Mind the Trap: Yield Curve Estimation and Svensson Model

Transcription

1 Mind the Trap: Yield Curve Estimation and Svensson Model Dr. Roland Schmidt February 00 Contents 1 Introduction 1 Svensson Model Yield-to-Duration Do Taxes Matter? Forward Rate and Par Yield Curves 6 Emerging Market Debt 7 Conclusion 7 1 Introduction Yield-to-maturity charts offer a tempting solution to investors. Draw a smooth line through the data points and invest in bonds with yields above the curve. The purpose of this article is threefold. First, we will explain what s wrong with this reasoning. Then we will show that replacing maturity by duration is not a remedy. Finally, we will estimate forward rate and par yield curves to infer an unbiased view of the term structure. There is more than one suitable model for estimating yield curves. Arbitrage models offer the advantage that they re consistent with how traders price fixed income derivatives. Unfortunately there isn t yet a standard model for doing so. Most arbitrage models are not able to fit even very common derivatives simultaneously. On the other hand, spline techniques quite often create oscillations that look counterintuitive and imply unreasonable forward rates. Nonparametric approaches like kernels have similar pitfalls. For these reasons we will follow the lines of Nelson and Siegel who used an exponential polynomial. Their approach has been extended by Svensson for a second exponential term. This so-called Comments are welcome at my address quant.analyse@t-online.de. For more information please visit 1

2 Svensson model today is used by most central banks to infer yield curves from market data 1. We will illustrate our arguments utilising data for German government bonds. The respective yield curve is shown in Figure 1.. Yield Curve German Government Bonds (6th February 00) Yield-to-Maturity Maturity Figure 1 From a first inspection we conclude a slight inversion at the short end but a normally sloped yield curve for maturities beyond 1 year. A few investment opportunities seem luring as well. For example, at a maturity of around 10 years there s a bond offering some yield pickup. However, we will soon see that this bond isn t attractive at all, letting the uncautious investor falling into a trap. Svensson Model The model can be best described by the (instantaneous) forward rate curve in equation (1): τ f τ = β 0 + (β 1 + β )e τ τ κ 1 + β e τ κ (1) κ 1 κ The paramaters β 0, β 1, β, κ 1, and κ are estimated from market data. From equation (1) follows that the short-term rate is β 0 + β 1 and the long-term rate β 0 (provided κ 1, κ are positive). The overall shape of the forward rate curve is determined by the β s. The two parameters κ 1 and κ are scaling factors for the time variable τ and affect the horizontal position of the curve. After estimating the parameters we can compare the market yields to the yields predicted by the model (see Figure ). Two observations are striking. First, the fit is very good, even at the short end where the yield curve is inverted. Second, the fitted line has kinks whenever we presumed investment opportunities to exist. The bond mentioned before in fact has a yield slightly below the yield predicted by the model. The reason 1 There is a link between the Svensson model and the extended Vasicek model which is among the most popular one-factor arbitrage models. But its discussion is beyond the scope of this paper. We minimised squared yield differences to obtain the estimates.

3 becomes clear when we look at the coupon of the bond. The coupon is.7%, compared to.% of the bond next to the right (which trades at a lower yieldto-maturity). The duration of the bond in question is increased by the lower coupon and distorts the comparison based on yield-to-maturities. The same happens at maturities near years. The bond with the highest yield has a.7% coupon against 6.% of the bond with the next higher maturity.. Market and Svensson Yield Curves German Government Bonds (6th February 00) Yield-to-Maturity Maturity Figure Yield-to-Duration In the last chapter we argued that coupon differences create a bias in yield-tomaturity charts. Suppose we use yield-to-duration charts in place. Would yields then become visually comparable without the need to estimate a model? Figure gives the answer.. Yield-to-Duration Curve German Government Bonds (6th February 00) Yield-to-Duration Duration Figure The quote for the.% bond was made after p.m. while most other bond prices stem from auctions between 11 a.m. and 1 p.m. As the German bund future soared by more than 60 ticks in the afternoon the.% bond had a yield below the Svensson curve.

4 Though the kinks have become less pronounced they didn t vanish. Yieldto-durations mitigate the problem but don t solve it. This means that circumventing the trap requires the estimation of a model. The main result so far is: The estimation of the Svensson model helped us to identify bonds that only seemingly have attractive yield-to-maturities. Investors should have a look on spreads between market and Svensson model yields. Do Taxes Matter? Do deviations of market from model yields have a random pattern? So far, we ignored tax effects in our analysis. According to the German tax code capital gains on bonds are tax-free if they are not realised within 1 year and the bond was issued at or near par. This means that, other things being equal, bonds with small coupons are more attractive to investors. Depending on demand and supply forces this may cause small-coupon bonds to trade at lower pre-tax yields and the spread between market and model yield to be positively correlated with the coupon. In Figure we explore this case. 7. Impact of Coupons Market-Model Spread vs. Coupon Spread in bps Coupon Figure In fact, investors in low-coupon bonds get penalised. They lose on average basis points of yield for each percentage point reduction in the coupon. The analysis of the preceding chapters could now be repeated for after-tax yields, searching for the (marginal) tax-rate that is reflected in market prices. However, this extension is beyond the scope of this article. The Svensson model can be used as well to determine fair values of illiquid bonds or OTC-transactions priced off a given yield curve. Another area are bonds with default risk. If hazard rates are constant the Svensson model can deal with credit spreads by varying the coefficient β 0 in equation (1). The correlation coefficient is 0.. The correlation between the spread and tax-free capital gains is 0.0. Though tax-free capital gains are more directly linked to tax effects than coupons, the corresponding correlation coefficient might be misleading due to what econometricians call simultaneous equation bias. This is the reason why we prefer exogenous coupons to measure the tax effect.

5 Forward Rate and Par Yield Curves The Svensson model allows the direct computation of other interest rate curves that show the overall shape of the term structure from another perspective. Figure depicts the forward rate curve, (annually compounded) spot rates and the par yield curve. Forward Rates, Spot Rates and Par Yields German Government Bonds (6th February 00) Forward Rate Spot Rate Par Yield Swap Rate Figure The forward rate curve is quite steep in the to year range, levelling off after 7 years. The slope at short maturities reflects the current market consensus that the European Central Bank will raise its repo rate at the end of 00 or early next year at the latest. But will the ECB really raise interest rates by more than 00 basis points over the next 7 years? Investors who think the German government will restore its financial discipline after the blast of the stability pact might want to ride the yield curve betting that bonds with maturities beyond 7 years will roll down the curve. We further added EUR swap rates to the chart. The swap rate curve is virtually parallel to the government par yield curve with the spread ranging from 0 (at 0 years) to 16 basis points 6. The positive swap spread reflects a (negative) convenience yield on German government bonds and higher credit risks of banks paying Libor flat. The latter effect is partially compensated by the fact that interest rate swaps don t require an exchange of principal. 6 Emerging Market Debt The German government bond market turned out quite efficient in the sense that deviations of market and model yields were small. Does this observation also hold true for emerging market debt? We will briefly look at Brazil and Argentina to scrutinise the capabilities of the Svensson model in more extreme situations. Let s start with the term structure of Brazil. In Figure 6 we see a normal yield curve. Yields occasionally differ by large margins from their fair (model) 6 The rates are not perfectly comparable because of different day count conventions and payment frequencies.

6 values. For example, the 8% bond maturing 01 changed hands at 97.% though its fair value was 9.01% only. 11 Brazil Government Bonds (1th February 00) Figure 6 We now turn to Argentina. Because Argentina defaulted on its payment obligations all bonds are traded flat. At the time of writing the government offers bondholders only cents on the dollar. For early investors the loss becomes even larger as they won t recoup unpaid interest under the proposal. The pressure on Argentina has been quite low because the country runs a current account surplus and only recently its properties in foreign countries have been seized to meet payment obligations. Under these circumstances it seems questionable whether the Svensson model can fit the Argentinian term structure. Figure 7 shows the results. 180 Argentina Government Bonds (1th February 00) Figure 7 As in the case of Brazil, the Svensson model can match the overall shape of the yield curve. 6

7 7 Conclusion The Svensson model fits the term structure of German government bonds very well. The results are reasonable also for high-yield bonds. The Svensson model is a powerful tool to adjust yield-to-maturities for different coupons and to derive forward rate and par yield curves. 7