A Comparison of Market and Model Forward Rates

Transcription

1 A Comparison of Market and Model Forward Rates Mayank Nagpal & Adhish Verma M.Sc II May 10, 2010 Mayank nagpal and Adhish Verma are second year students of MS Economics at the Indira Gandhi Institute of Development Research. 1

2 Contents 1 Introduction 3 2 Methodology 4 3 Analysis & Results Foreign Exchange Market Equity Market Conclusion 11 5 References 13 2

3 1 Introduction Forward rates are perhaps the most common measure of expected future interest rates. Forward rates are the interest rates for future periods that are implicitly incorporated within today s interest rates for loans of different maturities. The forward rates are derived from the yield curve by using the risk free interest rate for relevant maturities. For example, suppose that the interest rate today for borrowing and lending money for six months is 6% per annum and that the rate for borrowing and lending for twelve months is 7%. Taken together, these two interest rates contain an implicit forward rate for borrowing for a six-month period starting in six months time. To see this, consider a borrower who wants to lock in to todays rate for borrowing Rs. 100 for that period. He can do so by borrowing Rs for a year at 7% and investing it at the (annualized) six-month rate of 6%. In six months time he receives back this sum plus six months of interest at 6% (Rs. 2.92), which gives him the Rs. 100 of funds in six months time that he wanted. After a year he has to pay back Rs plus a year of interest at 7% (Rs ). In other words, the borrower ensures that his interest cost for the Rs. 100 of funds he wants to borrow in six months time is Rs He manages to lock in an interest ratethe forward rate (2) of 7.77% now for borrowing in the future. If there were no uncertainty about the path of future interest rates then forward rates would equal expected future interest rates. If this were not the case it would be possible to make unlimited riskless profits. Suppose, for example, that the borrower above knew for certain that six-month rates would be 8% in six months time. But if todays six-month and twelve-month rates are 6% and 7%, then it is possible to lock in to borrowing now at 7.77%, knowing that one can then lend these funds out at a higher rate in six months time to make a guaranteed riskless profit. Such an arbitrage opportunity would not persist long in a world of rational investors. As they exploited this situation, the configuration of interest rates would change until the implicit forward rates equaled expectations of future rates. Future interest rates are, of course, not known with certainty. Nevertheless, if forward rates differ from expected future short rates, an investor will be able to create a position that has positive expected profits. The presence of interest rate uncertainty means that the actual profits from these trades may be positive or negative. Risk-averse investors will then require a risk premium to bear this interest rate risk. In Equilibrium this will drive a wedge between the forward rates and expected short rates so that the expected profits incorporate the risk premium. The premise that the forward curve represents the path of expected future interest rates is known as the expectations hypothesis. But in practice there are a number of factors that may drive a wedge between forward rates and what the market expects future rates to be. For instance, if market participants are risk averse they are likely 3

4 to require a term premium as compensation for the uncertainty about future interest rates. Participants may also demand a liquidity premium to hold instruments that are difficult to trade at times of market stress. Both these factors, which we collectively refer to as the risk premium, will tend to push the forward curve above the path of expected future interest rates. Ideally, one would test the expectations hypothesis by comparing forward rates directly with expectations However, expectations are unobservable. To get round this problem, empirical studies often assume that expectations are rational. By assuming rational expectations hold, any systematic difference between forward rates and out turns can be interpreted as reflecting the risk premium. These differences can also be attributed to market imperfections such as taxes and transaction costs. Transaction costs produce a band around the equilibrium forward and expected forward differential within which arbitrage is not profitable Market expectations of the forward rate are derived from the currency futures and equity futures market. However, the existence of term premium, arising from interest rate uncertainty and investor risk aversion, means that derived forward rates will not in general equal expectations of future rates obtained from the term structure. Differences in credit quality, liquidity and contract specifications of the instruments as well as the kind of parties involved result in difference between the forward rates and the expectation of the forward rates. We compare the forwards rates derived from equity futures and those derived from currency futures with the forwards rate obtained from the term structure of interest rates. We study the differences between the expected forward rates obtained from the futures markets and the actual forwards interest rate. By using data from the foreign exchange market, we are able to eliminate the tax effect and reduce the impact of transaction costs. Furthermore, the uncertainty surrounding the likely path of interest rates is greater the further ahead one looks, so this term premium is likely to increase with maturity. Hence the longer the horizon, the larger the difference between forward rates and expected rates. In this paper we use time series econometrics analysis methods to see if the market expectations take the model forward rates into account and whether future expectations of the forward rates can be better predicted using past values of the model forward rate. 2 Methodology Daily data of currency futures prices of different maturities for the period of September to March-2010 is used to calculate the currency market expectations of forward interest rates. The three month forward rate is calculated for t time period ahead, where t varies daily from two months to eight months. A similar analysis is also per- 4

5 formed for equity futures for a one month forward rate. The expectations of interest rates calculated are compared with the corresponding forward rates obtained from the yield curve. The forward rate implied from future prices represent the market expectations of the future interest rate, whereas those calculated from the yield curve represent the model forward rate. We perform the Granger causality test to see if the market participants take into account the forward rate information implicit in the yield curve while forming expectations of future interest rates. We forecast the expected value of the interest rate to see if the past values of model rates contain information relevant for market participants over and above that in the previous values of the expectations. Given a discrete term structure of interest rate r 1, r 2..., the n period forward rate at time t is defined as: ( (1 + r t+n) t+n (1 + r t ) t ) 1/n 1 = r f t,n The data for futures prices and the Zero Coupon Yield Curve (ZCYC) is obtained from the National Stock Exchange (NSE) website. 3 Analysis & Results 3.1 Foreign Exchange Market We use daily data for futures prices of currency prices to obtain values of the implicit market expectations of future interest and compare these values with the corresponding model forward rates implicit in the NSEs ZCYC. The time plot below compares the rates obtained from the market with those obtained from the model. It can be clearly seen that the market expectations of future interest rates are consistently less than the model forward rate. This is due a number of factors such as the term premium which is the premium for interest rate uncertainty, the liquidity premium and transaction costs. Presence of all these factors drives a wedge between forward rates and what the market expects future rates to be. In addition, expectations about the future exchange rates also play an important role in determining the futures prices. The Granger causality test is performed to test for a causal link from the model forward rate to the expected future rate. The objective is to see if the market participants take into account the forward rate information implicit in the yield curve while forming expectations of future interest rates. For this purpose we plot the cross 5

6 Time Plot Forward Rate Market Expectation of Future Rate Model Forward Rate Sep Nov Jan Mar Time Figure 1: Time Series plot of Model and Market Forward Rate correlogram for market expectations against model forward rate for 10 lags. The graph shows that the correlation is significant at lag one and two suggesting that the model rate leads the market rate by two periods. This would imply that the forward rate obtained for the last two periods from has information which can be used to predict the present expectation of the rate from the market. market & model ACF Lag Figure 2: Cross Correlation: Market against Model 6

7 This would imply that the first and second lags of the model rate contain information explaining the market expectations. If the model rate Granger causes the expectations of the market there should be information in past values of the model rate over and above information in the past values of the rate expected by the market participants. To identify the number of relevant lags in the univariate model for the series of market expected rate we plot the ACF and PACF for the series. The Augmented Dickey Fuller unit root test for the series indicates a stationary series. The ACF and PACF also suggest stationarity in the series. The Acf and Pacf given below suggest an AR(1) model for the expected rate obtained from the futures market. Series market ACF Lag Series market Partial ACF Lag Figure 3: ACF and PACF plots of MArket Forward Rates To test if the forward rate from model granger causes the market rate we estimate two separate models with the expected rate from the market as the dependent variable. AR(1) model for the market Forward Rate Series. Forward Rate from Market with first lag of the market series (Determined from the ACFs PACFs plot) and the first two lags of the model series(determined from the Cross Correlation graph). 7

8 Model 1: Mkt t = a + bmkt t 1 + ɛ t Model 2: Mkt t = a + bmkt t 1 + c 1 Mod t 1 + c 2 Mod t 2 + ɛ t Null Hypothesis: H 0 = c 1 = c 2 = 0 The Results from the two models are given below. Market Estimate Std..Error t.value Pr...t.. 1 (Intercept) market.l Market Estimate Std..Error t.value Pr...t.. 1 (Intercept) market.l model.l model.l To Test for Granger causality we perform an F test for the joint significance of the estimates of the coefficients of the lags of the model series. The results of the test reject the null of no granger causality with a p value of Thus we conclude that our results suggest that the expected rate obtained from the model granger causes the forward rate from the market, which would mean that the foreign exchange market participants take into account the past values of the model forward rate while forming expectations about future interest rates. It is expected that the if we try to do an in-sample forecasting of expectations of the future rate we would get a better forecast in case of the second model than the first model as the granger causality test results suggest. This is further reinforced by the analysis done to evaluate the forecasting accuracy of the fitted models. When we forecasted the values using the two models even though, the first model was a better predictor according to two out of the three criterions used, i.e. RMSE and MAD, the 8

9 differences between the values of these two measures for the two models are very small. The second model performs better according to the correlation between the predicted and actual values criterion. The table below shows a comparison of the forecast from the two models. X Correlation RMSE MAD 1 Model Model Equity Market We analyze the NSE Nifty futures market for the time period September 2009 to February There are a total of 115 data points which is the number of days trading has taken place during the time span chosen. There are contracts with three different maturities trading daily, each with term ending on the last Thursday of that month, the next month and next to next month. The market forward rates have been calculated using the closing values of the traded futures each day. And the model forward rates have been calculated using the Zero coupon Yield Curve (ZCYC) as formulated by the National Stock Exchange of India (NSE). The daily forward rates used are one month (approximately; actual is 28 days) rates t days ahead, where t varies from on an average from 60 days to 30 days. Given below is the time series plot of the market and model forward rates. Time Series Plot Forwad Rates Market Expectation of Future Rate Model Forward Rate Sep Nov Jan Time Figure 4: Time Series plot of Model and Market Forward Rate 9

10 Looking at the above figure, we can infer that the market forward rates are consistently lower than the model forward rates. We next check for the cross correlation between the two series to find out if model rates can explain the market rates in some way. The figure below shows the cross-correlation graph of the two forward rates. We see that the two rates are correlated only at lag zero i.e. there is only contemporaneous correlation but no lead-lag relationship. market & model ACF Lag Figure 5: Cross Correlation: Market against Model Forward Rate Even though we do not find any cross correlation between the two rates, to cross verify, we proceed to check for the existence of Granger Causality. To do that, we begin by looking at the ACF and PACF of the time series of the market forward rates. The following figure shows the results: The figure below suggests that the time series of market forward rates follows AR (2). So, we go ahead to check for Granger Causality by regressing market forward rates on its own two lags and on two lags of the time series of model forward rates. We find that the coefficients for the later two are insignificant as shown in the table below. Model: Mkt t = a + b 1 Mkt t 1 + b 2 Mkt t 2 + c 1 Mod t 1 + c 2 Mod t 2 + ɛ t Null Hypothesis: H 0 = c 1 = c 2 = 0 10

11 Series market ACF Lag Series market Partial ACF Lag Figure 6: ACF and PACF plots of MArket Forward Rates X Estimate Std..Error t.value Pr...t.. X.1 1 (Intercept) market.lag ** 3 market.lag ** 4 model.lag model.lag Thus we can say that there is no Granger causality between the market and the model forward rates. 4 Conclusion The study analysed the relationship between the forward interest rates implicit in the yield curve and the market expectations of the forward interest rate for both the currency and the equity futures market. It was found that the forward rates found using the yield curve were consistently higher than those expected in the markets. This can be attributed to various factors such as the risk premium, liquidity premium and 11

12 transaction costs. Further, it was found that there was correlation between the expected forward rate from the market and the lagged values of the model forward rates in the currency futures market. However, no such correlation was found for the equity futures market. This led to the granger causality test which gave significant coefficients for the lagged model rates for the currency futures market, indicating that the model rates lead the market expectations, whereas no such relationship was found for the equity futures market. This can be attributed to the fact that the proportion of participation by banks in the currency futures market is larger than that in the equity futures market. Moreover, an important participant in the foreign exchange market is the central bank, which intervenes with the purpose of managing the exchange rate. Thus, there is less information asymmetry in the currency market due to the characteristics of the majority participants involved. The granger causality tests indicate presence of causality between the two rates. This is further reinforced by the analysis done to evaluate the forecasting accuracy of the fitted models. The three measures used are Mean Square Error (MSE), Mean Absolute Error (MAE) and correlation between the forecasted and actual values (COR). Two models tested are the following: Model 1, following AR1 process and Model 2, having lagged market (up to lag1) and model (up to lag 2) rate values as suggested by granger causality test. Model 2 performs better according to COR and on the basis of other two measures, both models have similar performance. This finding has to be taken with cognizance since forecasting forms an important part of any such study. 12

13 5 References BENNINGA, S. and Z. WIENER (1996). An Investigation of Cheapest to Deliver on Treasury Bond Futures Contracts. Mimeo. COX, J.C., J.E. INGERSOLL, and S. A. ROSS (1985). A Theory of the Term Structure of Interest Rates. Econometrica 53, DOTHAN, L. URI (1978). On the Term Structure of Interest Rates. Journal of Financial Economics 6, FISHER M. and D. ZERVOS (1996), Yield Curve, in Computational Economics and Finance, Modeling and Analysis with Mathematica, ed. H. R. Varian, TELOS/Springer-Verlag, pp J. C. Cox, J.E. Ingersoll, and S. A. Ross. The Relationship Between Forward and Futures Prices. Unpublished manuscript, University of Chicago, HEATH D., R. JARROW and A. MORTON (1992), Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation, Econometrica 60, HO, T. S. Y. and S.B LEE (1986). Term Structure and Pricing Interest Rate Contingent Claims, Journal of Finance 41, R. W. Lang and R. H. Rasche. A Comparison of Yields on Futures Contracts and Implied Forward Rates. Federal Reserve Bank of St. Louis Monthly Review 60 (December 1978), VASICEK, OLDRICH A. (1977). An Equilibrium Characterization of the Term Structure. Journal of Financial Economics 5 (November), pp