**Abstract**

The current study looked at the relationship between risk free rate and stock market return. A five year monthly basis time series data from 2003-2007 of T-bills and KSE-100 index were taken for research study. For the analysis of data, simple regression model approach was applied. Stock market return was taken as dependent variable whereas Risk free rates as independent variables. Also, Pearson Correlation Matrix was also obtained through correlation model. The results suggested that risk free rates had no effect on dependant variable. Furthermore, no correlation between risk free rate and stock market return was found. Consequently, a bivariate relationship cannot exist between risk free rate and stock market return. A multiple regression model of the risk free rate and stock market return exhibits a strong autocorrelation, indicating that the stock market return is a function of more variable than risk free rate.

**1. ****Introduction:**

The risk free rate is the return on the security or a portfolio of securities that is free from default risk. Theoretically, the return on a zero-beta portfolio is the best estimate of the risk free rate. The CAPM predicts the relation ship risk of an asset and its expected return. This relationship is very useful in two important ways. First, it produces a benchmark for evaluating various investments. Second, it helps us to make an informed guess about the return that can be expected from an asset that has not been traded in the market.

Risk free rate is an increasingly essential ingredient of every return computed on financial assets. The security market line (SML) predicts a simple linear relationship between expected return and standard deviation while capital market line (CML) contributes a relationship between risk free rate and straight line emanating from risk free rate(Rf) to tangential to the efficient frontier.

Investors combine their uncorrelated securities help to lesson the risk of a portfolio. They want to know the reasonable level of risk reduction about their portfolios. Research studies look at what happens to portfolio risk as randomly selected stocks are combined to form equally weighted portfolios. When we begin with single stock, the risk of the portfolio is only the standard deviation of that one stock. As the number of randomly selected stocks held in the portfolio is increased, the total risk of the portfolio is reduced.

The total risk of comprise systematic risk and unsystematic risk. Systematic risk is due to risk factors that affect the overall market- such as changes in the nation’s economy, world energy situation, world political and economic situation. This kind of risk is not diversifiable even the well-diversified portfolio expose to this type of risk. The second component, unsystematic risk, is unique to particular company. It is independent to all factors regarding systematic risk. Investors always want to be compensated for taking systematic risk. They should not, however, expect the market to provide any extra compensation for bearing avoidable, diversifiable, unsystematic risk. It is this logic that lies behind capital asset pricing model (CAPM).

**2. ****Significance of study:**

This study aims to investigate the relationship between risk free rate (T-bills) and market return of Karachi stock exchange KSE-100 index. There was a controversy among the investors; some were of the view that Risk Free Rate affects the market positively while others were of the view stock market return moves independently irrespective of Risk Free Rates.

Thus in order to resolve this controversy, current study was conducted with the following objectives.

**3. ****Objectives of study:**

The following objective would be fulfilled during the study:

· To see quantitative impact of Risk Free Rate on Stock market return.

· To workout the correlation between risks free rate and stock market return.

· Suggestions and recommendation for investors.

**4. ****Literature Review:**

**Peter Easton at el (July 2000**) elaborated the empirical estimation of the expected rate of return on a portfolio of stocks. They inverted residual income valuation model to obtain an estimate of the expected rate of return for a portfolio of stocks. They used analogous approach in estimation of internal rate of return on a bond using market value and coupon payments. They contributed through the use of stock price and accounting data to simultaneously estimate the unique implied growth rate and internal rate of return. They recommended adjusted growth rate for valuation return of stocks. They proved that estimated market premium over the risk free rate is closer to the historical premium that that obtained by other studies using earning forecast data.

**Roger G. Ibbotson (July 2002)** estimated long run stock market return participating in the real economy. He decomposed the 1926-2000 historical equity return into supply factors including inflation, earnings, dividends, price to earning ratio, dividend payout ratio, book value, return on equity and GDP per capita. He concluded that the growth overall economic productivity is in line with the growth of corporate productivity measured by earnings. The bulk of return comes from dividend payment and nominal earning including inflation and earning growth. In order to calculate incremental risk and return, bonds have been used as reference point.

**Christian Lundblad (February 2004)** discussed risk-return tradeoff which is fundamental to finance. Previous studies found weaker relationship between the risk premium on the market portfolio and variance of its return in spite of the positive relationship. He explained this weakness is due to the fact of small nature of available data, as an extremely large number of time- series observations are required to precisely estimate this relationship. His main focus was on large span of data of each component required to compute the risk-return trade off which is indispensable for theory of finance.

**Hui Guo and Robert F. Whitelaw (April 2005**) developed evidence of intertemporal capital asset pricing model (ICAPM) and proved with the positive the relationship between stock market risk and return and the extent to which stock market volatility moves stock prices. They provided new evidence on the risk-return relation by estimating a variant of Merton’s (1973) intertemporal capital asset pricing model (ICAPM). They identified the two components of expected return- the risk component and the component due to the desire to hedge changes in investment opportunities. They proved that the estimated coefficient of relative risk aversion positive, statistically significant.

**Rong Huang at el (May 2005)** in the study of BM company, used residual-income valuation model simultaneously to estimate relationship between long term growth rate in abnormal earnings and cost of capital. They related forward, earnings-to- price (FEP) and book- to-market ratio in a linear fashion. The slope coefficient on BM is the long-term growth rate of abnormal earnings (g), and the constant term is the effective cost of capital, i.e., the difference between the cost of capital (r) and the growth rate in abnormal earnings. To empirically implement this valuation representation, they used the analysts’ one-year-ahead earnings forecasts to compute FEP and regressed the difference between FEP and the risk free rate (rf) on BM diminished by one, such that the intercept captures the firm-specific risk premium (rp) and the slope coefficient ca

ptures the firm-specific, long-term growth in abnormal earnings (g). They extracted the risk-free rate from FEP to account for the covariance in FEP and the risk-free rate.

**Mika Vaihekoski**** ****(2007)** discussed how to compute risk free rate from money market instruments, especially for test of capital asset pricing model and event studies. He used US T-bills and CDs for calculation. He presented two alternative approaches: the interest compounding approach and price difference approach. He concluded that the price difference approach is superior to commonly used compounding method. He did event studies and time series with the help of US T-bills whereas they are used for calculation of risk free rates.

**Tamal Datta Chaudhuri (April 2008)** used a structural approach to stock market return, risk-free rate and Capital Asset Pricing Model (CAPM). He developed a structural model, which shows interdependent relationship between risk free rate and stock market returns. It gives a new macroeconomics structural features which shape the price movement in stock exchange. He used a Granger test and a Sims test to prove the interdependence of two variables. He suggested that instead using of exogenous values of stock market returns and risk free rate, one should use estimated values of these variable form reduced form equation of Capital Asset Pricing Model (CAPM). He tested and proved with the data of individual companies.

**5. ****Methodology:**

5.1 Data collection

In order to conduct the current study all the stock markets of Pakistan were proposed, to be taken for study purpose. The stock markets in Pakistan were Lahore stock exchange (LSE), Islamabad stock exchange (ISE) and Karachi stock exchange (KSE) with different indices. Among these all, KSE-100 index was most important and working at top level in Pakistan. One hundred top companies from Karachi Stock Exchange comprise KSE-100 index. Historical data indicated that most of the investors were investing in the KSE-100. The performance of the total businesses of Pakistan can be viewed by the movement of KSE-100 index. Keeping in view, the importance of KSE-100 index, a sample of indices from (2003-2007) was selected for data collection and was taken as dependent variable.

Similarly T-Bill is an important instrument of monetary policy, operated by State Bank of Pakistan. Through the T-bills, the central bank of Pakistan controls the economy and interest rate of the country. T-bill rates were collected from the State Bank of Pakistan for same period and were taken as independent variables. Then the data were feed into the computer software in the work sheet form.

5.2 Hypothesis Formulation

**Ho: The risk free rate has no impact on market return.**

**Ha: The risk free rate has impact on market return.**

5.3 Hypothesis Testing

In order to test these hypotheses, simple regression model in the following form was applied. The regression model was as under:

**Y = ? +? X1 + €**

Where

X1 = values of risk free rate

? = Y intercept

? = Slope coefficient

Y = values of stock market return

€ = Error Term

It is estimated by the regression equation.

Where

? = values of stock market return in the sample

a = y intercept

b = slope coefficient.

x = values of risk free rates in the sample.

Whereas

b = slope of estimated regression equation

X = values of the risk free rates

Y = values of the stock market return

= Mean of the risk free rates.

= Mean of the Stock market return

n = Number of observations in the sample

Whereas

= Mean of the Stock market return

= Mean of the risk free rates

a = y intercept

b = slope coefficient.

The coefficient of determinant, R2 measures how well independent variable explains the dependent variable, that is, the degree of association between dependent variable and independent variables.

The applied model included one dependent variable and one explanatory variable. In the current study the risk free rate was considered as explanatory variable while market return as dependent variable.

**6. ****Results and discussions:**

Data collected from 2003-2008 years on monthly basis was analyzed by applying Simple Regression Model Approach in the following form.

Y = a +b X1 + €

Whereas

Y = Stock Market Return

X1 = Risk Free Rate

b = Coefficient of X1

a = intercept

€ = Error

6.1 Empirical Results

Empirical results drawn through regression model approach are given in the table below.

**Table -1**

**Variables**

**Coefficients**

**t-Stat**

Intercept

0.0399

2.782

Risk free rate

-0.0055

-0.843

R Square

0.012

6.2 Hypothesis Testing:

**Ho = 0**

**Ha ? 0**

Data in the table (1) revealed that there was negative association between risk free rate and market return. But statistically this variable was found insignificant. Thus null hypothesis was accepted that risk free rate was not significant explanatory variable. Alternative hypothesis was rejected. Figure (a) also indicates that there is no relation between them.

6.3 Coefficient of determination (R2)

It is the primary way, we can measure the extent or strength of association that exists between two variables, dependent and independent variables or in other way the coefficient of determination is developed to measure the amount of variation in dependent variable that is explained by the regression line.

Data given in the table-1 indicated that the estimated value of R2 was 0.0123 showing that the strength of association between stock market return and risk free rates was very poor or in other words, only 1.2% of the total variation in stock market returns was being explained due to independent variable.

**Figure a**

6.4 Correlation Coefficient

Coefficient of correlation is the second measure that can be u

sed to describe how well one variable is explained by another variable. When study is based on some sample date, then coefficient of correlation is denoted by (r) and statistically is the square root of sample coefficient of determination.

**Coefficient of correlation (r) = 2 —————— (b)**

When the slope of estimation equation (b) is positive r is the positive square root but if (b) is negative, r is the negative square root. Thus sign of r indicates the direction of relationship between two variables-stock market return and risk free rate.

In the present study scenario the value of (r) coefficient of determination was found

**r = -0.11**

Thus relationship between two variables was negative indicating that slop is negative. The amount of r was 0.11 which indicated that risk free rate was poor explanatory variable for stock market return.

In order to see the two way relationship between the two variables that is RFR and market return. Pearson Correlation Matrix was obtained by analysis the data, through correlation model.

The results obtained through this analysis are given in the table-2.

**Table-2**

Correlation

RFR

M Return

RFR¹ Pearson Correlation

Sig. (2-tailed)

1.000

-0.110

0.403

M Return² Pearson Correlation

Sig. (2-tailed)

-0.110

0.403

1.000

¹ Risk Free rates

² Stock Market Return

The above table-2 indicates that the correlation between risk free rate and stock market return is negative. The correlation -0.110 is in-significant as the P value is 0.403 > 0.05.

Data given in the table-2 indicated that there was found no significant relation between these two variables; it was found that RFR and Market return move independently with each other.

**7. ****Conclusion:**

Correlation coefficient is a standardized statistical measure of linear relationship between two variables. A positive correlation coefficient indicates that the returns from two securities generally move in the same direction, while a negative correlation coefficient implies that they generally move in opposite direction. A zero correlation coefficient implies that implies that the returns from two securities are uncorrelated; they show no tendency to vary together in either positive or negative linear fashion.

The current study had the prime objective to identify the some relationship between risk free rate and stock market return. It was concluded that the risk free rates had no effect upon stock market return. These variables move independently ineffective from each other as there was very poor correlation and weak association between the two variables. These results also consistent with the study of **Confidence A. Amadi, **(Associate Professor of finance at Florida A &M University) who conducted the study on the relationship between the market risk premium and risk free interest rate.

**8. ****Recommendations:**

In the current scenario of Pakistan, it was the dire need of the investors to find the securities having less correlation that can be used to diversify their portfolios for investments. In the volatile markets like Karachi Stock Exchange (KSE), the T-bill is a useful instrument for investors who want to shuffle and readjust their portfolios. Keeping in view the findings and conclusion of the current study, it was proposed and recommended for the investors that they may include T-bills in their investment portfolios in order to save their investments from total collapse. Such diversified investment reduces the risk and increase returns comparatively more. The applied regression model also supports this recommendation.

**References:**

1. Empirical estimation of the expected rate of return on a Portfolio of stocks by Peter Easton 2000.

2. Stock market returns in the long run: participating in the real economy by Roger G. Ibbotson, PhD. July 9, 2002.

3. A structural approach to Stock Market Returns, Risk Free Rate and CAPM Tamal Datta Chaudhuri Investment Bank Of India, ltd. – Ibs Kolkata the ICFAI journal of applied finance, vol. 14, no. 4, pp. 21-31, April 2008.

4. The risk return tradeoff in the long-run: 1836-2003 Christian Lundblad¤ October 2004 uncovering the risk–return relation in the stock market Hui Guo and Robert F. Whitelaw working paper 2001-001c January 2001 revised April 2005.

5. An Intertemporal capital asset pricing model by Robert c. Merton Econometrica, vol. 41, no. 5. (Sep., 1973), pp. 867-887.stable URL: econometrica is currently published by the econometric society.

6. On The Calculation of the Risk Free Rate for Tests of Asset Pricing Models Mika Vaihekoski* Comments are Welcome 01-03-2007.

7. BM Company, Residual-Income Valuation Model to Estimate Relationship between long term growth rate in abnormal earnings and cost of capital. ** **Rong Huang at el (May 2005) ** **Accounting Association, Goteborg

8. Wadhwani, S.B., (1999) “The US Stock Market and the Global Economic Crisis,” National Institute Economic Review, 86-105.

9. Stock Market Risk-Return Inference, an unconditional non-parametric approach by Thomas Mikosch and C¸At¸Alin St¸Aric¸a and the Danish Research council grant no 21-01-0546.

10. Bond Portfolio Optimization A Risk-Return Approach by Olaf Korn and Christian Koziol Prof. Dr. Olaf Korn of Corporate Finance, ( March,2002) Aduate School of Management, Burgplatz 2, D-56179 Vallendar, Germany Dr. Christian Koziol, Chair of Finance, University of Mannheim, D-68131 Mannheim, Germany.

11. On the relationship between the market risk premium and the risk-free interest rate by confidence w. Amadi (Sep, 2005) Finance at Florida A&M University.