Universal Journal of Accounting and Finance 2(4): 69-76, 2014 http://www.hrpub.org DOI: 10.13189/ ujaf.2014.020401

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

Ming-Chang Lee1,*, Li-Er Su2

1National Kaohsiung University of Applied Science, Taiwan 2Shih Chien University, Kaohsiung Campus, Kaohsiung Taiwan *Corresponding Author: [email protected]

Copyright © 2014 Horizon Research Publishing All rights reserved.

Abstract Capital Asset Pricing Model (CAPM) is a diversifiable portion of the total (i.e., systematic general equilibrium model. It not only allows improved risk). This risk measure is call coefficient, and it understanding of market behavior, but also practical benefits. calculates the level of a security’s compared However, there exists a risk-free asset in the assumption of to that of the . One of the first assumptions the CAPM. are able to borrow and lend freely at the of the CAPM was that could borrow and lend any rate may not be a valid representation of the working of the amount of money at the risk free rate. marketplace. Therefore, in this paper, it studies that the CAPM uses the (SML), which is a efficient frontier of portfolio in different borrowing and trade-off between expected return and security’s risk (beta lending rate. This paper solves the highly difficult problem risk) relative to market portfolio. Each investor will allocate by matrix operation method. It first denotes the efficient to the market portfolio and the risk free asset in according to frontier of Markowitz model with the matrix expression of own risk tolerance. Line (CML) represents portfolio. Then it denotes the (CML) the allocation of capital between risk free securities and risky with the matrix expression too. It is easy to calculate by securities for all investors combined. An investor is only using Excel function. The aim of this study is to develop the willing to accept higher risk if the return rises proportionally. mean- variance analysis theory with regard to market The optimal portfolio for an investor is the point where the portfolio and provide algorithmic tools for calculating the new CML in tangent to the old efficient frontier when only efficient market portfolio. Then explain that the portfolio risky securities were graphed. frontier is hyperbola in mean-standard deviation space. It CAPM discovered by Shap (1964), Lintner (1965), constructs CML in order to get more returns than that of Treynor (1965) and Mossin (1966) is a general equilibrium efficient frontier if risk-free securities are included in the model for portfolio analysis. It derived from two fund portfolio. A proposed step for CML on efficient frontier of separation theory. Some important researches in CAPM are portfolio with borrowing and lending rate is presented. Mas-celell et al. (1995), Constantinides and malliarias Under these tools, it is easy calculation SML and CML by (1995) , Elliott and Kopp (1999) , Merton (1992) , Duffie using Excel function. An example show that proposed (1992), and Cochrance (2001). method is correct and effective, and can improve the Markowitz (1952) firstly offers the Portfolio theory. This capability of the mean-variance portfolio efficiency frontier theory is a model of modern . In ordering model. to introduce Markowitz’s Portfolio theory, Jarrow (1988) first proposed a general equilibrium mode, and then Keywords Efficient Frontier, Capital Market Line, proposed the concept of mean variance efficient frontier. Portfolio, Security Market Line, Capital Asset Pricing Markowitz’s Portfolio theory has somewhat cumbersome Model (CAPM) expressions. Levy and Sarmat (1970) study the optimal portfolio under different rate in Markowitz mean-variance model. LeRoy and Werner (2001) use the Hibert Space and orthogonal solution to prove mean variance Portfolio 1. Introduction analysis. It denotes the efficient frontier of Markowitz model with the weight vector of portfolio (Constantinides and Capital market theory represented a major step forward in Malliaris, 1995). Xuemei and Xinshu (2003) use geometric how investors should think about the investment process. In way (linear equation group) in solving the optimal portfolio discussion the Markowitz portfolio is its average covariance choice weights. An analysis of Canadian farmland risk and with all other assets in the portfolio. The CAPM extends return on investment shows that a Farmland Real Estate Capital market theory in a way that allows investors to Investment Trust (F-REIT) would have been a reasonably evaluate the risk-return trade-off for both diversified good investment over the past 35 years (painter, 2010). portfolios and individual security. The CPAM redefines the Painter (2010) study that shows that for period 1990-2007, relevant measure of risk from total volatility to just the no international portfolio investment performance was 70 Capital Market Line Based on Efficient Frontier of Portfolio with Borrowing and Lending Rate

significantly improved with the addition if Canadian rf : the return on the risk free asset farmland. Mangram (2013) found that diversification cannot eliminate all risk, i.e. it cannot eliminate systematic risk but 2 σ i : variance of the return on asset i unsystematic risk can be eliminated to a large extent by diversification. Mitra and Khanna (2014) present a σ ij : covariance between the return on assets i and j simplified perspective of Markowitz’s contributions to . It is to see the effect of duration of ρij : correlation between the returns on assets i and j historical data on the risk and return of the portfolio and to see the applicability of risk-reward logic. Chen et al. (2010) Let r = (r1, r2 , ... ,rn ) denotes the return on asset. V introduce the theory and the application of computer denotes covariance matrix. It is the n × n program of modern portfolio theory and introduce the variance-covariance matrix of the n assets returns. mean-variance spanning test which follows directly from 2 the portfolio optimization problem. σ 1 σ 12 ... σ 1n  Since the assumption of the CAPM that there exists a   σ σ 2 ... σ risk-free asset and investors are able to borrow and lend =  21 2 2n  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 2 σ n1 σ n2 ... σ n  financial practice become significance. So it is necessary to V discus the portfolio problem in different borrowing and The Markowitz model (mean- variance analysis method): lending rate. (Huang and Litzenberger, 1988) The following statement described the structure of this σ 2 = T study. Section 2 deals with Markowitz efficient frontier min p W VW portfolio. In this section it proposed a procedure for T mean-variance efficient portfolio. It will examine the S. t. W r = rp (1) portfolio frontier is a hyperbola in mean-standard deviation T space, and finding the minimum variance portfolio of risky W 1 = 1 T assets. Different borrowing and lending rates’ capital market Where 1 = (1,1, ... ,1) , is a 1× n column vector of 1’s. line is discussed in section 3. Section 4 gives an illustration. V is × matrix, and assumes that V is nonsingular matrix It help in understanding of this procedure, a demonstrative n n and positive definite matrix. Since min (x) = max (-x). case is given to show the key stages involving the use of the introduced concepts. Section 5 is discussion. Section is Using the Lagrangian method with multipliers λ1 and λ2 , conclusion. it is

T T T La = −W VW + λ1 (W r − rp ) + λ2 (W 1− 1) (2) 2. Markowitz Efficient Frontier It takes the derivate of it with respect to λ λ , set the Portfolio W , 1 , 2 derivate equation to zero. The derivates are: The Markowitz portfolios (1959) is that one that gives the ∂La highest expected return of all feasible portfolios with the = 2VW − λ r − λ 1 = 0 (3) same risk. It is called the mean-variance efficient portfolio. ∂W 1 2 The Markowitz model describes a set of rigorous statistical ∂ procedures used to select the optimal project portfolio for La T (4) = rp −W r = 0 wealth maximizing /risk-average investors. This model is ∂λ1 described under the framework of risk-return tradeoff graph. ∂La T Investors are assumed to seek our maximizing their return = 1 −W 1 = 0 (5) while minimizing the risk involved. ∂λ2 Solving W from Equation (3) yields. 2.1. Markowitz Portfolio Efficiency Frontier Model λ  This paper use the following notations 1 −1 1 −1 1 (6) W = V (λ1r + λ21) = V [r 1]  n: number of available assets 2 2 λ2  r : the return (sometimes called ) i Solving and rearranging from Equation (4) yields. on asset i T W [r 1] = [r 1] (7) rp : the (rate of ) return of portfolio or the mean of the p portfolio return T rp  [r 1] W =   (8) wi : the weight (share) of asset i in the portfolio 1  Universal Journal of Accounting and Finance 2(4): 69-76, 2014 71

−1 Substituting W from Equation (6) into Equation (8) yields. V 1 (17) λ  c rp  = 1 T −1 1 (9)   [r 1] V [r 1]  1  2 λ2  Let a = r TV −1r, b = r TV −11, c =1T V −11,

d = ac − b 2 a b r TV −1r r TV −11 = = [r 1]T V −1[r 1] (10)    T −1 T −1  b c r V 1 1 V 1  Equation (9) can be written as

rp  1 λ    = D 1 (11) Figure 1. The Minimum Variance Portfolio Hyperbola and Efficiency 1 2     λ2  Frontier Solving the Equation (1) yields. 2.3. Tangent Line of Portfolio Efficiency Frontier Model. cr − b λ1  -1 rp  1 p   =2D   =   (12) The CML leads all investors to invest in the in the sample λ2  1  2 a − brp    risky asset portfolio WT (see Figure 2). It uses the tangent concept to find the weight of tangent point of the portfolio of rp  W = V −1[r 1]D-1   (13) assets consisting entirely for risk. 1   Let rp = rf + kσ p be a tangent line of portfolio 2 Substituting W from Equation (13) into σ p yields Efficiency Frontier model. K is slop and rf is borrowing

2 T -1 rp  rate or lending rate. Substituting rp = rf + kσ p into σ p = W VW = [rp 1]D   = 1  Equation (16) yields. 2 2 2 1 c − b rp  − σ − − σ − − (14) (d c k ) p 2k(rf b c) p ((rf b c) [rp 1] 2     ac − b − b a 1  + d c2 ) = 0 (18) − + 2 2 a 2brp crp 1 2 Since Equation (18) is a quadratic equation with one σ = = (cr − 2br + a) (15) p − 2 d p p σ ac b unknown concerning p . σ p has only one root. Equation (15) can be written as: According to the extract root formula, it find the values of 2 2 σ and σ (r − b c) p, k p − p = 1 (16) k(r − b c) 1 c d c 2 σ = f (19) p (d c − k 2 ) It explains that the portfolio frontier is a hyperbola in mean-standard deviation space. 2 2 2 k = c((rf − b c) + d c ) (20) The tangent point is 2.2. Global Minimum Variance Portfolio Point WT

k(rf − b c) Find the minimum variance portfolio of risky assets in this ( , rf + kσ p ) (21) section. The minimum variance portfolio of risky assets (d c − k 2 ) denotes asWg . Differentiating Equation (16) with respect to Assume that ri < rg (= b c ), then the tangent line has r and setting it equation to zero yields. Figure 1 denotes p T the minimum variance portfolio hyperbola and efficiency only one tangent point WT . It satisfied 1 WT = 1, the Frontier tangent point of the portfolio of assets consisting entirely for ξ b risk. The excess return is i r = (r ) = , σ 2 = (σ 2 ) = 1 p g c p g c ξ TV −1ς ξ = − = (22) i ri rf T −1 −1 ς −1 rg  V [r 1] c − b b c 1 V = D-1 = = Wg V [r 1]   2     1  ac − b − b a 1  V −1ς W = (23) T 1Y V −1ξ 72 Capital Market Line Based on Efficient Frontier of Portfolio with Borrowing and Lending Rate

that is different fromσ 2 ; it measures the no diversifiable Where ,ς i = ri − rf Figure 2 denotes the tangent line of i part of risk. More generally, for any portfolio portfolio efficient Frontier model p = (α1,α 2 ,...,α n ) of risk assets, its beta can be computed as a weighted average of individual assets beta: 2.4. Capital Market Line and Security Market Line (Sharpe, 1964; 2000) rp − rf = β p (rM − rf ) (26) In this section, it finds Capital market line and Security where market line. SML shows that the trade-off between risk and n expected return as a straight line intersecting the vertical σ M , p β = = α β (27) axis (i.e., zero-risk point) at the risk free rate. CML efficient p 2 ∑ i i σ M i=1 investment portfolios were those that provided that highest return for chosen level of risk, or conversely, the lowest risk Note that when β p =1 then rp = rM ; the expected rate for a chosen level of return. of return in the same as for the market portfolio.

When β p >1 , then rp > rM ; when β p <1 , then rp < rM . Also note that if an asset i is negatively correlation with M, σ M ,i < 0 , then βi < 0 and ri < rf ; the expected rate of return is less than the risk-free rate. In the space of expected return and beta, the line ri − rf = β i (rM − rf ) is called Security Market Line (SML). Figure 3 denotes Security Market Line with Beta portfolio.

Figure 2. The Tangent Line of Portfolio Efficiency Frontier Model 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 providing their financial capital ( , and bearing σ rf ) p (r − r ) units of risk M f . Figure 3. Security Market Line with Beta portfolio δ M The availability of this zero-beta portfolio will not affect Let (σ M , rM ) denote the point corresponding to the the CML, but it will allow construction of a linear SML. market portfolio M. The investor will have a point (σ ,r ) The combination of this zero-beta portfolio and the market p p portfolio will be a linear relationship in return and risk, on the capital market line. The capital market line (CML) because the covariance between the zero-beta portfolio and is: the market portfolio is similar to what it was with the risk-free assets. rM −r f r = r + σ (24) p f σ p M 3. Different Borrowing and Lending rM −r f Rates’ Capital Market Line is called price of risk, also is the slope of the σ M This section uses the tangent line of portfolio Efficiency CML, which represents the change in expected return Frontier model to calculate Capital Market Line CMLb rp (borrowing rate capital market line) and CMLl (lending rate per one-unit change in standard deviation . capital market line). σ p In common case, borrowing rate (rb ) is large to lending rate Theorem 1 (CAPM formula) For any asset i, (rl ) . The tangent line of portfolio efficient frontier model r − r = β (r − r ) with borrowing rate and lending rate, which is called Capital i f i M f Market Line (CML) denoted as CMLb and CMLl. where r = r + k σ T σ CMLb: The equation is p b b p , tangent point is b . β = M ,i − − i 2 (25) The tangent line denotes as D Tb B . σ M CMLl: The equation is rp = rl + klσ p , tangent point is T . is called the beta of asset i. This beta value serves as an l important measure of risk for individual assets (portfolios) The tangent line denotes as C − Tl − L . Universal Journal of Accounting and Finance 2(4): 69-76, 2014 73

Step1: Calculate the efficient frontier inputs Where, kb and kl can obtain by Equation (20). Use Excel function AVERAGE ( ) calculation mean Investor’s optimal portfolio may be located at any point on return. Use Excel function STDEVP ( ) calculation standard line T − B (borrowing funds), may be located at any point b deviation, for variance it uses VARP ( ). Use Excel function on line C − Tl (lending funds), and may be located on the CORREL ( ) and COVAR ( ) calculation correlation co-efficient and the covariance. Table 1 denoted as three curve Tl −Tb , this is neither the lending nor borrowing. Figure 4 denotes borrowing and lending rates’ capital market ’ mean return standard deviation and covariance. line Table 1. Three Stocks’ Mean Return Standard Deviation and Correlation Co-Efficient. Mean Standard return deviation Correlation co-efficient (%) (%) A 15.5 30.3 1.0 0.56 0.22 B 12.3 20.5 0.56 1.0 0.14 C 5.4 8.7 0.22 0.14 1.0 Step 2: Calculate the efficient frontier by Markowitz portfolios mean- variance analysis method r represents that 1 x 3 mean returns column vector of 3 Figure 4. Borrowing and Lending Rates’ Capital Market Line stocks. V =[σ ij ] , represents that 3 x 3 variance- covariance matrix of 3 assets returns. W represents that 1 x 3 column In Figure 4, the CML is made up of C −T −T − B ; that i b vector of the portfolio shares. V-1 represents the inverse of is, a line segment ( − ), a curve segment ( − ); and C Ti Ti Tb matrix V. 1 represents that 1 x 3 column vector of 1’s. For − another line ( Tb B ). CMLb is slightly kinked as the −1 borrowing rate is higher than the risk-free lending rate and calculation V and V , it uses Excel function COVAR ( ) small part of the concave efficient frontier is a segment of and MINVERSE ( ) From table 1, r = [15.5, 12.3, 5.4]T. the CMLb . 918.090 347.844 57.9942 V = σ =   4. Illustration [ ij ] 347.844 420.250 24.9690 57.9942 24.9690 75.6900 4.1. The Steps of this Proposed Method 0.0016400 − 0.001305 − 0.000823 −1 Portfolio optimization involves a mathematical procedure V   called quadratic programming problem (OPP). There = − 0.001305 0.0034680 − 0.000144 considered two objectives: to maximize return and minimize − 0.000823 − 0.000144 0.01389  risk. The OPP can be solved using constrained optimization techniques involving calculus or by computational algorithm Calculate the value of a, b, c, and d by using Equation (10) T −1 applicable to non- linear programming problem. This paper a = r V r, It uses Excel function MMULT use matrix operation, it includes matrix inverse, matrix −1 multiplication, and matrix transpose. The objective trace the (TRANSPOSE (r), MMULT ( V ,r) ) and obtain the portfolio frontier in mean –standard deviation space and T −1 identify the efficient frontier, the minimum variance value of a 0.668237. b = r V 1, It uses Excel function −1 portfolio and borrowing and lending rates’ capital market MMULT (TRANSPOSE (r), MMULT ( V ,1) ) and line. T −1 The steps of this proposed method. obtain the value of b 0.08699. c =1 V 1, It uses Excel −1 Step 1: Calculate the efficient frontier inputs function MMULT (TRANSPOSE (1), MMULT (V ,1) ) Step2: Calculate the efficient frontier by Markowitz and obtain the value of b 0.018926. portfolios mean- variance analysis method, 2 d = ac − b = 0.00508. Step3: Calculate global minimum variance portfolio From Equation (16), the portfolio frontier is a hyperbola in point mean-standard deviation space. Step 4: Build the capital market line under different 2 2 σ p (rp − 4.59632) borrowing and lending rate − =1 52.8373 14.1817 Step 5: Analyze different borrowing and lending rate in Step 3: calculate global minimum variance portfolio point portfolio Efficiency Frontier. The return on the minimum risk portfolio is rg and the

4.2. Example minimum standard deviation isσ g Selected three returns A, B, C. in this illustration. 74 Capital Market Line Based on Efficient Frontier of Portfolio with Borrowing and Lending Rate

b optional for investors to hold. r = (r ) = = 4.596% p g c

σ p = (σ g ) = 1 c = 7.268% V −11 From Equation (17), calculation Wg = , it uses Excel c −1 function MMULT (V ,1) , and obtain the weight (22.56%, 4.10%, 73.34%). Therefore, the minimum variance portfolio is characterized by an expected return of 4.596% and risk of 7.268% with portfolio weights of 22.56% for stock A, 4.10% for stock B and 73.34% for stock C. Step 4: Build the capital market line under different Figure 5. The Efficient Frontier and Capital Market Line borrowing and lending rate The investors have three available investment options in The capital market line under different borrowing and expected return. (1) When investor’s expected return rate is kb lending rate are CMLb and CMLl. Calculate the slope of less than rl , investor can lend riskless assets to invest risk k assets. (2) When investor’s expected return rate is between r and l by using Equation (20). l with rb , investor can invest risk assets. (3) When investor’s 2 2 expected return rate is greater than , investor can borrow k = c((r − b c) + d c ) = 0.813526 rb b b riskless assets to invest risk assets. From the Equation (25),

2 2 the beta of rl is 0.4508 < 1 and the he beta of rb is 0.4507 kl = c((rl − b c) + d c ) = 0.81533 < 1. rate of return. Since the tangent portfolio has the maximum slope, one can directly obtain the portfolio Calculate the tangent point by using Equation (21). Calculate the portfolio weight by using Equation (23). weights of the tangent portfolio by maximizing the slope of the line joining a portfolio in the portfolio frontier and the Therefore, the equation of CML is = + σ = 3.7 + b rp rb kb p risk free rate of return (Mitra and Khanna, 2014). There are two important differences between the CML and SML. First, 0.813526σ p . The tangent point T b is the CML measures risk by the standard deviation of the (δ p ,rp ) = (9.42785%, 7.70681%) . We find that investment while the SML considers only the systematic the portfolio weights of 19.9% for stock A, 42.3% for stock component of an investment’s volatility. Second, as a B and 37.8% for stock C. The equation of CMLl is consequence of the first point, the CML can be applied only to portfolio holdings that are already fully diversified, rp = rl + klσ p = 2.0 + 0.815336σ p . The tangent point whereas the SML can be applied to any individual asset or T l is (δ p ,rp ) = (9.4136%, 7.6952%) . We find that collection of assets. the portfolio weights of 10% for stock A, 30.4% for stock B The contributions of this study: (1) A portfolio with the and 59.6.8% for stock C. characteristic is known as an efficiency portfolio and lie in Step 5: Analyze different borrowing and lending rate in the solution set of the model (see Equation 1). The solution portfolio Efficiency Frontier. σ 2 (r − b c)2 The line tracing the points from the minimum variance of his model denotes as p − p = . 2 1 portfolio A to B is the efficient frontiers. A risk adverse 1 c d c 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 5. Findings and Discussions expected return than C with the same portfolio risk. More risk adverse investors would choose points on the efficient It is noticed that if the risk free lending and borrowing frontier that is closed to point A. Similarly, an investor who rates are equal, the optimum risky portfolio is obtained by is less risk adverse or can handle more risk would choose a drawing a tangent to the portfolio frontier from the risk free portfolio closed to point B. Where a, b, c denoted as matrix operation. It is easy It traces the portfolio frontier by selecting different value calculation by Excel function. (2) The CML is denoted as of expected return and calculating the corresponding weight vector too. (3) By the definition of CML, we are standard deviation by the Equation (16). Figure 5 shows the able to find the efficient market portfolio. (4) By the graph of the frontier. concept of tangent line of portfolio efficiency frontier model The capital market line under different borrowing and to calculate Capital Market Line CMLb (borrowing rate lending rate are CMLb and CMLl. Broken line E -T1 -T2 –B in capital market line) and CMLl (lending rate capital market figure 5, it constitutes with line (CMLl and CMLb) and an arc line). (5) To find out the weight of securities which are (T, Tb). The investors can lend at one rate but must pay a there in the portfolio in order to invest in those securities. (6) different and presumably higher rate to borrow. The To construct CML in order to get more returns than that of efficient frontier would become E -T1 -T2 –B in Figure 5. efficient frontier if risk-free securities are included in the Here there is a small range of risky portfolios that would be portfolio. Universal Journal of Accounting and Finance 2(4): 69-76, 2014 75

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. = i = 1, 2,..., n − 2, j = 1, 2,..., n − 2 ∑aij w j b j ,

Table 2. A Comparison between this Study and Xuemet and Xinshu’s Research Xuemet and Xinshu (2003) This study

The efficient frontier of Markowitz can represent as linear equation group (simultaneous equation) that is The efficient frontier of Markowitz can composed by n-2 linear equations. The efficient represent as frontier of Markowitz can represent as σ 2 (r − b c)2 Design p − p = 1 = , 2 ∑aij w j b j 1 c d c i = 1, 2,..., n − 2, j = 1, 2,..., n − 2 By the definition of the CML, the efficient market portfolio thus can be identified.

Use the matrix operation concept to calculate Use different geometry concept to calculate normal efficient portfolio. vector. Methodology Use the concept of tangent line to calculate Construct a simultaneous equation in order to obtain Capital Market Line CMLb (borrowing rate efficient portfolio. capital market line) and CMLl (lending rate capital market line)

Algorithmic tool MATLAB software EXCEL software

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

6. Conclusions optimization models and mean variances spanning test, Handbook of quantitative and risk management, pp. 115-184, This paper has discussed an algorithm (matrix operation) 2010. to look for the efficient market portfolio of CAPM and a [2] Cochrance , J. H. “Asset pricing, Princeton”, N. J.: Princeton new method (tangent line) to obtain the CML. This University Press, 2001. algorithm applied in different borrowing and lending rate [3] Constantinides, G. M. and malliarias, A. G. “Portfolio theory, portfolio problems. Under this algorithmic tool, this paper Finance”, Springer Velag, pp. 1-30, 1995 have finished the following works (1) Prove portfolio frontier is a hyperbola in mean-standard deviation space (2) [4] Duffie, D. and Rahi, R. “ innovation and security design: An introduction”, Journal of Economic CML is also expressed as the portfolio weight vector (3) In Theory, Vol. 65, pp. 1-42, 1996.. mean-variance portfolio efficiency frontier model, calculate different borrowing and lending rate’s CML (4) calculate the [5] Elliott, R. J. and Kopp, P. E., “Mathematics of Financial tangent point and slop of CML (5) The steps of this proposed Markets”, Springer, 1999. method [6] Huang, C. F. and Litzenberger,, R. H. “Foundations for Excel is far from the best program for generating the Financial Economics”, N. J., Prentice-Hill, Englewood Cliffs, efficient frontier and is no limited in the number of assets it 1988. can handle. It finds that Excel, the computation of the [7] Jarrow, R. “Finance Theory”, Prentice Hall Englwoos Cliffs, efficient frontier is fairly easy. It finds that this paper is NJ, 1988. helpful to the correct application of the capital market line [8] LeRoy, S. F. and Werner, J. “Principle of Financial based on efficient frontier of portfolio with borrowing and Economics”, Cambridge: Cambridge University Press.2001. rate, enriching the theory and method of invest management. The further research will focus on performing the [9] Levy, H. and Sarnat, M., “Internationally diversification of investment portfolios”, American Economic Review, Vol. 60, calculation of this algorithm tool such as Excel and. September, pp. 668-675, 1970. [10] Lintner, J. “The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets, review of Economics and Statistics”, V0l. 47, No. 1, pp. REFERENCES 13-37, 1965. [1] Chen, W. P., Chung, H., Ho, K. Y. and Hsu, T. L., Portfolio [11] Mangram, M. E., A simplified perspective of the Markowitz 76 Capital Market Line Based on Efficient Frontier of Portfolio with Borrowing and Lending Rate

portfolio theory, Global Journal of Business Research, Vol. 7, Management and Farming System Journal , Vol. 7, No. 1, pp. No. 1, pp. 77-91, 2013. 11-15, 2010. [12] Markowitz, H. M. “Portfolio selection: efficient [19] Painter, M. J., the portfolio diversification impact of a diversification of investment”, Wiley, New York., 1959. farmland real estate investment trust, International Business & Economics Research Journal, Vol. 9, No. 5, pp. 115-123, [13] Markowitz, H. M., “Portfolio selection”, Journal of finance, 2010, March, pp. 71-91, 1952. [20] Roll, R. and Ross, S. A. “An empirical investigation of the [14] Mas-celell et al. “Microeconomic theory”, New York: Oxford ”, Journal of Finance, Vol. 35, pp. University Press, 1995. 1073-1103, 1980.

[15] Merton, R. and Bodie, Z. “Deposit insurance reform: a [21] Sharp, W. F. “Capital asset price: a theory of market functional approach”, in conference series on public policy, equilibrium under condition of risk”, Journal of Finance, Vol. 1992. 19, No. 3, pp. 425-442, 1964.

[16] Mitra, A., and Khanna, P., A dynamic spreadsheet model for [22] Sharp, W. F. “Portfolio Theory and Capital Markets”, New determining the portfolio frontier for Bes 30 stocks, York, McGraw-Hill, 2000. Independent Journal of Management & Production (IJM&P), Vol. 5, No. 1, pp. 106-120, 2014.. [23] Treynor, J. I. “How to rate the management of investment funds”, Harvard business review, Vol. 43, pp. 63-75, 1965. [17] Mossin, J. “Equilibrium in a Capital asset market”, Econometrica, Vol. 34, No. 4, pp. 768-783, 1966. [24] Xuemei, A. and Xinshu, ,The identifying of the efficient market portfolio, Chinese Business Review, Vol. 2, No. 5, [18] Painter, M. J., Equity financing and investment opportunities pp. 70-75 , 2003. in Canadian primary agriculture, Agriculture Business