Microsoft Word - RE-Non-Diversifiable Risk in Investment Portfolios an Aid to Investment Decision Making Received 1 January 2015 Accepted 5 February 2015 Non-Diversifiable Risk in Investment Portfolios --- an Aid to Investment Decision Making Emma Anyika Department of Accounting and Finance, Mount Kenya P.O. Box 342-01000 Thika, Kenya E-mail: mmnk55378@gmail.com Abstract Modeling Non - Diversifiable risk in investment portfolios is undertaken in this paper together with redefinition of estimators of diversifiable risk and portfolio expected returns to reflect normal market conditions. GARCH (General Auto - Regressive Conditional Heteroskedasticity) models are then used to make forecasts of given time series, from which future predictions of Non - Diversifiable risk, Diversifiable risk and portfolio expected returns are made. The required investment decisions are then made. In making investment decisions several factors are considered. These include profits, dividend yield, price earning ratios, and expected future performance of financial institutions. This paper has considered expected future performance of financial institutions. In particular the paper derives a method of determining non - diversifiable risk in investment portfolios that enables investors and investment managers make viable investment decisions. This study is expected to improve the accuracy of predicting future expected performance of financial institutions. Investment analysts can now rely on the predictions to make good investment decisions. Non-diversifiable risk, Diversifiable risk, GARCH, portfolio 1. Introduction Risk underlies most investment decisions. This is because it is not possible to anticipate the occurrence of possible future events with certainty and hence, making any correct predictions about the cash flow sequence is not possible. The simplest expression for risk in investment is the standard deviation of returns on investments. For single investments other risk determining expressions include coefficient of variation and the Beta (B) factor method, Wilkes (1996) and Value at Risk, Vlaar (2000). When dealing with investment portfolios, return on investment is a weighted average of the expected returns of the individual investment. This alone will not allow one to make investment decisions since one needs to determine the best portfolio by looking at that which will give maximum returns at the lowest risk. This necessitates determination of portfolio risk, which is divided into non–diversifiable and diversifiable risk. Diversifiable risk is that which the investor can eliminate if he held an efficient portfolio. The non– diversifiable risk on the other hand is that risk which still exists in all well diversified efficient portfolios. The investor therefore seeks to eliminate the diversifiable risk. Nevertheless non–diversifiable risk permeates most areas of investment. These are particularly those characterized by unstable rates of returns due to the erratic nature of market forces. These include interest related portfolios and also exchange rates related ones. For new, short term, and long term players in investment markets there is need for proper determination of non diversifiable risk as a basis for investment decision making. This paper focuses on the problem of limited tools for estimation and prediction of non–diversifiable risk in investment portfolios that has lead to indecisiveness by investors and incorrect investment decisions. Non–diversifiable risk is not normally determined since most estimators present are Journal of Risk Analysis and Crisis Response, Vol. 5, No. 1 (April 2015), 31-46 Published by Atlantis Press Copyright: the authors 31 E. Anyika. used to determine diversifiable risk which can on the onset eliminate some risk by combining several investment portfolios until the best one is realized i.e. one with the highest returns at the lowest risk. Thus new and long–term investors who want to enter risky businesses, i.e. risk seekers, lack risk benchmarks on which to base their investment decisions. Many investors congest short–term investment markets believing that they have high returns and are less risky. This eventually brings down the earnings from these investments. For example short–term interest earning government bills and bonds will have their interest earning rates decreased if we have many investors in the market. Thus investors are left with only long term investments to consider. The situation could at least be made bearable if they knew precisely how much risk they are bound to encounter and how it will affect their rates of return. The aim of this research was to determine a method of estimating non–diversifiable risk to enable investment managers make viable investment decisions. This estimator is that which accounts for the normal market conditions. This will decrease losses made by financial organizations due to inability to predict non– diversifiable risk accurately and reduce drastically the large provisions set aside to hedge and manage risk. The financial environment is full of risk hedging techniques and risk management measures for systematic risk. These are mostly done on speculative terms Jennifer (2003). There are no real concrete and long lasting solutions to the problems of non–diversifiable risk. Most financial market players blame it on the volatility of financial markets, i.e. that one cannot determine and predict non–diversifiable risk since the forces determining it keep on fluctuating. The proper determination of non–diversifiable risk in this paper is expected to, prevent the making of wrong investment decisions by, for example, relying only on an investment managers belief or experience in risk management, boost the confidence of new, short - term and long–term investors since they would know the level of risk to be encountered. It would also enable investors have a broad spectrum of portfolios to invest in since most of them just engage in short term investments where non–diversifiable risk can be speculated easily, facilitate development of suitable risk managing and hedging techniques to minimize non– diversifiable risk and set free colossal amounts of money from elaborate schemes hatched to hedge and manage portfolio risk thus enabling reinvestment to improve or increase the profitability of financial organizations. The uncertainty of non–diversifiable risk, i.e., the fact that it is believed not to be exactly known Robert (1993), always makes investment markets so jittery such that any small force acting on these markets sends them tumbling. The proper determination of non– diversifiable risk will go a long way into stabilizing the investment markets. Basic concepts applied include, diversifiable risk (also known as non –systematic or specific risk) which is the risk that can be eliminated by diversification. It is unique to the company or it’s industry and includes management competence and shifts in demand for company products, non–Diversifiable risk (also known as systematic or market risk) which is related to fluctuation of the market as a whole and cannot be eliminated by diversification, Portfolio expected returns a weighted average of the expected returns of each asset held in isolation and Investment portfolio which is a combination of more than one investment with a belief that risk shared is risk less. 2. Literature Review Originally portfolio expected return was the only criterion for making investment decisions. This has been lacking particularly in areas where the future of the expected returns was uncertain. Thus other methods which incorporated determination of future uncertainties have to be sought. Markowitz (1952) identified statistical measures of dispersion as measures of risk to be a way of indicating future uncertainties in investment decision making. Later on he modified his risk estimator to include correlation coefficient. This was significant since for example if investments are perfectly negatively correlated i.e. their correlation coefficient is negative one, and then holding them in a portfolio greatly reduces their risk. This is synonymous to the idea of not carrying all your eggs in one basket, i.e. in different baskets risk is shared thus less. Researchers then began to examine the implications of all investors using this approach. This led to the development of the Capital Asset Pricing Model (CAPM), Sharpe, et al (1964), which Published by Atlantis Press Copyright: the authors 32 Non-Diversifiable Risk in Investment Portfolios gave a simple relationship between expected returns and risk in a competitive market. It also provided further insight into the process of diversification. Although there is a great deal of evidence which supports CAPM, there are major difficulties in testing the model. First it is stated in terms of investors’ expectations rather than historic returns. Secondly the market portfolio (M) should include all risky investments, whereas most market indices contain only a sample of shares. Furthermore research in the 1980’s mainly on U.S.A stocks revealed a number of stock market anomalies which are inconsistent with CAPM. These include “the small firm effect” and “calendar effects” Dimson (1988). The model should therefore not be relied upon for more than general indications of the market pricing mechanism. Currently, research in this area is centered on how to use the Markowitz estimator of risk and the CAPM in investment decision making. Less emphasis is being placed on further development of better risk estimators and models. Stulz (1999) argues that total risk is often costly and discusses how taking total risk into account in capital budgeting is necessary to make capital budgeting and capital structure decisions consistent. Harper (2003) discusses risk and returns in balanced portfolio and views the efficacy of cash and cash equivalents in investor portfolios. All the above researchers do not account for non–diversifiable risk as properly defined in their risk estimation thus rendering their estimators inaccurate. This paper seeks to remedy this by determining total risk which accounts for diversifiable and non–diversifiable risk according to its definition. 2.1. Research methodology The population for this study was "Turnover of shares in Kenya shillings (Ksh) on the Nairobi Stock Exchange (NSE) from the year 2009 to 2013”. NSE is a Kenyan Capital Market. Currently there are fifty six financial institutions listed on the NSE. Every day the total turnover of shares in terms of volume and Ksh. is recorded for each one of the fifty six financial institutions. Each month a bulletin is released showing the ten or twenty leading companies in terms of turnover in Ksh. These companies differ from one period to another. Thus data from NSE is first analyzed. Returns for shares are derived by taking a base month and the weighted average share price of an investment stock for that month and considering it as a buying price Subsequent share prices are taken as the selling prices and the returns are thus calculated. Simple random sampling is then used to pick the companies with the best returns over the five year period. This entails ranking the companies in terms of one with the highest turnover in Ksh for the five year period. It also involves ensuring that the companies selected are those that are making positive returns frequently. This is due to the fact that this study determines methods that will enable investors make the best investment decisions, thus picking a financial institution with the highest and most frequent positive returns is the first step towards making good investment decision. Turnover from shares is derived by multiplying the weighted average share price per month by the turnover of shares in volume per month. Simple random sampling is then used to pick a number of companies from those with the highest turnover over the five year period. Assumptions made include the following; the sample data size is representative of the population, the values of weights range from negative infinity to positive infinity. It is important to note that the estimators derived and tested are used to predict future performance of certain financial institutions from the rest of the population. These estimators are expected to be used for prediction of future performance of financial institutions situated in any location, the only limitation to the use of the estimators being the accuracy of the forecasting techniques. S–Plus version 2000 and Mat lab version 6.1 software are used for mathematical and statistical calculations, problem solving, graphing and forecasting. 3. Results 3.1. Derivation of True Functions of Diversifiable and Non–Diversifiable Risk Let the Markowitz portfolio expected returns and diversifiable variance be given by (1) and (2) respectively. See Andrew (1995) and Alexander (1974). Published by Atlantis Press Copyright: the authors 33 E. Anyika.     1 )()( i iiu RExRE (1)       j ijji ii iim xxxP  11 222 2 (2) Non–diversifiable variance is computed as     1 222 i iiq xP  (3) where,  ix ,  2 0 i , and ix = Weight of an investment i iR = Return of investment i  uRE = Expected returns 2 i = Variance of investment i ij = Covariance of investments i and j and u, m & q are arbitrary symbols differentiating i & u expected returns and non–diversifiable & diversifiable variances respectively. To find the weight of investment i that will maximize expected returns and minimize total risk we apply the classical optimization method with no constraints See Rao (1994). We thus differentiate the expression; jixxxRxERE i j ijji i ii i iimu                    ,2)( 1 11 22 1 2  with respect to ix i.e.  2 2 1 ( ) ( ) 2 2 0, u m i i i j ij ji E R P E R x x i j x              (4) and differentiate 2 2 2 2 2 2 1 1 1 1 2 ,m q i i i j ij i i i i j i P P x x x x i j                   with respect to ix , i.e.  2 2 2 1 4 2 0, m q i i j ij ji P P x x i j x             (5) where   2u mE R P = maximum returns (derived by subtracting diversifiable variance from expected returns) 2 2 m qP P = total variance (derived by adding diversifiable variance to non–diversifiable variance) Note: i) The second derivative of (4) is equal to 22 i implying that ix obtained will always maximize returns. ii) The second derivative of (5) is equal to 24 i implying that ix obtained will always minimize risk. See Rao (1994). Equate (4) to (5) to get, the expected returns of investment i . 2 2 1 1 ( ) 2 2 4 2 ,i i i j ij i i j ij j j E R x x x x i j                2 1 6 4 ,i i i j ij j E R x x i j       2 1 1 ( ) 3 2 , 2 i i i j ij j E R x x i j       (6) Hence for infinite investments, values of ix are given by the expression                                                    RE RE RE x x x      2 1 2 12 1 2 1 2 2 221 112 2 1 32 232 223    (7) Therefore, it follows from the derived values of ix in equation (7) and substituting them in the square root of equations (2) & (3) that the true function for portfolio diversifiable risk is given by 1 2 2 2 1 1 1 ,i i i j i j i i j x x x i j                  (8) and that for portfolio non-diversifiable risk is   2 1 1 22  i iix  (9) 3.2 Empirical Study Using Sampled Data In this section, an empirical study is done to check if the true functions developed in the previous section work as estimators. i) The data used was turnover of shares for Kenya Commercial Bank (KCB) given in Appendix 5 and East African Breweries Limited given in Appendix 6 First we normalize /standardize the data by taking logarithms and getting the first difference i.e. log Published by Atlantis Press Copyright: the authors 34 Non-Diversifiable Risk in Investment Portfolios 1 1 2 t t t t y y y y          , whereby the sum of the ratios of the values of the difference between consecutive periods and the earlier of these periods will give the trend line, see Figure 1 given byMurray in1961 . Final values for finding the risks are given by the weights,                              142371.0 158996.0 )590005.0(3)167223.0(2 )167223.0(23621844.03 552.0 2 1 2 1 2 1 x x where, variance of KCB = 0.3621844 variance of EABL = 0.590005 covariance of KCB & EABL = 0.167223 expected returns of KCB = - 0.158996 expected returns of EABL = - 0.142371 Therefore Diversifiable risk of a portfolio of KCB & EABL = 0.082687821 or 8.268% and its non – diversifiable risk = 8.88%. ii) If we assume that 0ix implying that sales are always positive and 1 1   i ix implying perfect market conditions for the same data as in i) above (Markowitz assumptions) then 2 2 2 2 2 1 1 2 2 1 2 122pP x x x x     where 2 pP = portfolio variance. To find the value of 1x that minimizes portfolio risk we differentiate 2pP with respect to 1x and equate it to zero 2 2 1 1 2 12 1 2 2 pP x x x         121 2 11 122  xx  = 0 12112 2 11 222  xx  = 0 0.724369 1x + 0.33444 - 0.33444 1x = 0 0.389929 1x = - 0.33444 1x = 389929.0 33444.0 Note: The negative numerator value 1x indicates short selling. (Act of selling securities you do not own) since 0ix  ， 1x becomes 389929.0 33444.0 1 x = 0.85769 KCB turnover of shares - 1.5 - 1 - 0.5 0 0.5 1 1.5 1 2 3 4 5 6 7 8 9 1 11 Months T u rn o v e r Series1 Figure 1: A plot of normalized turnover per month. Published by Atlantis Press Copyright: the authors 35 E. Anyika. since 1 1i i x    ， 1423053.01 12  xx and Diversifiable risk = 85.769 % while Non–diversifiable risk = 14.23 % iii) When we use returns of shares of KCB from Appendix 1 and EABL (Figure 2) from Appendix 4 we are not making any of the above assumptions The final results are as follows, the weights 13 14 09 1 23 14 12 07 2 3.3922 10 3.2582 10 6.8657 10 7.20592 10 3.2582 10 1.8238 10 1.5087 10 x x                                  where 3 (variance of KCB) = 3.3922 1310 3 (variance of EABL) = 1.8238 1210 2 (covariance of KCB & EABL ) = 3.2582 1410 ½ ( value for expected returns of KCB) = -6.8657 0910 ½ ( value for expected returns of EABL) = -1.5087 0710 9026.198436,5220 21  xx Therefore diversifiable risk = 5.310 and non– diversifiable risk = 17.215 % THEOREM 1: Non-diversifiable risk,   1 22 2 i ix  of a given investment i remains unchanged for i =1, 2, 3, …,  , investment of a given portfolio. We make the following assumptions; i) ix    ii) For diversifiability, 2 20 i ix    and for non - diversifiability, 2 2i ix z  , where  z0 . iii) Sampled data is normally distributed and representative of the population. iv) Variances of investments i 1, 2, 3, …,  . are uncorrelated for non–diversifiability. Proof: From equation (5), the value of 1x that minimizes risk for a two sampled investment portfolios is given by, 2 2 2 1 1 2 12 1 2 1 1 2 12 2 1 1 2 12 2 1 1 2 12 2 2 1 1 ( ) 4 2 4 2 0 4 2 4 2 4 4 m nP P x s x s x x s x s x s x s x s x s s s            2 12 1 2 1 1 2 x s x s   (10) For three investments the value of 1x that minimizes risk is given by, 2 2 2 1 1 2 21 3 31 1 ( ) 4 2 2 0m n P P x s x s x s x        NORMALISED RETURNS FOR EABL -0.000003 -0.000002 -0.000001 0 0.000001 0.000002 0.000003 0.000004 0.000005 1 2 3 4 5 6 7 8 9 10 11 Months Returns Series1 Figure 2: Normalized / Standardized Returns of EABL (derived by taking the first differences of the returns of shares, then the reciprocal of these differences). Published by Atlantis Press Copyright: the authors 36 Non-Diversifiable Risk in Investment Portfolios which gives 2 1 1 2 21 3 314 2 2x s x s x s   so that 3 312 21 1 2 2 1 1 1 1 2 2 x sx s x s s    (11) For four investments the value of 1x that minimizes risk is given by, 3 312 21 4 411 1 1 1 2 2 22 2 2 1 1 1 x sx s x s x s s s     (12) continuing in the same manner we see that for infinite investments the value of 1x that minimizes risk is given by, 3 312 21 4 41 11 1 1 1 1 2 2 2 22 2 2 2 1 1 1 1 ... x sx s x s x s x s s s s      (13) Similarly 2 3, ,...,x x x can be obtained. Taking expectations of equations (10), (11), (12), (13), to remove bias in the sample variance and considering the assumption (iv) we obtain           2 1 1 1 2 212 2 1 1 1 2 212 2 1 1 1 22 2 1 1 1 0 0 E x s E x s x E s x E s n x x n as n x               where n is the sample size and n>1. NOTE:  ijE  = 0 from assumption (v) . For equation (11), Similarly for investments 4, 5, 6,…, , 2 0i ix   .Thus non–diversifiable risk estimator (since we are using sampled data) is   122 2q i iP x  of investment i = 1, 2, …, , for a given portfolio remains unchanged. iv) Analysis of returns of shares of three investments (see Appendix 1, 2 and 3) gives the following results, 18 20 15 1 21 20 15 21 2 15 21 20 3 08 07 07 1.14589 10 2.82915 10 9.10521 10 15097 10 2.82915 10 9.10521 10 6.83994 10 9.10521 10 6.83994 10 4.85532 10 6.86872 10 1.50874 10 3.32536 10 x x x                                                         (14) where the right hand side is the product of the reciprocal of determinant of variance covariance matrix, inverse of variance covariance matrix and values representing means of the three investment returns. 719814553.0 581127467.1 484843067.3 3 2 1    x x x Thus diversifiable risk of a portfolio of KCB, EABL &STAN CHART = 2.54382 0610 %, and its non– diversifiable risk = 8.93518 0610 %. For a two investment portfolio of KCB and EABL diversifiable risk for KCB is 0.01755 and that of EABL is 0.15460. For an investment portfolio of KCB, EABL and STANCHART diversifiable risk for KCB is 2.15374 0910 , EABL is 8.71066 0810 and that of STAN CHART is 9.14787 1110 . Clearly portfolio risk has been diversified but non– diversifiable risk for KCB and EABL is not the same as that one for two investments as per the definition of non–diversifiable risk and the findings of theorem one. This is also true for turnover of a three and four investment portfolio. Thus there is some white noise in the data analyzed. We therefore remedy our estimator for non–diversifiable risk to include the random error. This is done by adding   122 1 ie i s    to the non– diversifiable risk estimator. We thus denote this model estimator as,           2 1 1 1 1 2 21 3 312 2 2 1 1 1 1 2 32 2 2 1 1 , 1 0 0 , 0 . E x s E x s E x s n x x x n as n x                 Published by Atlantis Press Copyright: the authors 37 E. Anyika.     1 2 1 22 2 2 1 1 iG i i e i i p x s s        where ie is an independent random variable with mean zero and variance 2 1 eis n  (i.e. sample variance) THEOREM 2: The non–diversifiable risk model estimator GP is a consistent estimator of non– diversifiable risk qP . We make the following assumptions; i) ie is an independent random variable with mean zero i.e.   02  ie E  and variance 1 2 n ie  ii) Variances of investments i 1, 2, 3, …,  . are uncorrelated. iii) Sampled data is representative of the population and is normally distributed. Where G & q are arbitrary symbols. Proof: By definition we note that     1 1 2 2 2 2 2 1 1 iG i i e i i P x s s        Thus     1 21 22 2 2 2 2 2 2 1 1 1 1 2 i iG i i i i e e i i i i s x s x s s s               (15) The expectation of equation (15) is     2 2 2 2 1 1 1 2 i iG i i i i e e i i i E P E x s E x s s E s                             Hence  2 2 2 1 1 G i i i n E P x n             as n  2 2 2 1 ( 0)G i i i E P n x             Note : Since 2 is is the sample variance of sampled data of investment i which is normally distributed thus there is bias in the unmodified form of the sample variance. Therefore   1 22 2 1 G i i q i P x        This proves that GP is an unbiased estimator of qP . From (15), observe that       11 22 1 1 2 2 2 2 2 2 2 2 2 1 1 1 4 4 4 4 4 1 1 1 2 2 2 1 1 var var var var 2 2 2 1 1 1 var i i i i G i i i i e e i i i i i i i e i i i i i e i i P x s x s s s x x n n n x                                                                                        1 1 2 2 4 1 4 4 4 4 4 2 2 2 4 1 1 1 1 1 1 2 1 2 2 var 1 1 i i i i e i i i i i e i i e e i i i i i i n x x x n n                                                                                   (16) Note: In equation (16),   122 2i ix  is rewritten as    1 22 2 2 2 i i i ix x   and   1 22 ie  as    1 22 2 i ie e    to assist in simplification. Also  2 42var 1 s n   as proved by Tobago (2010) Recalling that as n    G qE   , it follows from this result (i.e.  var 0G  ) that GP is a consistent estimator of Non–diversifiable risk. 4. Discussion 4.1. Presentation of Results 4.1.1 Forecasts of Time Series In forecasting we used GARCH (Generalized Auto- Regressive Conditional Heteroskedastic) models. This is because the time series values have low valued correlations. In GARCH modeling the true GARCH (A, B) model parameters of the time series are entered. These parameters correspond to a given GARCH (A, B) model for the conditional variance F (t) and innovations Y (t), sequences Published by Atlantis Press Copyright: the authors 38 Non-Diversifiable Risk in Investment Portfolios                           2 2 2 2 2 1 2 2 ( ) F t = L t + S 1 * F t - 1 + S * F t - + S P * F t - P + W 1 Y t - + W Y t - + W Q Y t Q      for time steps t = 1,2, … N, where S = order of AR W = order of MA A & B = model order determined by the number of elements of S & W S, W = coefficient. t = the current time index. Y (t) = square root F (t) L (t) where L (t) is an identical and independent sequence N (0, 1) and Y and F are related. The forecasts for the investment time series from Appendix 1 through to Appendix 3 are given in Table I and Table II. 4.1.2 Prediction of Expected Returns Diversifiable Risk and Non–Diversifiable Risk Using the forecasts from Section 4.1.1 above and equations 7, 8 and 9 the predictions for expected returns, diversifiable risk and non–diversifiable risk are made as Table III and Table IV. 4.2. Result Discussion 4.2.1 Derivation of Forecasts In forecasting using GARCH models we let t be the current time index, {F (t)} be the return series of interest, {Y (t)} the innovations noise process and L(t) be an identical and independent sequence. The input coefficient vectors AR (Autoregressive) and MA (Moving Average) are specified exactly as they would TABLE I: The table below represents forecasted returns of shares of the given companies in Kenyan shillings ( ksh). COMPANY EABL ICDC KCB STAN CHART 1 55660000 3881300 38460000 32670000 2 42710000 3590200 29520000 25070000 3 36010000 3320800 24880000 21130000 4 31720000 3071700 21920000 18620000 5 28680000 2841300 19810000 16840000 6 26370000 2628200 18230000 15480000 7 24550000 2431000 16960000 14400000 8 23050000 2248600 15930000 13540000 9 21810000 2080000 15070000 12790000 10 20740000 1923900 14330000 12180000 11 19820000 1779600 13690000 11630000 12 19000000 1646100 13130000 11160000 13 18290000 1522600 12640000 10730000 14 17650000 1408400 12200000 10360000 15 17070000 1302800 11790000 10020000 16 16540000 1205000 11430000 9710000 17 16060000 1146000 11100000 9420000 TABLE II: The table below represents forecasts of turnover of shares of the given companies in ksh. COMPANY EABL ICDC KCB STAN/ CHART 1 105230000 8609700 406500000 69829000 2 99830000 7718800 385640000 66245000 3 94710000 6920100 365850000 62846000 4 89850000 6204100 347080000 59621000 5 85240000 5562100 329270000 56561000 6 80860000 4986600 312370000 53659000 Published by Atlantis Press Copyright: the authors 39 E. Anyika. be read from the ARMA (Autoregressive Moving Average) (R,M) model equation when solved for F (t): F (t) = L (t) + AR (1) F (t-1) +… + AR(R) F (t-R) + Y (t) MA (1) Y (t-1) +… + MA (M) Y (t-m) Note that the coefficients of F (t) and Y (t) are assumed to be 1, and are not part of the AR / MA input vectors. For the following ARMA (2, 2) model, of ICDC F (t) = 0.036Y (t-1) - 0.002 Y (t-2) + Y (t) + 0.402 Y (t- 1) + - 0.107 Y (t-2) AR = [0.036 – 0.002] and MA = [0.402 – 0.107]. The first 20 weights of the infinite order AR approximation may be found as follows: ZI = garchar ([0.036 – 0.002], [0.402 – 0.107], 20); where ZI represents weights of the polynomials generated and garchar converts j–th lag of the return series and innovations processes F (t-j) and Y (t-j), respectively. To maintain consistency, the j-th element of the truncated infinite–order auto- regressive output vector, ZI (j), is the coefficient of the j–th lag of the observed return series, F (t-j), in the infinite order representation of the input ARMA (R, M) process. The AR and MA input vectors differ from the corresponding AR and MA ‘polynomials’ formally presented in time series. To estimate, and make forecasts from the GARCH (A, B) parameters from the equation: Y (t) = square root (F (t)*L (t)) where Y (t) represents innovations noise process, you simulate (i.e. , to reverse–engineer the process for comparison) (see Bollersler (1986), Box, Jenkins (1994), Engle and Robert (1982), Hamilton 1994). Figures 3 and 4 show that the forecasted time series is normally distributed thus there is no need for its standardization before making predictions. This also explains the downward trend of the forecasts i.e. to maintain the bell like shape of the curve indicative of normally distributed data. (Note that the plots show values for half the distribution because both halves are identical). See Lucey (2000). Returns of shares are forecasted in the long term since these data were prepared for long term investors who are not ready to sell their shares in the near future. Short term forecasts are meant for speculative buyers who are risk averse and can sell their shares anytime the prices go up. The estimators used to predict future expected returns, diversifiable risk and non– diversifiable risk are applicable within the normal ranges of negative infinity to positive infinity. This is reflected in Tables III and IV where values range from the tens to hundreds to thousands etc, compared to Markowitz estimator of diversifiable risk whose weight values ix must be greater than or equal to zero i.e. 0ix . The sum of these weights for any number of investments in a portfolio must be equal to one i.e. TABLE III: Seventeen month predictions of returns of shares of the given companies Portfolio Expected returns Diversifiable risk Non–diversifiable risk EABL & ICDC 0.559 or 55.9 % 0.219 or 22 % 0.223 or 22.32 % ICDC & STANCHART 1.27 or 127 % 0.503 or 50.3 % 0.5104 or 51.04 % EABL & STANCHART 1.14 or 114 % 0.478 or 47.8 % 0.478 or 47.84 % EABL, ICDC & STANCHART 1.34 or 134 % 0.532 or 53.2 % 0.552 or 55.21 % TABLE IV: Six month Predictions of turnover of shares of the given companies Portfolio Expected returns Diversifiable risk Non–diversifiable risk EABL & ICDC 1822 % 150.82 % 255 % ICDC & STANCHART 1821 % 150.81 % 262 % EABL & STANCHART 2063 % 203 % 203.2 % EABL, ICDC & STANCHART 474 % 83.79 % 95.3 % Published by Atlantis Press Copyright: the authors 40 Non-Diversifiable Risk in Investment Portfolios     1 1 i ix . The first assumption limits the data set being used since if it is one with negative returns the estimator collapses. This means that one assumes positive returns always and yet many a time financial institutions experience losses or negative sales. The second assumption implies perfect market conditions, i.e. the portfolio is efficient with no systematic risk present. Under normal financial environments this is not true since fluctuations in returns do occur frequently. The development of an estimator for non– diversifiable risk in this study is an indication of the presence of market risk. For kR the return of k CAPM is given by          2 m f km k f m E R E R E R E R      where m= market portfolio ( )kE R = expected returns of the risky security k  fE R = risk free rate of return  mE R = expected returns of the market portfolio m 2 m = variance of the market portfolio m km = covariance between the single risky security k and the market portfolio m This model works under the following assumptions, See Andrew (1993): i) All investors are risk–averse and measure risk in terms of standard deviation of portfolio return (as for the Markowitz model). ii) All investors have a common time horizon for investment decision making ( e.g. one month or two years ). iii) All investors have identical subjective estimates of future returns and risks for all securities. There exists a risk–free asset and all investors may borrow or lend unlimited amounts at the risk–free nominal rate of interest. iv) All securities are completely divisible, there are no transaction costs or differential taxes, and there are no restrictions on short–selling. Information is freely and simultaneously available to all investors. Although there is a great deal of evidence which supports CAPM, much courage is required to develop a model on the basis of these assumptions. Many of them are clearly unrealistic, for example the assumption that there exists a risk–free asset and all investors may borrow or lend unlimited amounts at the risk–free nominal rate of interest. The model should therefore not be relied upon for more than general indications of the market pricing mechanism as seen in section 2.1. The estimators developed in this study have addressed the above short comings such that none of the above assumptions are made in deriving these estimators. Furthermore the Beta factor i.e. 2 km k m B    is an indicator of non–diversifiable risk. This is not a good indicator since from the definition of non–diversifiable risk, there should not be any correlation between the various investments, yet the expression km in the beta factor formulae represents covariance between single risky security k and the market portfolio M which counterfeits the definition of non–diversifiable risk, thus giving weight to the doubts cast on the CAPM practical applicability by researchers in the 1980’s. The predictions in tables III and IV indicate that the higher the risks the higher the expected returns. This is expected since high risks imply venturing into unknown areas. Thus one can only be motivated to do so if he/she expects big rewards for taking these risks. This is experienced in financial markets where monopolists venture into new business areas. If their ventures are successful they expect to reap maximum benefit since there is no competition from other financial institutions. If they fail in their ventures they stand to incur heavy losses since they will not be able to recover their base capital among other costs incurred. There would be no other financial institution trading in the same items thus they cannot sell their business to them or merge with these financial institutions in order to diversify their risk thus minimizing it and reaping the benefits of economies of scale. The predictions show a clear distinction between diversifiable risk and non–diversifiable risk, whereby non–diversifiable risk is always the higher of the two risks. Table IV which has six month predictions of turnover of shares shows this observation clearly. Unlike the data for returns of shares, the data for Published by Atlantis Press Copyright: the authors 41 E. Anyika. turnover of shares is not derived using a common reference price. Variances in the time series tend to be huge leading to higher risks. The length of time the forecasts postulate, i.e. six month have an impact on risk. Whereby investors can sell their investments easily when the prices are high thus returns will be high. The high returns will thus be accompanied by high risks as experienced by monopolists. 5. Conclusion and Recommendations 5.1. Conclusion In this study a non–diversifiable risk model estimator, diversifiable risk and expected return estimators have been developed. In doing so we have done away with the many theoretical like assumptions that are usually employed in similar studies as seen in Section 4.3.1 The presence of white noise in non–diversifiable risk has been established which confirms the fact that such a risk is independent i.e. cannot be diversified. This paper confirms that the risks are high as it should be in the stock market due to the high volatility experienced. Moreover, our study indicates that diversification reduces diversifiable risk but not non– diversifiable risk. Investment analysts have relied on Markowitz estimator of risk Markowitz (1952) , the market model and the CAPM to analyze the effect of risk in investment decision making. These have had various shortcomings as evidenced by earlier researchers (Section 2.0) and this study’s findings (Section 4.3). Development of risk estimators and expected return estimators that have addressed these shortcomings in this study is expected to improve the accuracy of predicting future expected performance of financial institutions. Investment analysts can now rely on the predictions to make good investment decisions. Financial institutions can now venture into the unknown future financial environment knowing what to expect. Predictability of the future of financial institutions is expected to spill over to investment markets. Overtime investment markets have been known to be unstable, particularly stock, credit, bond and foreign exchange markets. It is hoped that a systematic method will be established which will ensure frequent determination and publication of non–diversifiable risk for respective financial institutions, i.e., those on NSE and other capital markets with a view to reducing their future uncertainties. Stabilized investment markets mean increased investor confidence. We should therefore expect to see an influx of investors in the investment markets. A thriving investment market bolsters the economy. Such that we have increased levels of foreign exchange which strengthens the value of the Kenyan shilling. Imports and exports then become cheap thus influencing the economy positively. New financial institutions will be started and the existing ones sustained. Employment opportunities will be created leading to increased purchasing power, money supply and Gross Domestic Product. All these improve the economy and eventually the way of life of individuals in this economy. Lending and borrowing of funds is hoped to increase due to stable interest rates. Small scale businesses will thrive, improving the standards of living of individuals in these business environments. Determination of non–diversifiable risk and prediction of future performance of financial institutions does not eliminate risk. Nevertheless their proper estimation enables financial analysts make accurate current and future business plans (Paul 2003). Expected future cash flows are made accurately by incorporating this determined risk in their discounting methods. Correct capital investment decision making is facilitated. Required levels of cash and cash equivalents will always be maintained leading to high profitability of the financial institutions. Capital allocation will also be made accurately, since the optimum portfolio will be accurately identified. Despite the good investment decision making and proper financial planning anticipated portfolio risk has to be addressed as an entity. If left unchecked it can cause unexpected losses. This is particularly for financial institutions which trade in interest related portfolios and foreign exchange trading portfolios. With the determination of total risk, risk management techniques are expected to be properly initiated and established, Jennifer (2003). Hedging schemes such as derivatives, options and futures are hoped to cost less, karithi (2003). Savings are anticipated in risk management departments which should be used to increase the wealth of owners of the financial institutions. These will then motivate them to continually invest in them thus sustaining their growth and development. Published by Atlantis Press Copyright: the authors 42 Non-Diversifiable Risk in Investment Portfolios Having discussed the positive impact this study is expected to have on the general environment, unstable or risky environments should be a thing of the past, while stabilized environments should be the norm. 5.2. Recommendations From the above conclusive results this paper recommends the use of all derived estimators in both capital allocation in investment portfolios and net present value investment decision making criterion. Risk is not only experienced in financial environments. It is also encountered in such areas as medical practices, whereby there is some risk attached to various curative methods. Notably are the radiotherapy and chemotherapy methods of curing cancer. The environment also experiences some health hazards due to environmental pollution or poor sanitation, i.e. environmental risk. It would be challenging to adapt the risk determining methods developed in this study in determining medical and environmental risks with a view to reducing them. References Andrew, A., Della, B., Philip, B., and Peter, E. (1995) Investment Mathematics and Statistics, pg. 208–257. Kluwer Law International Box, G., Jenkins, G., Reinsel, G. (1994) “Time Series Analysis; Forecasting and control” Prentice Hall Bollersler, T. (1986) “Generalized Autoregressive Conditional Heteroskedasticity’’ Journal of econometrics, vol. 31, pp 307–327 Dimson, E. (1988) Stock Market Anomalies, Cambridge University Press Engle, R. (1982) “Autoregressive Conditional Heteroskedasticity with Estimates of the variance of United Kingdom Inflation “ Econometrica, vol. 50, pp.987–1007. Hamilton, J. (1994) ‘Time Series Analysis’ Princeton University Press. Harper, R. (2003) Asset Allocation, Decoupling and the opportunity cost of cash, Journal of portfolio management, Vol. 29 Issue 4, p25. Jennifer, J. (2003) Risk Management at the World Bank Global Liquidity Portfolios, pp 11 - 17. Karithi, M. (2003) Management of Foreign Exchange Risk, KASNEB Newsline pg 19-25 Lintner, J. (1965) The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets, Review of Economics and statistics, 47, pg 13–37. Markowitz, H. (1952) Portfolio Selection, Efficient Diversification of Investments, Basil Blackwell. Mossin, J. (1966) Equilibrium in a capital asset market, Econometrica, 34, pg 768–783, October Murray, R.(1961) Theory and Problems of Statistics SI units, pg. 231, McGraw–Hill Book Company Nicholas, F. and Laverne, S. (1989) Quantitative Forecasting Methods, pg. 117 &118, Pws-Kent. McClure, P. (2003) Management’s Discussion and Analysis, Financial Statements and Investment Portfolios International Finance Corporation 2003 Annual Report Volume 2 pg 11–15. Macharia, P. (2001). NSE, Bulletin pg, 2–4 NSE Rao. S. S. (1994) Optimization theory and application Wiley Eastern Limited New Age International p.gs. 37–40. Raymond, B. (2000) Financial Management 6th edition pg, 123-134 Sharpe, W. (1964) “Capital Asset Prices: A Model of Market Equilibrium under conditions of Risk”, Journal of Finance, Vol. 19, pp. 425-442. Stulz, R. (1999). What’s wrong with Modern Capital Budgeting? Financial Practice & Education, Vol. 9 Issue 2, pp7, DP Publications Taboga, M. (2010) ‘Lectures on probability and statistics’, http://www.statlect.com. Vlaar, P. (2000) Value at Risk Models for Dutch Bond Portfolios. Journal of Banking and Finance, 24: 1131–1154. Wilkes, F., Samuels, J., Brayshew, R. (1996) Management of company Finance, pg 262–265. International Thomson Business Press. Published by Atlantis Press Copyright: the authors 43 E. Anyika. APPENDICES Appendix 1: Appendix 2: Table V: Returns For KCB Shares from January 2009 – December 2013 Base Price April 2009 in Kenyan Shillings (Ksh) YEAR MONTH 2009 2010 2011 2012 2013 JAN -254977217 -4909732 -48293377 -28599388 -176773446 FEB -11858496 -18114517 -35537228 -86093014 -27392971 MAR -20843003 -11145914 -20663935 -235975275 -235975275 APR -35789720 -20446319 -16526970 -249196506 -201979585 MAY -12178766 -72691432 -44283975 -12896377 -72325367 JUN -31529069 -69476411 -12607319 -24309319 -3479830 JUL -8193042 -15160040 -94872136 -28713343 -80326198 AUG -8925414 -78123604 -73423079 -16496101 -30548808 SEP -27908155 -35911665 -7891213 -48476514 -44356839 OCT -6450228 -7641496 -23904507 -60275745 -31303040 NOV -19916397 -19712658 -10867080 -140518771 -39819260 DEC -6620513 -2055670 -4412974 -16975041 -61658147 Table VI: Returns For Standard Chartered Shares from January 2009 – December 2013 Base Price April 2009 in Ksh YEAR MONTH 2009 2010 2011 2012 2013 JAN -10356582 2465938 529514 1111462 7044060 FEB 114576 6572177 3748395 1485758 7846250 MAR 328887 938150 4481874 393326 21271177 APR -260376 811697 6695297 -82382 21024330 MAY -951995 473787 910638 713692 40984696 JUN 435601 -239449 2715433 1376041 53320718 JUL 2712442 698182 3454344 2466946 14738214 AUG 2979260 170156 2664586 3125082 45759692 SEP 3627526 11828563 788886 789498 30389636 OCT 2478381 6078253 2120709 2634216 250200283 NOV 4948147 1699087 618373 4069588 83728614 DEC 1474898 -832158 543333 6367901 15751289 Stan chart is one of the three companies with the highest returns and least negative sales thus analyzed for investment purposes. Published by Atlantis Press Copyright: the authors 44 Non-Diversifiable Risk in Investment Portfolios Appendix 3: Appendix 4: Table VII: Returns for ICDC (Industrial Commercial Development Corporation) Shares From January 2009 – December 2013 Base Price April 2009 1n Ksh. YEAR MONTH 2009 2010 2011 2012 2013 JAN 11935779 834948 3910778 -661320 -4407558 FEB 418644 1115537 2724842 -379153 -1360183 MAR 1346134 748676 3267626 -614656 -137580 APR 3239774 130815 486736 -2644600* 89436 MAY 971928 936465 3403992 -980303 1238786 JUN 731335 -356349 9535859 -1628149 3385594 JUL 850178 458268 185853 -10295746 -510603 AUG 1439235 344772 961124 -32399844 15592897 SEP 4163159 4570032 -652724 -236345 5216308 OCT 2460198 945932 73658 -552725 6314293 NOV 774004 2910810 -666640 -572520 3275379 DEC 1459217 507676 -96840 -148838 13410155 * This value is calculated from Table XV as 20.875 – 38.375 = - 17.5 X 151120 = - 2644600 Table VIII: Returns For East African Breweries Shares From January 2009 – December 2013 Base Price April 2009 in Ksh. YEAR MONTH 2009 2010 2011 2012 2013 JAN -10106177 -173378 4433440 980372 22434220 FEB 2715705 -1470663 17493087 1298971 49769005 MAR -9821081 -830676 5217513 1314180 50841507 APR -283444 -914474 0* 856305 90741122 MAY 1748910 -2648273 2786088 3520085 137930125 JUN 948375 -7199135 459345 2908110 199612113 JUL 489416 1106750 676910 13447207 22265976 AUG 16667673 -3985070 -756480 14598393 323570569 SEP 4112532 1861435 322465 22556215 155506703 OCT 386565 3300665 4528066 8768165 162636994 NOV -321409 -9450660 1236078 29758680 177642340 DEC 503414 1054648 4122929 60118940 82052882 * is a rare value which indicates that the share price of April 2011 is at par with that of April 2009. Published by Atlantis Press Copyright: the authors 45 E. Anyika. Appendix 5: APPENDIX 6: TABLE IX: Turnover for KCB from shares for the years 2009 – 2013 IN KSH YEAR MONTH 2009 2010 2011 2012 2013 JAN 327531709 4107488 22019778 8954014 77310126 FEB 57563116 18114517 16772678 25391488 11980046 MAR 64039370 6958143 10979475 63794401 31424618 APR 6232688 14910085 32051257 67655160 148367720 MAY 518649381 43367048 21730863 3217234 3331709520 JUN 43872364 43372557 13403376 4093669 72749417 JUL 15061956 9443960 29123540 4765700 97403422 AUG 10745499 43918134 20571842 2858406 54024740 SEP 31055691 20150434 3966706 6897065 85512669 OCT 6136675 4387974 6767486 10770750 80214040 NOV 16424561 12306168 3605193 30860459 137787282 DEC 5838395 12496811 1241589 4357636 79203149 All values are greater than zero indicating presence of trading for the shares of KCB within this period. TABLE X: Turnover for east african breweries from shares for the year 2009 – 2013 in KSH. YEAR MONTH 2009 2010 2011 2012 2013 JAN 456462305 4629179 57827475* 15273156 60558630 FEB 14000968 9845272 228170700 18433014 96149482 MAR 108458889 32040360 55053761 11714976 86315305 APR 78230475 30749188 44099577 8507807 171045565 MAY 42119583 40502995 33655937 33067468 66678263 JUN 14083369 39176690 5991450 18107100 295773078 JUL 10172851 7581238 7888605 108469564 37608452 AUG 121598251 59206758 3609040 93887704 429389988 SEP 34900943 26414646 2949602 108146238 205652636 OCT 3955849 46838001 30125497 36685194 201641710 NOV 12397185 45087524 11018748 98166600 212222050 DEC 20424215 26285067 71302424 153152160 233933799 This value is derived as follows: From Appendix 10 the row for EABL we have 771033 X 75 = 57827475 Published by Atlantis Press Copyright: the authors 46