Are implied forward rates good measure of future spot rates?

Submitted by : Group 11 Ameya Tikekar 004, Amit kumar 005, Derrick 24, Alok Kumar 258, Vaibhav Mahske 255

ABSTRACT

Report analyzes the relationship between implied forward interest rates and future spot interest rates. We try to find that if the implied forward interest rate is a good predictor for the future spot rate and what is relationship between future spot rate and implied forward interest rate. We have tried to explain the reasons for the difference between them.

Table of Contents

1. INTRODUCTION......................................................................................................................41. 1 MONEY MARKET.............................................................................................................................41.2 INTEREST RATE.................................................................................................................................51.3 YIELD CURVE....................................................................................................................................51.4 SPOT RATE......................................................................................................................................61.5 FORWARD RATE...............................................................................................................................6

2. METHODOLOGY......................................................................................................................7

3. ANALYSIS................................................................................................................................73.1 ANALYSIS FROM 91-DAYS INTEREST RATE......................................................................................103.2 ANALYSIS FROM 182-DAYS INTEREST RATE..........................................................................................11

4. CONCLUSIONS.......................................................................................................................12

1. Introduction

Government uses various monetary policies to control interest rates. Implied forward rates provide prediction of future spot rates. We will examine this hypothesis.

1. 1 Money marketThe money market is part of the fixed income market. However unlike the bond market, money market specializes in very short-term or maturity less than one year debt securities. Hence the investments made here, due to their short term maturity is also referred to as cash investments. It also provides a mechanism through which institutions can execute their trade of money as well as manage their cash needs. The securities traded here are essentially IOU or informal document acknowledging debt issued by governments, financial institutions and large corporations. So they are very liquid and are almost risk free. Due to the same fact the returns offered by them are much lower than other securities.

The function of money market is balanced by two forces: supply and demand of money.

By demand we refer to the demand for funds by individuals for financing purchases or expansion of existing facilities or acquiring other firms. But again the supply of money is provided by the individuals.

There is a big difference in terms of denominations of securities being traded in money market and stock market. Money market trades in high denominations and hence is quite distinct in the profile of players involved in it. Also the money market can be considered as a dealer market, hence trading happens in investor account and the investors face all the risks, if any unlike the stock market which has a broker who receives commission for being the agent. Another difference between money market and stock market is that there is no central trading floor or exchange. Transactions take place over the phone or via electronically.

Entry into money market happens through a money market mutual funds, or money market bank account. But one money market instruments, Treasury bill, can be purchased directly or through other large financial institutions with direct access to these markets.

There are several types of money market instruments, each differing in terms of returns and different risks. The differences become key when investors plan to invest in money market Various types of money market instruments are essential in our analysis are:

Treasury Bills (T-bills)

Government Bonds (G- secs)

Treasury Bills (T-bills) are short-term securities that mature in less than one year usually they do not pay coupons and are sold below par and repurchased at par value by government to maintain the positive yield. They are the security with the least risk. On the basis of maturity T-bills can be classified into: 3-month, 6- month and 1-year T-bills. T-bills excel in their marketability and affordability to the individual investors. Largest purchasers of T-bills are Banks and financial institutions especially the primary dealers. They are sold at periods when the treasury cash balances are very low.

Government bonds are another kind of securities issued by the government. They offer a fixed rate of interest over a fixed period of time. Like treasury bills, government bonds are risk-free and regarded as one of the safest types of investments.

1.2 Interest rateAn interest rate is simply defined as the cost of borrowing or return on lending. It is the rate at which interest is paid by a borrower to the lender for the money borrowed. In return for lending money to the borrower, the lender will expect some extra as compensation, known as interest rate. Generally stocks & bonds are simple contracts / agreements between the borrower and the lender. Interest rate is a very important part of money market, more over fixed income securities move with respect to any type of interest rate.The economics glossary defines the interest rate as:“The interest rate is the yearly price charged by a lender to a borrower in order for the borrower to obtain a loan. This is usually expressed as a percentage of the total amount loaned.” Government and firms borrow by issuing bonds at a particular interest rate to the lenders, usually the households. The firms can also borrow the loan from the bank and pay the interest to the bank. The equilibrium of the interest rate in economy is determined by the intersection where the money supply equals the money demand. Usually when market conditions change the government is forced to change the interest rate to control the economy. The government does it through the central bank by changing the interest rate by changing the money supply. The interest rate is inversely proportional to the quantity of money. When interest rate is low then money supply is high and when money supply is low; interest rate increases. Interest rate also influences how much money each house hold or firm wish to save with itself, more the interest rate, less the households wish to hold. This can be proved by assuming that the economy holds wealth in 2 types: bonds and money. A high interest will increase the demand for bonds, while a low interest rate will decrease it as more and more people would be willing to exchange the bonds for money. Thus bonds are less attractive and there is no incentive for keeping the money as bond and extent your consumption. Thus we see that a high interest rate lowers the spending since people reduce spending so that they can do it later. Therefore interest rate is a tool that by itself is necessary for the government to influence both money market and goods market.

1.3 Yield curveYield curve is the relation between the rate of borrowing or interest and the time to maturity of a particular debt. The description of this relation is often called the term structure of interest rates. Yield can also be considered as a benchmark for debt.

The total rate of interest is denoted by the yield. In general the yield is dependent on the duration of the bond. This function Y, called the yield curve is often but not always an increasing function of time. In most of the cases, yield increase with a diminishing marginal growth. Investors would only invest their money now, if they expect to receive a better return in the future. The rise of yield curve can be considered as a compensation for the investors. This is the reason why the long-term investments have relatively higher interest rates. This phenomenon can also be explained as that, when maturities are longer, the risk taken by investors are higher and hence the further future is more uncertain than the near future. The yield curve may also have flat or hump shape if the interest rates are stable or their short-term volatility outweighs their long-term volatility.

The shape of the yield curve does not have to be increasing. There are times when short term interest rates are higher than the longer term interest rates. This happens because expectations of people to have a decreasing interest rates due to macroeconomic factors. However if long term investors tend to dominate the market, the risk premium can even end up negative. But none of the theories predict a negative risk premium. Hence the only way of an inverted yield curve occuring is caused only by the expectations of decreasing interest rates. The analysis of yield curves is done by fixed income analysts. It helps them to get an idea of the existing market conditions in financial market and the opportunities that exist in market. Economists also use the curve to understand the movement of economy: predicting changes in economic output and growth.Generally values of certain points of yield curve are known and curve fitting is used to analyze the other points. Hence the yield curve function Y is a discount factor function.

1.4 Spot rateThe spot rate is defined as the theoretical rate of interest when invested in a zero coupon bond. In a coupon paying bond trading at par; this rate becomes the IRR. We use this rate to calculate the amount we will receive at maturity if we invest a particular amount today. Usually bonds are traded by mentioning their spot rate. If we invest $ D today and receive $ A at the end of n years; then spot rate (r) isr = (A/D) ^ (1/n)

1.5 Forward rateThe forward rate is the future yield on a bond. It is calculated using the yield curve. The yield curve usually expresses the interest rate at both current and future. Due to this reason we can determine the possible implied future date. Unlike spot rate this rate can be only defined by specifying the

periods between which it is defined, usually both of these periods are tow future dates and hence it is also called future rate. To extract the forward rate, one needs to know the term structure of interest rates. Generally the formula used to calculate the forward rate is:

Where we assume that all involved tenors are at most 1 year (short term rate /linear discount) Use actuarial form for long term rate(>1 year).

: The forward rate between term t1 and term t2,d1: The time length between time 0 and term t1 (in years),d2: The time length between time 0 and term t2 (in years),r1: The interest rate for the period time 0 to term t1 ,r2: The interest rate for the period time 0 to term t2 ,

2. Methodology We have taken the data for Treasury bonds (risk free interest rate) for Indian market from NSE site. We considered data for last 5 years (1-Jan 2007 to 5 Sep 2011) and took 20 instances where we have tried to find the co-relation between dependent and independent variable. We used the yield to maturity of 3-month, 6-months and 12-months T-bill. We have selected all interest rates of first day of each quarter from 2007 to 2011. From this data we calculated implied forward rates for a period of 3 months and 6 months for entire period. Then we did regression analysis between implied forward rates and actual future spot rates. We used SPSS tool for our analysis. We consider implied forward rates as the independent variable and Spot rates as the dependent variable.

So the regression equation becomes:

Spot Rates = β0 + β1 (Implied forward rate)

Theoretically β0 should be 0 and β1 should be 1.

Assumptions: We took some assumptions to bring data into workable format. Since multiple securities with same maturity date were being issued at different YTM’s on the same date, we have taken average of these different YTM’s and used it for calculation of implied forward rates. This data is compared with the spot rates to find the regression.

3. Analysis

Data of T-bills (91 days, 182 days and 364 days) was collected (period, interest rate) for 5 yrs (Jan 2007 to June 2011). The whole data is attached in the Appendix A at the end of the report.

The data gathered is the spot rate for the given period of the T-bills. Implied forward rates are calculated for the given T-bills i.e. 91 days and 182 days.

The calculated implied forward rate for 91 days and 182 days is shown below.

Quarters Interest rates Forward rate for 91 days

Difference 91 days:

Spot – forward

rate

Forward rate for

182 days

Difference 182 days:

Spot – forward

rate

91 days 182 days 364 days

3-Jan-07 7.0747 6.99 7.46 - - - -4-Apr-07 7.56 7.57 7.43 6.91 0.65 7.93 -0.364-Jul-07 6.19 6.65 6.56 7.58 -1.39 7.29 -0.643-Oct-07 6.72 6.86 6.94 7.11 -0.39 6.47 0.392-Jan-08 6.35 6.8 7.31 7.00 -0.65 7.02 -0.222-Apr-08 6.74 6.81 7.01 7.25 -0.51 7.82 -1.012-Jul-08 8.43 8.8 8.82 6.88 1.55 7.21 1.591-Oct-08 8.6 8.75 8.25 9.17 -0.57 8.84 -0.09

31-Dec-08 4.83 5.01 5.01 8.90 -4.07 7.75 -2.741-Apr-09 4.6 4.62 4.79 5.19 -0.59 5.01 -0.391-Jul-09 2.97 3 3.75 4.64 -1.67 4.96 -1.96

30-Sep-09 2.99 3.37 3.5 3.03 -0.04 4.51 -1.1430-Dec-09 3.52 3.2 3.82 3.75 -0.23 3.63 -0.4331-Mar-10 3.84 4 4.49 2.88 0.96 4.44 -0.4430-Jun-10 5.35 5.23 5.4 4.16 1.19 4.98 0.2529-Sep-10 6.25 6.3 6.28 5.11 1.14 5.57 0.7329-Dec-10 7.05 7.22 7.2 6.35 0.70 6.26 0.9630-Mar-11 7.09 7.17 7.49 7.39 -0.30 7.18 -0.0129-Jun-11 8.01 8 8.19 7.25 0.76 7.81 0.19

Table: Implied forward rates

We plot the spot rates and implied forward rates for 91 days T-bills

Figure: Comparison – spot rate vs. forward rate for 91 days

In the above figure for 91 days T-bills, spot rate curve follows forward rate curve with some difference

We plot the spot rates and implied forward rates for 182 days T-bills

Figure: Comparison – spot rate vs. forward rate for 182 days

In the above figure for 182 days T-bills, spot rate curve follows forward rate curve with some difference

Excel sheet with analysis can be found here.

3.1 Analysis from 91-days interest rate

We have run the regressive data model on the basis of 91-days spot rate and 91-days implied forward rates. It should be noted that there is a 91-day time lag.

Regression Statistics

Multiple R 0.742172255R Square 0.550819656Adjusted R Square 0.522745884Standard Error 0.012968305Observations 18

ANOVA

df SS MS FSignificance

F

Regression 1 0.003299705 0.00329970519.620436

6 0.000420688Residual 16 0.002690831 0.000168177 Total 17 0.005990536

Coefficients Standard Error t Stat P-value

Intercept 0.015536218 0.01080007 1.4385293680.1695578

3

X Variable 1 0.771213191 0.174108557 4.4294962060.0004206

9

From the output, our β1 = 0.015536218 and β2 = 0.771213191

Therefore, the equation becomes spot rate = 0.015536218 + 0.771213191*(implied forward rate).

3.2 Analysis from 182-days interest rate

We have run the regressive data model on the basis of 182-days spot rate and 182-days implied forward rates. It should be noted that there is a 182-day time lag.

Regression StatisticsMultiple R 0.543810219R Square 0.295729554Adjusted R Square 0.248778191Standard Error 0.013035935Observations 17

ANOVA

df SS MS FSignificance

F

Regression 10.00107036

3 0.0010703636.29863617

9 0.024040486

Residual 150.00254903

4 0.000169936

Total 160.00361939

7

CoefficientsStandard

Error t Stat P-value

Intercept 0.0367022270.01089559

9 3.3685368870.00422170

6X Variable 1 0.437036312 0.17413828 2.509708385 0.02404048

4 6

From the output, our β1 = 0.036702227 and β2 = 0.437036312

Therefore, the equation becomes spot rate = 0.036702227 +0.437036312*(implied forward rate).

4. Conclusions Under risk neutrality condition forward rate should be accurate predictor of future spot rate. If we consider risk factor people will prefer certain investment over risky investment that’s why forward rate are higher than the future spot rate. Longer is the period there will be more risk so difference between implied forward rate and future spot rate is higher. For shorter period forward rates are more accurate predictor of future spot rate.The results are validated by the regression analysis. Future spot rate = b1+b2* implied forward rate. B1 found from regression is very close to zero and b2 is less than one. This indicates that the future spot rates are smaller than the forward rates. Also, the estimated 3-month regression coefficient is larger than the estimated 6-month regression coefficient, and the 6-month’s larger than 1-year’s. From this we can conclude that the shorter maturity implied forward rates are more close to the future spot rates. Above results conclude that in practice liquidity theory is more applicable, which states that people need some risk premium for uncertainty for longer period

References[1] http://nseindia.com/marketinfo/wdm/wdm_hist.jsp, 06-09-2011[2] http://nsdp.rbi.org.in/, 06-09-2011