operation, it includes matrix inverse, matrix multiplication, and matrix transpose. The objective trace the portfolio frontier in mean -standard deviation space and identify the efficient frontier, the minimum variance portfolio and borrowing and lending rates' capital market line.
The steps of this proposed method.
Step Step 5: Analyze different borrowing and lending rate in portfolio Efficiency Frontier.

Example
Selected three stock returns A, B, C. in this illustration.
Step1: Calculate the efficient frontier inputs Use Excel function AVERAGE ( ) calculation mean return. Use Excel function STDEVP ( ) calculation standard deviation, for variance it uses VARP ( ). Use Excel function CORREL ( ) and COVAR ( ) calculation correlation co-efficient and the covariance. Table 1 denoted as three stocks' mean return standard deviation and covariance. Step 2: Calculate the efficient frontier by Markowitz portfolios mean-variance analysis method   Calculate the tangent point by using Equation (21). Calculate the portfolio weight by using Equation (23). Therefore, the equation of CML b is Step 5: Analyze different borrowing and lending rate in portfolio Efficiency Frontier.
The line tracing the points from the minimum variance portfolio A to B is the efficient frontiers. A risk adverse investor will never hold a portfolio which is to the south-east of point A. Since any point to the south-east of A such as C would have a corresponding point like D which has a higher expected return than C with the same portfolio risk. More risk adverse investors would choose points on the efficient frontier that is closed to point A. Similarly, an investor who is less risk adverse or can handle more risk would choose a portfolio closed to point B.
It traces the portfolio frontier by selecting different value of expected return and calculating the corresponding standard deviation by the Equation (16). Figure 5 shows the graph of the frontier.
The capital market line under different borrowing and lending rate are CML b and CML l . Broken line E -T 1 -T 2 -B in figure 5, it constitutes with line (CML l and CML b ) and an arc (T , T b ). The investors can lend at one rate but must pay a different and presumably higher rate to borrow. The efficient frontier would become E -T 1 -T 2 -B in Figure 5.
Here there is a small range of risky portfolios that would be optional for investors to hold.

Findings and Discussions
It is noticed that if the risk free lending and borrowing rates are equal, the optimum risky portfolio is obtained by drawing a tangent to the portfolio frontier from the risk free Where a, b, c denoted as matrix operation. It is easy calculation by Excel function. (2) The CML is denoted as weight vector too. (3) By the definition of CML, we are able to find the efficient market portfolio. (4) By the concept of tangent line of portfolio efficiency frontier model to calculate Capital Market Line CMLb (borrowing rate capital market line) and CMLl (lending rate capital market line). (5) To find out the weight of securities which are there in the portfolio in order to invest in those securities. (6) To construct CML in order to get more returns than that of efficient frontier if risk-free securities are included in the portfolio.
Xuemet and Xinshu (2003) use the concept of differential geometry to solve an efficient portfolio model. The efficient frontier of Markowitz can represent as linear equation group (simultaneous equation) that is composed by n-2 linear equations. A comparison between this study and Xuemet and Xinshu's research denotes as Table 2.

Results
It is difficult to understand the algorithm.
The computation of efficient is not easy.
Use Excel software, the computation of the efficient frontier is fairly easy.

Conclusions
This paper has discussed an algorithm (matrix operation) to look for the efficient market portfolio of CAPM and a new method (tangent line) to obtain the CML. This algorithm applied in different borrowing and lending rate portfolio problems. Under this algorithmic tool, this paper have finished the following works (1) Prove portfolio frontier is a hyperbola in mean-standard deviation space (2) CML is also expressed as the portfolio weight vector (3) In mean-variance portfolio efficiency frontier model, calculate different borrowing and lending rate's CML (4) calculate the tangent point and slop of CML (5) The steps of this proposed method Excel is far from the best program for generating the efficient frontier and is no limited in the number of assets it can handle. It finds that Excel, the computation of the efficient frontier is fairly easy. It finds that this paper is helpful to the correct application of the capital market line based on efficient frontier of portfolio with borrowing and rate, enriching the theory and method of invest management. The further research will focus on performing the calculation of this algorithm tool such as Excel and.