Forward Rate Bias

Trading the Forward Rate BiasMay 15 2008 Trading the Forward Rate Bias The so-called Unbiased Expectations Hypothesis (UEH) suggests that de-convexity-adjusted forward rates are unbiased predictors of future spot rates. A well-known phenomenon in the front end of the yield curve, however, is a systematic bias in forward rates: On average, forward rates over-estimate future spot rates. While academic research does not offer a conclusive explanation of the forward rate bias puzzle, some of the recent work may be worth reviewing. The current slope of the money market curve, now implying some 25bp of Fed tightening until year-end, is at odds with what many (maybe even the majority of) market participants expect. Some of this disconnect is now attributed to the systematic bias in forward rates. We discuss some basic trading strategies. Predictive Power of Fwd Rates The so-called Unbiased Expectations Hypothesis (UEH) suggests that de-convexity-adjusted forward rates are unbiased predictors of future spot rates. A well-known phenomenon in the front end of the yield curve, however, is a systematic bias in forward rates: On average, forward rates over-estimate future spot rates. Besides the inherent bias, the predictive power of forward rates is scarce. To illustrate the limited informational content of (convexity-adjusted) forward rates, we consider the following naïve forecasting model: Spott+1 = α + Fwdt + εt Fwdt represents the 1-year swap rate, 1-year forward, at time t, while Spott+1 stands for the subsequently realized 1-year swap rate, 1 year later. A regression of daily data going back 10 years yields the following results: Fidelio Tata, Ph.D. Head of IR Derivatives Strategy +1 (203) 618-2116 [email protected] (1) α = -80bp; R-square = 17.3% Notice that, on average, 1-year-into-1-year forward rates overestimate subsequent 1-year spot rates by as much as 80bp! At the same time one needs to admit that the coefficient of determination (R-square) is 8% vs. A regression of daily data going back 10 years yields the following results: α = -13bp.8% What is striking is that the goodness of fit is even lower than with forward rates (R-square of 15. more often than not. To evaluate the forecasting accuracy of spot rates. are higher than the subsequently realized spot rates (on average 80bp for the past 10 years).65% on 5/14/08).3%).9% on 5/14/98 to 3. This also affects the forward rate analysis (equation 1) and ideally one should choose a period without such trend. making spot an even worse predictor of the future. The fact that spot rates also overestimate future realized rates (although to a much smaller degree) indicates that the time series chosen here (past 10 years) captures a drift lower in yield (1-year swaps from 5. Chart 1: Systematic bias in forward rates 7 6 5 4 3 2 1 0 -1 -2 Dec-99 Dec-00 Dec-01 Dec-02 Dec-03 Dec-04 Dec-05 Dec-06 Dec-07 Jun-99 Jun-00 Jun-01 Jun-02 Jun-03 Jun-04 Jun-05 Jun-06 Jun-07 Jun-08 1y1y fwd minus subsequent 1y spot rate Fed Funds Target Rate Average fwd minus spot Source: RBS Global Banking & Markets Spot rate bias An interesting question is whether spot rates make for a better predictor of the future. R-square = 15. To test this assumption.The Royal Bank of Scotland very. Combining spot and forward Often. Forward rates. Forward rate bias Chart 1 illustrates the forward rate bias for 1-year-into-1-year swap rates. This suggests that the statistical model (1) explains only very little of the variability in the data set. practitioners expect realized rates in the future to come in somewhere between spot and forward. we consider the following model: Spott+1 = α + β Spott + (1-β) Fwdt + εt (3) 2 Trading the Forward Rate Bias | May 15 2008 % . very low. 17. we consider the following model: Spott+1 = α + Spott + εt (2) Spott represents the 1-year swap rate at time t. while Spott+1 stands for the same 1-year swap rate. 1 year later. on the other hand. 680-692.94% (median of 2%). there is widespread acknowledgement of some informational value beyond what can be extracted from a history of spot rates. Trading the Forward Rate Bias | May 15 2008 3 Disconnect between expectations and forwards The current market environment suggests that there is somewhat of a disconnect between what market participants would verbally express in terms of rate expectations and what is implicitly priced into the money market curve. The information in long maturity forward rates. R.4% vs. there appears no widespread expectation of a Fed rate hike between now and year-end. for the most part. 1987. the goodness of fit is higher than with either spot or forward rates alone (R-square of 20. For example. have been skeptical about the Fed hiking implied for later this year and in 2009. Alfonso.4% While still not particularly high. 77. implied a roughly 30% chance of a 25bp hike during the survey period (since then. Vol. This is maybe most obvious when looking at the Fed fund target rate at year-end. Fama and Bliss (1987) concluded that.. Year-to-date. 2002. R-square = 20. if any. Taylor and Francis Journals. market participants.2 While few. Fed fund futures contracts..1 When looking at the incremental informational value of forward rates. However. Dominguez. A survey of 54 Street economists conducted by Bloomberg between May 2 and May 8 resulted in an average target rate of 1. 1-year forward rates on Treasury bills contained information on expected returns on bills one year in advance. Can Forward Rates Be Used to Improve Interest Rate Forecasts?. given the fragile economy. respectively).8% and 17. 15. E.” often vaguely attributed to a term-risk premium. studies suggest that forward rates produce better one-step-ahead forecasts. Applied Financial Economics. this has increased to as high as 100%). studies suggest that forward rates are good predictors of the future. Main theme of 2008 Already for several months. In other words. there is also a “bias. and Bliss. Empirical evidence The results presented here appear consistent with the majority of studies conducted in academic research. which greatly reduces the value of forward rates as a signal. over 1964-1985. 12(7). July 2002. pages 493-504. as well as better once-and-for-all forecasts of interest rates over a full year horizon than those obtained form the own past of interest rates.The Royal Bank of Scotland A regression of daily data going back 10 years yields the following results: α = -47bp.3%. the liquidity crisis in the financial industry and the weak housing market. The beta of 47% suggests that roughly the average of spot and forward makes for the best predictor of the future. Emilio & Novales. . β = 47%. American Economic Review. it has been a recurring market theme among many market strategists (including our 1 2 Fama. From that point on. In other cases. 3 We highlighted this before in our 2/1/08 edition of U. Derivatives Strategy Weekly. investors can pick the part of the yield curve that is believed to reflect the highest bias. By choosing which contract to buy and by setting the frequency of rolling the contract. the Eurodollar curve continues to price for 2-3 25bp Fed tightening moves over a 1-year period starting in 8-month’s time. 1.The Royal Bank of Scotland own) to (explicitly or implicitly) suggest variations of the Forward Rate Bias trade. Buying Eurodollar/Fed fund future contracts Investors who believe that there is a systematic bias in short-dated forward rates are known to be active participants in Eurodollar and Fed fund futures trading. investors only enter into them if their specific view about future rates conflicts with what forwards are implying. those trades are part of a longer-term trading program. Chart 2: Fed tightening implied for 1yr period starting 8 months forward Trading the Forward Rate Bias | May 15 2008 4 Fed starts tightening Fed starts easing 200 175 ED5-ED8 spread [bps] 150 125 100 75 50 25 0 -25 Jan-00 Jul-00 Jan-01 Forward getting it wrong Jul-01 Jan-02 Jul-02 Jan-03 Jul-03 Jan-04 Jul-04 Jan-05 Jul-05 Jan-06 Jul-06 Jan-07 Jul-07 Source: RBS Global Banking & Markets Forward Rate Bias trades A number of generic trading strategies are based. We summarize and discuss the main variation of those trades below. a 3½-year period during which no Fed tightening had occurred. at least to some degree. Despite the potential of a prolonged recession. What is implied right now roughly corresponds to what was priced in for in March 2001. forward rates got it wrong for almost 3 years (see chart 2).3 The steepness in the front end of the yield curve is nothing unusual: Between January 2001 and June 2004. Often. on exploring the forward rate bias. Jan-08 Fed starts easing . forward rates were implying anywhere between two to eight 25bp tightenings over a 12-month period.S. thus. 2. 1 year forward) exhibited the highest forward rate bias and. February 2000.. vol. pages 239-279. see Gupta.4 4 On the de-convexity adjustment of forward rates. 5 Trading the Forward Rate Bias | May 15 2008 Chart 3 displays the cumulative P/L of being long one Eurodollar futures contract and rolling it every quarter (to keep it at constant maturity). this strategy would have been profitable. “Red” Eurodollar packs (roughly representing the 1-year swap rate. 2000. Anurag & Subrahmanyam. An empirical examination of the convexity bias in the pricing of interest rate swaps.000 $0 -$5. where the spot 2s10s swap slope is compared to what the 1year forward-starting 2s10s slope had implied a year before.000 Dec-98 Mar-99 Jun-99 Sep-99 Dec-99 Mar-00 Jun-00 Sep-00 Dec-00 Mar-01 Jun-01 Sep-01 Dec-01 Mar-02 Jun-02 Sep-02 Dec-02 Mar-03 Jun-03 Sep-03 Dec-03 Mar-04 Jun-04 Sep-04 Dec-04 Mar-05 Jun-05 Sep-05 Dec-05 Mar-06 Jun-06 Sep-06 Dec-06 Mar-07 Jun-07 Sep-07 Dec-07 Mar-08 ED1 to Exp ED5 to ED4 ED2 to ED1 ED6 to ED5 ED3 to ED2 ED7 to ED6 ED4 to ED3 ED8 to ED7 Source: RBS Global Banking & Markets According to the analysis presented in chart 3.500 from the terminal cumulative P/L). forward curve slopes tend to overestimate the degree of expected curve flattening.000 -$10. . Curve steepener trades Because short-dated rates (primarily the ones reflected by Red Eurodollar pack) appear to be plagued by a forward rate bias to a larger degree than long-dated rates.000 $20.000 $10. Elsevier.The Royal Bank of Scotland Chart 3: Cumulative P/L of rolling one Eurodollar futures contract $25. Journal of Financial Economics. Over a 10-year period. are good candidates to a potential trading program.000-$7. This is illustrated in chart 4. although part of the calculated gains are just due to the fact that short-dated interest rates dropped some 200-300bp since 1998 (to adjust for this.000 $15. one would have to subtract some $5.000 P/L $5. Notice that forward rates have been de-convexity adjusted to eliminate the convexity bias of (especially) longer-dated forward rates. 55(2). Marti G. The risk is measured as the standard deviation of the P/L. close to the 0. In any case.5 While evidence of a forward slope bias.47 Sharpe ratio of the outright 1-year-into-1-year trade displayed in chart 1. due to the bias in forward rates. in the eyes of many market participants. fwd from a year before Average spot minus forward slope Fed Funds Target Rate Source: RBS Global Banking & Markets During the past 10 years. may not be convincing enough for entering in a rolling curve-steepener position as an ongoing trading program. Similarly (and somewhat easier to analyze from an analytical point of view). to a lesser extent. Chart 5 illustrates this by looking at 1-year-into-1-year receiver swaptions. the goal is to explore the fact that strike levels for the options are biased to the upside. this expected compensation due to the forward bias is still attractive from a risk-return perspective. 6 Trading the Forward Rate Bias | May 15 2008 .adj. call spreads and call flies. The Sharpe ratio of the structure displayed in chart 4 is 0.The Royal Bank of Scotland Chart 4: Systematic bias in forward slope 7 6 5 4 3 2 1 0 -1 Jun-99 Jun-00 Jun-01 Jun-02 Jun-03 Jun-04 Jun-05 Jun-06 Jun-07 Jun-08 Dec-99 Dec-00 Dec-01 Dec-02 Dec-03 Dec-04 Dec-05 Dec-06 Dec-07 % 2s10s spot slope minus de-conv. receiver swaptions can be utilized. Receiver swaptions A variation of an outright long position in Eurodollars is to structure the trade conditionally through Eurodollar and Treasury future (and. an analysis into the attractiveness of receiver swaptions centers around how much roll-down from forward to spot is necessary to offset the costs of buying the option. Displayed is the ratio between the expected value of the receiver swaption at expiration assuming unchanged spot rates and the premium paid at trade 5 The Sharpe ratio measures the expected return per unit of risk. 3. Because curve slope positions are typically less volatile than outright interest rate exposure. 1 year forward.42. Typically. the bias helps making curve steepening trades look more attractive during times when investors are looking for curve steepening anyway (such as during periods of Fed easing). the average bias of forward slope overestimating the degree of flattening to come was about 30-35bp as far as 2s10s slope. is concerned. Fed fund future) calls. . only that the ratio between implied volatilities. A ratio of 1 suggests that forward rates need to converge to spot rates (i. not their outright level. There is little causation.6 Because the payout ratio has the expected return in the numerator and a risk measure (swaption price. .The Royal Bank of Scotland Chart 5: Payout Ratio of 1-year-into-1-year receiver swaption 1-year roll-down divided by upfront premium paid 5 4 3 Multiple 2 1 0 -1 -2 2000 2001 2002 2003 2004 2005 2006 2007 2008 Source: RBS Global Banking & Markets The attractiveness of receiver swaptions (and other conditional variations of forward rate bias trades) highly depends on the general level of implied volatility. 6 Unfortunately.e. The analysis is similar. as slope and vol are highly correlated with each other. As we have shown above (formula 3). any ratio above 2 looks attractive from a forward rate bias perspective. Conditional curve trades The same way outright interest rate positions can be structured bullishly through calls/receivers. Rather. because the expected roll-down results in a gain twice as large as the upfront premium. Thus. a 50% convergence of forward to spot rates is sufficient for the trade to break even. those trades are typically favored whenever the yield curve is steep and implied volatility is low. are often favored when implied volatility is trading at elevated levels. this is not often the case. historical evidence suggests that a rate halfway between spot and forward makes for the best predictor of future realized rates. though (“correlation does not imply causation”). 7 Trading the Forward Rate Bias | May 15 2008 inception. the attractiveness of receiver swaption trades based on this payout ratio is inherently measured on a risk-adjusted basis. is the second essential driver of the trade (besides the forward rate bias). Conditional curve steepeners. Therefore. as a consequence. Thus. complete roll-down on the yield curve) to offset the costs of the receiver swaption. the fraction is essentially a Sharpe Ratio. which is a function of implied swaption volatility) in the denominator. both are a function of general uncertainty. curve steepeners can be expressed conditionally as bull steepener trades. A ratio of 2 implies that. 31% 3.42% 2. Here.98% 3. the structure profits from a potential forward rates bias.78% 2.30% 2.86% 2. Here.67% 2.07% 1.30% 3.06% 2.53% 1.73% 2.16% 2.34% 2.90% 10y 3. Table 1 shows the break-even rates of various 1-by-2 receiver spreads as of April 11.63% 1.98% 1. but one of the most popular variations is the premium-neutral “1-by-2” receiver spread. the purchase of an at-the-money-forward (ATMF) receiver swaption is subsidized by selling out-of-the-money (OTM) receivers.e.07% 2.82% 2. the analysis of receiver spreads typically incorporates volatility skew (i.78% 1.59% 2.77% Source: RBS Global Banking & Markets.89% 1.53% 1.19% 3.98% 3.00% 2. 2008. Break-even rate Often. .81% 3y 2. or even zero).72% 2.61% 3. the difference between implied ATM and OTM swaptions).43% 5y 2. but also on some sort of comparison between expected payout and premium paid. Also.09% 3.63% 1.68% 3.86% 1. Strike levels and notional weights differ on this kind of structure. the notional amount of the OTM receiver sold is twice that of the ATMF receiver bought.85% 3.64% 1.45% 2.46% 2.87% 1.79% 1.55% 3.50% 2.92% 1.27% 3.98% 2.60% 2.76% 1.53% 2. Like in the outright receiver swaption.53% 1.05% 2.14% 2.97% 2.46% 3. Receiver spreads have more leverage than simple receiver swaptions (due to the fact that the initial upfront premium is reduced.89% 2. this is no longer premium-neutral and an analysis of this structure not only depends on volatility skew.23% 2. the attractiveness of receiver spreads is measured against the break-even rate.05% 2.36% 2.41% 2. Naturally. struck at a level that makes the package premium-neutral.77% 3.94% 2.31% 3. This rate shows how much rates can drop before the structure starts losing money (at swaption expiration) and can be compared to what economic analyses suggest the likelihood of such a rally would be.17% 2.83% 3.47% 2.25% 3.00% 4. Table 1: Break-even rates for 1-by-2 receiver spreads ↓Expiry Tail→ 1m 2m 3m 4m 5m 6m 7m 8m 9m 10m 11m 1y 1y 2.48% 2.10% 3.82% 3.75% 1.91% 3.05% 2.70% 2.24% 3.16% 4y 2.08% 4.45% 2.84% 1.55% 2y 2. as of 4/11/2008 Receiver butterfly A three-legged receiver spread is a receiver butterfly.58% 2.33% 4.64% 7y 3.55% 2.36% 3.36% 2.72% 3.16% 2.42% 2. the maximum downside of the two-legged 1-by-2 spread is reduced by buying back an even further-OTM receiver. but add downside risk to the trade.43% 1.The Royal Bank of Scotland 4. Receiver spreads Trading the Forward Rate Bias | May 15 2008 8 In a receiver spread.33% 2.13% 30y 4. only that the investor sees limited potential of interest rates falling below the strike of the OTM receiver.00% 1.37% 3.89% 1..09% 2. financial instruments or strategies mentioned herein. providing investment. Inc. Greenwich Capital Markets. 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