Drake DRAKE UNIVERSITY Fin 284 The Term Structure and Volatility of Interest Rates Fin 284

Drake Drake University Fin 284 Treasury Yield Curve The most commonly investigated and used term structure is the treasury yield curve. Treasuries are used since they are considered free of default, and therefore differ mainly in maturity. Also the treasury is the benchmark used to set base rates. The treasury market is also very liquid so there are no problems with liquidity

Drake Drake University Fin 284 Current Treasury Yield Curve The most straightforward way to represent the yield curve is by graphing the combinations of yield and maturity The problem with this measure is that it does not account for differences in coupon rates across bonds of similar maturities. Therefore there are some alternative methods we need to explore such as using the zero spot rates.

Drake Drake University Fin 284 Investigating the Yield Curve The best theoretical measure is looking at the spot rte (zero coupon) yield curve. However often the yields for the on the run treasuries are used as a proxy for this. The recent yield curve based upon the on the run yields are given in the next few slides.

Drake Drake University Fin 284 Yield Curves Previous 6 Months

Drake Drake University Fin 284 Yield Curves Previous 5 quarters

Drake Drake University Fin 284

Drake Drake University Fin 284 US Treas Rates Jan 1990 Dec 2003 Downward sloping yield curve

Drake Drake University Fin 284 Yield Curve 2000

Drake Drake University Fin 284 Shifts in the yield curve Using the Arbitrage free valuation approach of a security requires an estimation of the zero coupon treasury yield curve. As the shape of the yield curve changes so will the rate corresponding with each respective time frame.

Drake Drake University Fin 284 Shifts in the Yield Curve The shape of the yield curve and changes in the shape can provide information to the market concerning future interest rates. Want to investigate two things, overall shifts in the curve and changes in its slope.

Drake Drake University Fin 284 Parallel Shifts ShortIntermediateLong Maturity ShortIntermediateLong Maturity

Drake Drake University Fin 284 Approximate Parallel Shift Change in rates is approximately the same for all maturities

Drake Drake University Fin 284 Twists ShortIntermediateLong Maturity ShortIntermediateLong Maturity Flattening Twist Steepening Twist

Drake Drake University Fin 284 Flattening of the curve Change for short term is greater than for long term

Drake Drake University Fin 284 Steepening of the Curve Change for long term is greater than for short term

Drake Drake University Fin 284 Butterfly Shifts ShortIntermediateLong Maturity ShortIntermediateLong Maturity Positive Butterfly Negative Butterfly

Drake Drake University Fin 284 Positive Butterfly shift Increase for short term and long term is greater than for intermediate term

Drake Drake University Fin 284 Negative Butterfly Shift Decrease for short term and long term is greater than for intermediate term

Drake Drake University Fin 284 Why does the Yield Curve usually slope upwards? Three things are observed empirically concerning the yield curve: Rates across different maturities move together More likely to slope upwards when short term rates are historically low, sometimes slope downward when short term rates are historically high The yield curve usually slopes upward

Drake Drake University Fin 284 Three Explanations of the Yield Curve The Expectations Theories Pure Expectations Local Expectations Return to Maturity Expectations Segmented Markets Theory Biased Expectations Theories Liquidity Preference Preferred Habitat

Drake Drake University Fin 284 Pure Expectations Theory Long term rates are a representation of the short term interest rates investors expect to receive in the future. In other words, the forward rates reflect the future expected rate. Assumes that bonds of different maturities are perfect substitutes In other words, the expected return from holding a one year bond today and a one year bond next year is the same as buying a two year bond today. (the same process that was used to calculate our forward rates)

Drake Drake University Fin 284 Pure Expectations Theory: A Simplified Illustration Let r t = today’s time t interest rate on a one period bond r e t+1 = expected interest rate on a one period bond in the next period r 2t = today’s (time t) yearly interest rate on a two period bond.

Drake Drake University Fin 284 Investing in successive one period bonds If the strategy of buying the one period bond in two consecutive years is followed the return is: (1+r t )(1+r e t+1 ) – 1 which equals r t +r e t+1 + (r t )(r e t+1 ) Since (r t )(r e t+1 ) will be very small we will ignore it that leaves r t +r e t+1

Drake Drake University Fin 284 The 2 Period Return If the strategy of investing in the two period bond is followed the return is: (1+r 2t )(1+r 2t ) - 1 = 1+2r 2t +(r 2t ) (r 2t ) 2 is small enough it can be dropped which leaves 2r 2t

Drake Drake University Fin 284 Set the two equal to each other 2r 2t = r t +r e t+1 r 2t = (r t +r e t+1 )/2 In other words, the two period interest rate is the average of the two one period rates

Drake Drake University Fin 284 Applying the model The 2 year rate is an average of the current 1 year rate and the expected rate one year in the future. This implies that the yield curve will slope upward when the expected one year rate is expected to increase compared to the current one year rate. Similarly the yield curve will slope downward when the expected rate is less than the current rate.

Drake Drake University Fin 284 Expectations Hypothesis r 2t = (r t +r e t+1 )/2 If you assume that the expected rate is representative of the long run average and that rates will move toward the average, empirical fact two is explained. When the yield curve is upward sloping (R 2t >R 1t ) The current rate would be less than the long run average and it is expected that short term rates will be increasing. Likewise when the yield curve is downward sloping the current rate would be above the long run average (the expected rate).

Drake Drake University Fin 284 Expectations Hypothesis r 2t = (r t +r e t+1 )/2 As short term rates increase the long term rate will also increase and a decrease in short term rates will decrease long term rates. (Fact 1) This however does not explain Fact 3 that the yield curve usually slopes up. Given the explanation of Fact 2 the yield curve should slope up about half of the time and slope down about half of the time.

Drake Drake University Fin 284 Problems with Pure Expectations The pure expectations theory also ignores the fact that there is reinvestment rate risk and different price risk for the two maturities. Consider an investor considering a 5 year horizon with three alternatives: buying a bond with a 5 year maturity buying a bond with a 10 year maturity and holding it 5 years buying a bond with a 20 year maturity and holding it 5 years.

Drake Drake University Fin 284 Price Risk The return on the bond with a 5 year maturity is known with certainty the other two are not. The longer the maturity the greater the price risk. If interest rates change the return and the 10 and 20 year bonds will be determined in part by the capital gains resulting from the new price at the end of five years.

Drake Drake University Fin 284 Reinvestment rate risk Two new options: Investing in a 5 year bond Investing in 5 successive 1 year bonds Investing in a two year bond today followed by a three year bond in the future. Again the 5 year return is known with certainty, but the others are not.

Drake Drake University Fin 284 Local expectations Local expectations theory says that returns of different maturities will be the same over a very short term horizon, for example 6 months. This assumes that all the forward rates currently implied by the spot yield curve are realized.

Drake Drake University Fin 284 Local Expectations Previously we calculated the zero spot rates using the bootstrapping method for the on the run treasury securities given below. MaturityYTMMaturityYTM 0.54%2.55.0% %3.05.2% %3.55.4% % %

Drake Drake University Fin 284 Zero spot Review (local expectations example) Given the assumption that all of the on the run treasury securities were selling at par we found the 1.5 year zero coupon rate by discounting the coupons by the respective zero coupon rates.

Drake Drake University Fin 284 Zero spot curve (local expectations example) Continuing the same process for future rtes we started to build a zero spot yield curve TimeYTMZero Spot 0.54% 4% % 4.2% % % % %

Drake Drake University Fin 284 Forward Rates (local expectations example) Given the zero spot rates it is possible to find the forward rates. Let 1 f m be the 1 period (six month) forward rate from time m to time m+1. The forward rate can then be found as:

Drake Drake University Fin 284 Forward Rate (local expectations example) Given the 6 mo. zero spot rate of 4% and the 1 year zero spot rate of 4.2%, the one period (6mo.) forward rate from 6 months to 1 year would be: (1+.021) 2 = (1+.02)(1+ 1 f 1 ) 1 f 1 =.022 Similarly the 6 month forward rate from 1 year to 1.5 years can be found from the 1 year zero spot rate of 4.2% and the 1.5 zero spot rate of % (1+.021) 2 (1+ 1 f 2 ) = ( ) 3 1 f 2 = Likewise 1 f 3 =.02847

Drake Drake University Fin 284 Local expectations example Local expectations theory says that returns of different maturities will be the same over a very short term horizon, for example 6 months. The return from buying the 2 year 4.45% coupon bond that makes semiannual payments selling at par and selling it in six months should be equal to the return on a 1 year coupon bond with a YTM of 4.2% if you hold it 6 months.

Drake Drake University Fin mo return on 1 year bond The 1 year bond has a current YTM of 4.2%. This means that a equivalent bond selling at par would make a $2.10 coupon payment at eh end of 6 months and at the end of 1 year. If you bought this bond at t = 6 months and sold it at t=1 year you should earn 1 f 1 (the 6 mo. forward rate) over the time you own the bond. The price of the bond at t=6 mos should reflect this.

Drake Drake University Fin mo return on 1 year bond At time t=1 year the bond makes payments of The total return from owning the bond is the capital gains yield and interest yield and it should equal 1 f 1 =.022

Drake Drake University Fin mo return on 1 year bond Buying the bond at time 0 and selling it at the end of the first 6 months would then produce a return of Which is equal to the spot (zero coupon) six month rate

Drake Drake University Fin 284 Return on the 2 year bond The price of the 2 year bond at the end of 6 months should also equal the PV of its expected cash flows discounted back at the forward rate (otherwise there would be an arbitrage opportunity). By finding the price at the end of 6 months we can again find the return from owning the bond for 6 months form t=0 to t=6 mos.

Drake Drake University Fin mo return on 2 year bond There are three coupon payments left from time t = 6 mos to t= 2 years, using the forward rtes the price of the bond at t=6mos should be:

Drake Drake University Fin 284 Total Return on holding 2 year bond for 6 months The 2 year bond originally sold for par and it made a $2.375 coupon payment at t=6mos. The total return from owning it is then: Which is the same as the 6 month return on the 1 year bond (and the same as the 6 month spot rate)

Drake Drake University Fin 284 Cox, Ingersoll, and Ross 1981 Journal of Finance Local Expectations Similarly owning the bond with each of the longer maturities should also produce the same 6 month return of 2%. The key to this is the assumption that the forward rates hold. It has been shown that this interpretation is the only one that can be sustained in equilibrium.*

Drake Drake University Fin 284 Return to maturity expectations hypothesis This theory claims that the return achieved by buying short term and rolling over to a longer horizon will match the zero coupon return on the longer horizon bond. This eliminates the reinvestment risk.

Drake Drake University Fin 284 Expectations Theory and Forward Rates The forward rte represents a “break even” rate since it the rte that would make you indifferent between two different maturities According to the pure expectations theory and its variations are based on the idea that the forward rte represents the market expectations of the future level of interest rates. However the forward rate does a poor job of predicting the actual future level of interest rates.

Drake Drake University Fin 284 Segmented Markets Theory Interest Rates for each maturity are determined by the supply and demand for bonds at each maturity. Different maturity bonds are not perfect substitutes for each other. Implies that investors are not willing to accept a premium to switch from their market to a different maturity. Therefore the shape of the yield curve depends upon the asset liability constraints and goals of the market participants.

Drake Drake University Fin 284 Biased Expectations Theories Both Liquidity Preference Theory and Preferred Habitat Theory include the belief that there is an expectations component to the yield curve. Both theories also state that there is a risk premium which causes there to be a difference in the short term and long term rates. (in other words a bias that changes the expectations result)

Drake Drake University Fin 284 Liquidity Preference Theory This explanation claims that the since there is a price risk and liquidity risk associated with the long term bonds, investor must be offered a premium to invest in long term bonds Therefore the long term rate reflects both an expectations component and a risk premium. This tends to imply that the yield curve will be upward sloping as long as the premium is large enough to outweigh an possible expected decrease.

Drake Drake University Fin 284 Preferred Habitat Theory Like the liquidity theory this idea assumes that there is an expectations component and a risk premium. In other words the bonds are substitutes, but savers might have a preference for one maturity over another (they are not perfect substitutes). However the premium associated with long term rates does not need to be positive. If there are demand and supply imbalances then investors might be willing to switch to a different maturity if the premium produces enough benefit.

Drake Drake University Fin 284 Preferred Habitat Theory and The 3 Empirical Observations Thus according to Preferred Habitat theory a rise in short term rates still causes a rise in the average of the future short term rates. This occurs because of the expectations component of the theory. Therefore the long and short rates move together and fact 1 is explained.

Drake Drake University Fin 284 Preferred Habitat Theory The explanation of Fact 2 from the expectations hypothesis still works. In the case of a downward sloping yield curve, the term premium (interest rate risk) must not be large enough to compensate for the currently high short term rates (Current high inflation with an expectation of a decrease in inflation). Since the demand for the short term bonds will increase, the yield on them should fall in the future.

Drake Drake University Fin 284 Preferred Habitat Theory Fact three is explained since it will be unusual for the term premium to be so small or negative, therefore the the yield curve usually slopes up.

Drake Drake University Fin 284 Measuring Yield Curve Risk Key Rate Duration, Calculating the change in value for a security or portfolio after changing one key interest rate keeping other rates constant. Each point on the spot yield curve has a separate duration associated with it. If you allowed all rates to change by the same amount, you could measure the response to the security or portfolio to a parallel shift in the yield curve.

Drake Drake University Fin 284 Key Rates and Portfolios By focusing on a group of key rates it is possible to investigate the impact of changes in the shape of the yield curve on specific parts of a portfolio, we will cover this in more detail in the portfolio section of the course.

Drake Drake University Fin 284 Yield Volatility One of the most important determinants of interest rate risk is the volatility of the yield. There fore it is important to measure the amount of volatility present, this requires a review of basic statistics.

Drake Drake University Fin 284 Measuring Historical Volatility The easiest way to measure the historical volatility is to calculate the variance and standard deviation of historical data. Variance of a random variable:

Drake Drake University Fin 284 Standard Deviation The standard deviation is then simply the square root of the variance. Generally, the variance of returns will be used to measure volatility.

Drake Drake University Fin 284 Change in Daily Yield We want to look at the daily change in yield, calculated as the percentage change in yield. This assumes simple compounding on the yield Assume that the yield on a 20 year treasury was 4% and it increased to 4.05% the next day. The percentage change in yield would be:

Drake Drake University Fin 284 Continuous Compounding The change could also be calculated assuming continuous compounding In this case the change in the daily yield would be equal to the natural log of the ratio of the two rates :

Drake Drake University Fin 284 Calculating Volatility Using historical data, the number of observations used will have a large bearing on the calculation. Often the choice depends upon the position that you are concerned with. For example a trader concerned with the change in value over a short period of time might look at a very short period such as 10 days. In contrast a portfolio manager might look at 20 or 30 days.

Drake Drake University Fin 284 Daily vs. Annual The daily standard deviation needs to be adjusted if you want to consider the annual volatility. The daily standard deviation can be scaled by multiplying it by the square root of the number of days in a year. (250, 260, or 365). The choice of the number of days to use will make a difference if the yearly standard deviation is used in valuation or measuring risk exposure.

Drake Drake University Fin 284 Interpreting Standard Deviation Assume that the annual standard deviation of a 20 year treasury is 10% and its yield is 5%. This implies that the annual standard deviation is 50 Bp ((.10)(.05) =.005) Assuming that distribution of returns is normally distributed there is a 68.3% probability that the yield will be within a range one standard deviation below and above the expected value.

Drake Drake University Fin 284 Prob Ranges for Normal Dist % 95.46% 99.74%

Drake Drake University Fin 284 Interpreting Standard Dev In our example the expected return was 5% with a standard deviation of 50Bp. This implies that there is a 68.3% probability that the yield will be be between 4.5% and 5.5%. And a 95.5% chance that he yield will be between 4% and 5%

Drake Drake University Fin 284 Implied Volatility Instead of calculating the volatility based upon past observations of the yield it is also possible to look at interest rate options available in the market and the volatility implied by their current price. Volatility is a key component in most option pricing models. This idea reverses the process by using the market price to calculate the volatility that would have needed to be used to arrive at the market price.