A Benchmark for Interest Rate Risk using a Markowitz Approach
J.V. Verheijen
Amsterdam, October 29, 2014
A thesis in partial fulfillment of the requirements for the degree of Master of Science in Financial Econometrics
Department of Quantitative Economics
University of Amsterdam
Principal Advisor: Dr. H.P. Boswijk Second Advisor: Dr. S.A. Broda Advisor ABN AMRO: V.M. van Rooijen Preface
This study was conducted at request of ABN AMRO, further referred to as ”the bank”. I would like to extend my sincerest thanks and appreciation to Wilson Jan Kansil and the Balance Sheet Analysis team, for providing the opportunity and their support. Further I would like to recognize Martijn van Rooijen, from ABN AMRO, and dr. Peter Boswijk, from the University of Amsterdam, for their guidance and input on the subject. Finally I would like to emphasize that the views expressed in this thesis are those of the author. No responsibility for them should be attributed to the ABN AMRO.
1 Contents
1 Introduction 5
2 Literature and background 8 2.1 Mismatch results ...... 8 2.1.1 Interest rate risk ...... 8 2.1.2 Balance sheet ...... 9 2.2 Current approach ...... 10 2.3 Mean-variance optimization ...... 11 2.3.1 Mean-variance optimization ...... 12 Collecting information on the investor and the market . . . . 13 Computing the optimal portfolio allocation ...... 14 2.3.2 Sharpe Ratio optimization ...... 15 2.4 Yield curve model ...... 16
3 Theory 18 3.1 Pricing Bonds in continuous time ...... 18 3.1.1 Change the probability measure for bond pricing ...... 20 3.2 Modelling yield curves and stylized facts ...... 22 3.3 Nelson Siegel Term Structure Models ...... 23 3.3.1 Nelson-Siegel model ...... 25 3.3.2 Dynamic Nelson-Siegel model ...... 25 3.3.3 The arbitrage-free Nelson-Siegel model ...... 27 “The Yield Adjustment Term” ...... 29
2 3.4 Forecasting ...... 29 3.5 Models for comparison ...... 30 3.5.1 Random Walk model ...... 31 3.5.2 Principal Component Analysis ...... 31 3.6 Monte Carlo simulation ...... 33
4 Empirical Results 35 4.1 Data description ...... 35 4.2 Specification model ...... 38 4.3 Empirical Results ...... 39 4.3.1 Forecasting Performance ...... 39 Fit of the Euro swap curve ...... 40 Fit of the zero coupon fixed-income yield curve ...... 44 Bond Prices ...... 46 4.3.2 Time Evolution ...... 47 4.3.3 Stability of the model parameters ...... 49 Dynamic Nelson Siegel model ...... 49 4.3.4 Monte Carlo Simulation ...... 49 4.4 Portfolios under different yield scenarios ...... 51 4.5 Alternative Benchmark ...... 54
5 Conclusion 55
Bibliography 56
A Example 60 A.1 Duration Gap analysis Balance sheet ...... 60 A.1.1 Net change with increasing yield curve rates ...... 60 A.1.2 Duration Gap ...... 60
B Derivations 63 B.1 Derivation Correction term of AFNS model ...... 63
3 C Theorems 65 C.1 Girsanov’s Theorem ...... 65 C.2 Itˆo’sLemma ...... 65
4 1. Introduction
The objective of this thesis is to calculate a benchmark to measure the performance of steering transactions on the interest rate mismatch. The mismatch naturally follows from the balance sheet of a bank, because it consists mostly of short-term liabilities and long-term assets. Since the yield curve is in general a monotonically increasing and concave function the long-term yields are higher than the yields for short maturities and therefore the mismatch (usually) generates positive results. However, because of this difference in interest rates and the mismatch in duration, the bank is exposed to interest rate risk. Interest rate risk is the bank’s exposure to adverse movements in the interest rates. There are four sources of interest rate risk, which are basis -, optionality -, repricing - and yield curve risk. Accepting interest risk is normal for banks and can be an important source of profitability and shareholder value. Nevertheless, excessive risk taking can significantly threaten the bank’s earnings and its capital. Therefore, effective risk management is needed to secure the safety and soundness of banks. To maintain effective risk management the bank for international set- tlements 1 (BIS) requires banks to have standards for Performance Measurement. Here lies the relevance of this thesis, which tries to obtain a benchmark for the steering transactions for the duration mismatch. From a management perspective the mismatch, which follows from the bal-
1The Basel Committee on Banking Supervision is a Committee of banking supervisory au- thorities which was established by the central bank Governors of the Group of Ten countries in 1975. It consists of senior representatives of bank supervisory authorities and central banks from Belgium, Canada, France, Germany, Italy, Japan, Luxembourg, Netherlands, Spain, Sweden, Switzerland, United Kingdom and the United States.
5 ance sheet, is steered via duration. Duration is the most commonly used measure of risk in bond investing. The duration mismatch is steered with swap transactions, which can be used to make the balance sheet more (or less) sensitive to changes in the interest rates. These transactions have an impact on the Net Interest Income (NII) and the development of the Market Value of Equity (MVE) of the bank. When the bank receives floating rates from a swap transaction increasing rates lead to a higher NII. The MVE is also affected by changes in the interest rates, an increase (decrease) of the yield curve leads to a decrease (increase) in MVE, since the MVE is the discounted value of all future cash flows that are discounted with lower (higher) yields. The benchmark needs to take the changes in NII and MVE into account. This thesis tries to obtain a benchmark for the NII and MVE by investigating the return and market value of an optimal bond portfolio, constructed by the Markowitz approach. This mean-variance approach aligns with the objective of the bank, which is maximizing its returns given a prespecified level of risk. Hence, when comparing the NII and the market value of the steering portfolio with the benchmark, the magnitude and structure of risk of the latter must correspond to the risk of the steering transactions. To perform a mean variance optimization the expected return and (co-) vari- ances of the available bonds are needed. The expected return and variance of (fixed-income) bonds follow from forecasts of the yield curve, hence a yield curve model is needed. The yield curve models used in this thesis are the Dynamic Nel- son Siegel model (DNS) and Affine Arbitrage Free Nelson Siegel model (AFNS). The empirical results of these models are compared with those obtained from the Random Walk model and a Principal Component Analysis. The benchmark for the return generated by the duration mismatch is ob- tained in four steps. First, different models are used to obtain the yield curve. Secondly, the zero coupon fixed-income yield curve is bootstrapped from the swap curve. Thirdly, the bond prices, and hence the period holding returns, are obtained using the latter yield curve. Finally, the bond portfolio returns are optimized with respect to their variance.
6 This thesis is divided into five chapters. The second chapter explains the framework used to obtain the benchmark. The third chapter elaborates on the different models in detail. The fourth chapter describes the empirical results of the followed framework. Finally in Chapter five the conclusions are given.
7 2. Literature and background
2.1 Mismatch results
As discussed in the introduction this thesis tries to obtain a benchmark for the results generated from the duration mismatch1. This mismatch arises primarily from the fact that the repricing period of the assets typically exceeds the repricing period of the liabilities. To understand the concept of mismatch results it is important to understand the basics of interest rate risk and the balance sheet. This section gives a short introduction to both concepts. The remainder of the chapter introduces the models needed for the benchmark portfolio.
2.1.1 Interest rate risk
Interest rate risk is the exposure of a bank’s financial condition to adverse move- ments in interest rates. Accepting this risk is a normal part of banking and can be an important source of profitability and shareholder value (Basel Committee on Banking Supervision, 2004). Banks are typically exposed to four sources of interest rate risk, which include basis -, optionality -, repricing - and yield curve risk. The interest rate risk that follows from the duration mismatch are repricing - and yield curve risk, hence a short introduction might be helpful. Repricing risk arises from the timing difference in maturity (for fixed-rate) and repricing (for floating-rate) of bank assets, liabilities, and off balance sheet positions. For instance a bank that funded a long-term fixed-rate loan with a
1These results are also known as mismatch results.
8 short-term deposit could face declining net interest income (NII) if interest rates increase. This decline follows from the fixed, and therefore unchanged (long-term) income together with the increased (variable) funding costs. The second source of interest rate risk, yield curve risk, follows from the same timing differences but arises from non parallel changes of the yield curve. For instance, the value of a position in 10-year bonds hedged by a position in 5-year bonds could decline if the yield curve steepens. In this case, the present value of the 10-year position decreases, because it is discounted at higher rates, which is not offset by the value change of the hedged positions because the corresponding rates did not change or changed less. Therefore the total position decreases in value when the interest rate curve steepens. These two sources of interest rate risk, repricing - and yield curve risk, affect the balance sheet of bank and the interest income. The next section elaborates the balance sheet to examine the exposure of the bank towards these two sources of interest rate risk.
2.1.2 Balance sheet
The bank fulfils a maturity transformation role by financing long term assets with short term liabilities. Under normal conditions this ensures a positive NII, since the interest income generated by assets (long-term) exceeds the interest expenses paid for liabilities (short-term). To connect the balance sheet of the bank to interest rate risk a duration gap analysis is often used. A duration gap analysis examines the sensitivity of the market value of the financial institutions net worth to changes of the interest rates. This analysis is based on modified duration, a modified version of the Macaulay model. Macaulay duration Modified duration = YTM . (2.1) 1 + n Here is n the number of coupon payments per year, YTM the yield to maturity and Macaulay duration is given by Pn t·C n·M t=1 t n Macaulay duration = (1+y) (1+y) , (2.2) Current bond price where C is the coupon payment, M the face value and y the periodic yield.
9 The modified duration, further referred to as duration, of an instrument is an important measure for investors to consider, as bonds with higher durations carry more risk and have a higher price volatility than bonds with lower durations. For zero coupon bonds the duration equals the time to maturity, for plain vanilla bonds, which offer coupon payments, the duration is shorter than time to maturity. An important fact of duration is that it is an additive measure, which implies that the duration of a portfolio is the weighted average duration of all individual assets. A positive interest mismatch is ensured when the duration of assets is higher than the duration of the liabilities. This implies that the liabilities are repriced more frequently than the assets on the balance sheet. Hence with a positive in- terest mismatch and increasing interest rates both the interest mismatch and the market value of equity decreases. The interest mismatch decreases since liabilities are repriced earlier than assets and the interest expenses are elevated because of increased rates. The market value of equity decreases since the market value of assets decreases more than the market value of liabilities. The asset value changes more because the duration of assets is higher, which implies has a higher sensitivity to changes in the interest rates. To make the concept of duration and duration gap more tangible, a fictive balance sheet is considered in Appendix A.1. The current approach of the bank is based on this gap analysis and is elaborated in the next section.
2.2 Current approach
The duration gap is managed by taking receiver - and payer positions in swaps. A net receiver position means that the bank receives fixed and pays floating rates. An increase of the interest rates would increase the rates payed (while not affecting the rates received) and therefore decreases the market value of the position. A net payer position means that the banks pays fixed and receives floating rates. In this case an increase of the interest rates would increase the market value of the position. With these positions it is possible to make the balance sheet less -, or more sensitive to changes of the interest rates. Net payer positions can be used to
10 decrease the duration of equity and therefore make the balance sheet less sensitive to changes in the yield curve. The interest mismatch is managed by the Asset & Liability Committee (ALCO) by means of duration, market value of equity-at-Risk (MVE-at-Risk) and Net interest income (NII). When the bank reduces the duration of the balance sheet, net payer swaps are needed for hedging. Given the level of duration the bank has a certain NII and MVE, which are both determined by developments of the interest rates. It is important to measure the NII and MVE given this level of duration and the steering actions. Especially the NII is affected by the steering transactions. The proposed benchmark tries to provide a performance measure for these two statistics. The calculation of the duration mismatch only contains an interest rate risk component, this should be reflected in the benchmark. Therefore, the benchmark should be a portfolio containing only an interest rate risk component, which can be done with a portfolio of bonds. Note, that the assumption is made that bonds are default free. This portfolio will result in coupon payments, which is comparable with the NII of the bank. Further the portfolio has a changing market value, the present value of all the future cash flows, which can be compared with the market value of the balance sheet. Secondly, the objective of the bank, generate the best results possible given the risk that is taken, should be reflected in performance measurement and therefore in the benchmark. This characteristic of the high- est return given the risk that is taken leads to a mean variance optimized bond portfolio as our benchmark. The next section will introduce the mean-variance optimization used for the benchmark.
2.3 Mean-variance optimization
The previous section elaborated on the need for a mean-variance optimized portfo- lio as the benchmark (portfolio). When performing a mean-variance optimization both the return and variance of available bonds are needed. Hence, this section gives a short introduction of the mean-variance approach.
11 2.3.1 Mean-variance optimization
The benchmark is an optimized fixed income portfolio using the mean-variance ap- proach proposed by Markowitz (1952), to be specific a Sharpe ratio optimization. The mean-variance optimization is widely used by managers for portfolio con- struction and to develop quantitative asset allocation strategies. However, these strategies are often restricted to equity portfolios. For the selection of fixed income portfolios managers often use duration. There are two reasons that explain why mean-variance optimization is rarely used in fixed income portfolio selection. The first argument is the relative stable behavior and low historic variability of bonds, which discouraged the use of ad- vanced techniques to exploit the risk-return trade-off. However the variability in bond markets has increased a lot since the crisis, even in markets with low default probabilities, see Korn and Koziol (2006). This increase in volatilities encourages the use of more sophisticated methods like a Sharpe optimization for bond portfolio selection. Secondly, difficulties in obtaining the expected returns and covariances of the fixed income portfolios has restrained the use of the mean-variance ap- proach in fixed-income portfolios. Fabozzi and Fong (1994) argued that if returns and covariances were easily available fixed income portfolio optimization would be equivalent to that of equity portfolios. Factor models like the DNS model have greatly simplified the computation of the expected return and covariance matrix. Together, the increase in volatility of the bond markets and the introduction of factor models encourage the use of mean-variance optimization techniques for fixed income portfolios. The mean-variance approach states how investors can maximize their returns and minimize their risks. The mean-variance optimization provides analytical solutions in a large class of models, restricting the investment constraints to be affine. The market considered consist of N bonds, with prices Pt at generic time t. The optimization provides an N-dimensional vector ω∗, the most suitable portfolio allocation for a given investor, which follows from the investor preferences and the information on the market.
12 Collecting information on the investor and the market
The information on the investor consist of knowledge of the investors’ current situation and the objective of the investor. The investors’ current situation can be summarized in a portfolio ω which corresponds to his wealth at the time the decision is made: 0 Wt = Pt x0. (2.3)
Note that at time t, the moment of optimizing the portfolio, the prices Pt, the initial amount of assets x0 and therefore the wealth of the investor are known. The objective of the investor is determined by his preferences. When using the mean-variance approach only the first two portfolio moment are considered in the optimization. This approach is justified if the individuals expected utility depend only on the mean and variance of the portfolio return. If the utility function is quadratic, which implies that all derivatives of order three and higher are equal to zero, it follows from a Taylor expansion that the expected utility given by
1 2 00 E[Rp(ω)] = U(E[Rp(ω)]) + E[(Rp(ω) − E[Rp(ω)]) ]U (E[Rp(ω)]) 2 (2.4) 1 = U( [R (ω)]) + [R (ω)]U 00( [R (ω)]). E p 2V p E p Therefore a quadratic utility function leads to an expected utility, which only de- pends on the first two portfolio moments, independent of the distribution of the portfolio returns. Recall that a power utility is assumed, which implies constant relative risk aversion. Levy and Markowitz (1979) showed that the mean-variance analysis can be regarded as an second order Taylor-series approximation of the standard utility functions, such as the power - and exponential utility function. Hence assuming the power utility justifies the use of the mean-variance optimiza- tion. In general the investor has multiple objectives, which depend on the alloca- tion ωt from (2.3). The assumption is made that the main objective of the investor is the return of the bond portfolio. Any objective of the investor is a linear func- tion of the allocations and of the market vector. In case of the main objective, the portfolio return is a linear combination of the returns of the N available bonds
13 over the investment horizon T of the investor.
0 R(ω) = RT ω, (2.5) where RT is an N-dimensional vector of returns of the available bonds. It is further assumed that the investors evaluates his net returns in terms of their value at risk, which means that the investor obtains a higher utility if the variance of his investment is smaller. Therefore the secondary objective of the investor is minimizing the variance of the portfolio, that is
0 S(ω) = −V ar(RT ω). (2.6)
The investors’ current portfolio, investment horizon, main objective and util- ity function are all information on the investor needed. To complete the informa- tion needed for the mean-variance optimization, information on the market is needed. The information needed from the market are the current prices of the N bonds and the future prices at the investment horizon T . The current prices are deterministic and known. The future prices PT are random variables and follow from the yield curve models. Combining the information from the investor and the market, allows to obtain the most suitable portfolio allocation, which is described in the next section.
Computing the optimal portfolio allocation
With the information on the investors and the market, the most suitable portfolio allocation can be found. Combining the the primary and secondary objective of the investor, the expected utility of each portfolio allocation can be approximated by
U(Rp(ω)) = F (E[Rp(ω)], V[Rp(ω)]) (2.7) The optimal portfolio allocation approach suggested by Markowitz is defined as:
ω(v) = argmax E[Rp(ω)], (2.8) V[Rp(ω)]=v
14 where v ≥ 0. The solution of the optimization in (2.8) is the (mean-variance) efficient frontier, which are all portfolio allocations that offer the highest expected return for a defined level of risk or the lowest risk for a given level of expected return. Given the efficient frontier the investor can obtain the most suitable portfolio allocation by selecting the allocation such that:
ω∗ = ω(v∗) = argmax U(ω(v)). (2.9) v≥0 This section described a two step procedure to obtain the most suitable portfolio allocation for a given investor. Note that only risky bonds were available, when there is also a risk free bond available additional allocations are available. The next section elaborates on the Sharpe ratio, which takes these additional allocations into account.
2.3.2 Sharpe Ratio optimization
The mean-variance approach in the previous section was a two step calculation. The first step was the computation of the efficient frontier and the second step was optimizing the investors’ utility given the efficient frontier. Note, that in the optimization only risky bonds were available. When considering a market with a risk free asset the set with possible portfolio allocations increases. The Sharpe Ratio optimization takes these additional allocations into account, which results in the capital market line. The allocations on the capital market line are a linear combination between the risk free rate and the optimal portfolio, which is the tangent of the capital market line and the efficient frontier. Adding a risk free bond shifts the objective from obtaining the efficient fron- tier to finding the optimal portfolio. This can be done by obtaining the tangent to the efficient frontier, that is finding the linear combination between the risk free rate and the set of feasible allocations with the highest slope. Following this procedure results in the Sharpe optimal portfolio
[Re(ω)] max S(ω) = E , (2.10) ω Sd(Re(ω))
15 where Re(ω) is the excess return relative to the risk-free (short) rate. The two components for portfolio selection are the expected return of the investment and the variance. For the returns of the bonds within the portfolio the simple holding-period returns are used that is the return of buying a (zero-) coupon bond with maturity τ at time t and selling this bond at time T , where T ≤ τ. Hence the holding-period return of a bond is given by P (T, τ − (T − t)) − P (t, τ) h(t, τ, T ) = , (2.11) P (t, τ) which shows that at time t, the moment of optimizing the portfolio allocation, P (t, τ) is deterministic term and P (T, τ − (T − t)) is a stochastic term. Therefore the expected return of a single bond is defined as P (T, τ − (T − t)) − P (t, τ) E[h(t, τ, T )|Ft] = E Ft P (t, τ) (2.12) [P (T, τ − (T − t))|F ] − P (t, τ) = E t . P (t, τ) The variance of the bond return is given by P (T, τ − (T − t)) − P (t, τ) V[h(t, τ, T )|Ft] = V Ft P (t, τ) (2.13) [P (T, τ − (T − t))|F ] = V t . (P (t, τ))2 Both (2.12) and (2.13) show that the distribution of the future price P (T, τ −(T − t)) is needed, which follows from a yield curve model. This section shortly introduced the mean-variance and Sharpe ratio optimiza- tion. To perform a Sharpe optimization both the expected return and covariances are needed, which follow from the forecast of the yield curve. The yield curve fore- cast depends on the yield curve model that is used. The next section introduces the DNS model, the yield curve model used in this thesis.
2.4 Yield curve model
To obtain a mean-variance optimized bond portfolio the expected bond returns and covariances are needed. The DNS model is used to fit and model the yield
16 curve. With these future yields (and their distribution) it is possible to obtain the distribution of the future prices and returns, needed for the mean variance optimization. The DNS model tries to explain the yield curve using three factors, which are level, slope and curvature. Once these factors are estimated based on historical rates, an auto-regressive (AR) structure is added to forecast them. Finally, a Monte Carlo simulation is used to obtain the distribution of the yield curve. The expected bond returns and covariances follow from the future yields and their distribution, obtained from the Monte Carlo simulation. The next chapter gives a detailed overview of bond pricing and the DNS - and AFNS model.
17 3. Theory
The proposed benchmark is a mean-variance optimized bond portfolio. To apply a mean-variance optimization the expected returns and covariances of the available bonds are needed, which follow from the distribution of the future yield curves. The distribution is obtained using the DNS- and AFNS model. In this thesis the empirically relevant assumption is made that the expectation hypothesis does not hold. The expectation hypothesis states that the expected holding-period returns on bonds of different maturities should be equal. However Engle et al. (1987) show that risk premia change systematically with the perceived uncertainty which lead deviations. Consequently, price dynamics under the risk neutral measure Q are different from price dynamics under the real measure P, which requires to know how to change the probability measure. This section elaborates on bond pricing, changing the probability measure and the yield curve models used in this thesis.
3.1 Pricing Bonds in continuous time
The affine term structure models following the work from Duffie and Kan (1996) all have closed form expressions for the price of zero coupon bonds. Zero coupon bonds pay a terminal notional at maturity date, often normalized to one, without intermediate coupon payments and disregarding default risk. A zero-coupon bond with maturity τ is currently traded at P (t, τ). Buying the bond at time t, holding it and selling it at time T the n-period holding-period return is given by
P (T, τ − (T − t)) − P (t, τ) h(t, τ, T ) = , (3.1) P (t, τ)
18 where n ≤ τ and T = t+n. When selling the zero-coupon bond before the maturity date the holding period excess return is usually random, because it depends on the unknown P (T, τ − (T − t)). At maturity the return of the bond is known. It follows that the price of a zero-coupon bond, and therefore the holding-period return, is a random variable until maturity, and is a deterministic quantity at maturity. This implies that the statistical properties of the price and return of bonds depend on their time to maturity. Therefore, the bond price and return are non-ergodic processes 1 and traditional statistical techniques do not apply (Meucci, 2009, p.110). A pricing problem involves conditioning on the current market data, through the fundamental theorem of asset pricing it is possible to price under an equivalent martingale measure or “risk-neutral-probability measure” Q. The fundamental theorem of asset pricing states that for a stochastic process, the existence of an equivalent martingale measure is essentially equivalent to the absence of arbitrage (Delbaen and Schachermayer, 1994). This allows pricing without knowing the exact risk preferences of the investors, in case of complete markets. Hence prices are future expected payoffs discounted at the risk free rate, where expectations are computed using the risk neutral measure Q. Usually the face value of a zero- coupon bond is normalized to one, hence the price is given by
h R t+τ i Q − t rudu P (t, τ) = Et e , (3.2) where rt is the instantaneous spot rate. Under the risk neutral probability measure the expected return on bonds is the risk free rate, which implies that the expected excess return is zero. However, in the case of interest rate risk prices are needed under the restriction of a certain exposure to the term structure of interest rates. This implies that the investors’ risk attitudes need to be considered, which requires the price dynamics under the probability measure P. Hence obtaining the expected bond prices consist of two steps. The first step is changing the probability measure P to Q. The second step 1A stochastic system is called ergodic if it tends in probability to a limiting form that is independent of the initial conditions, (Horst, 2007). Hence, non-ergodic implies path dependency, in our case time to maturity.
19 is determining the dynamics of the short rate r, which we will do with a factor model. That is, making r a function of a state vector x, and factor loadings. The assumption is made that the state vector x is a Markov process under Q. Doing so, it is possible to rewrite (3.2) as a function of these state vector and time to maturity, which leads to
P (t, τ) = f(xt, τ). (3.3)
When obtaining the evolution of the bond prices by rewriting (3.2), assumptions about the dynamics under the P- and Q measure are needed, which will be inves- tigated in the next section.
3.1.1 Change the probability measure for bond pricing
In this thesis the future bond prices are modelled with a DNS and an AFNS model. The DNS models the price dynamics directly under the P-measure but the AFNS model models these dynamics under the risk neutral measure Q. Since the empirically relevant assumption is made that the local expectation hypothesis2 does not hold, modelling with the AFNS model requires the change of probability measure. An additional advantage of pricing under the probability measure P, is that we have an intuition of the parameters, which we do not have under the risk neutral measure. This section elaborates on changing the probability measure required for the AFNS model. It is important to realize that under the risk-neutral measure the expected returns are always equal to the riskless rate that is
Q ∗ −r(t,T ) Et [h(t, τ, T )] = µf (x, τ) = e (3.4) where r(t, T ) is the yield at time t for maturity T and a function of the state variables x. The AFNS model assumes that, under the risk neutral measure Q, the state vector x solves
∗ ∗ ∗ dxt = µx(xt)dt + σx(xt)dzt (3.5)
2The Local Expectation Hypothesis states that the data generating measure P and the risk neutral measure Q coincide (Piazzesi, 2009).
20 ∗ where zt is a standard vector Brownian motion under the risk neutral measure Q. Now we can change the probability measure using Girsanov’s theorem3. Girsanov’s theorem states that, for a Brownian motion, an absolutely continuous change of measure is equivalent to change of drift. Note that changes of probability measure do not affect the variance on innovations of the state vector x. The dynamics under the P measure are obtained in four steps. First, (3.2) states that at maturity the bond price equals the payoff, which implies that f(x, 0) = 1 ∀x. Secondly, the exponential function within the expectation (3.2) implies a strictly positive price. Thirdly, Itˆo’slemma4 implies that f(x, τ) is also an Itˆoprocess, hence
df(xt, τ) ∗ ∗ ∗ = µf (xt, τ)dt + σf (xt, τ)dzt (3.6) f(xt, τ) with an instantaneous expected bond return
f˙ (x, τ) f 0(x, τ)> 1 f 00(x, τ) µ∗ (x , τ) = − τ + µ∗(x) + tr σ∗(x)σ∗(x)> , (3.7) f t f(x, τ) f(x, τ) x 2 x x f(x, τ)
˙ ∂f(x,τ) 0 ∂f(x,τ) 00 ∂2f(x,τ) where fτ (x, τ) = ∂τ , fτ (x, τ) = ∂x , fτ (x, τ) = ∂x∂x and tr denotes ∗ ∗ trace. The drift µx(x) and volatility σx(x) of the state vectors are still under the risk neutral measure. The fourth step, changing the measure, captures the risk adjustment of the future prices. This change of measure involves a strictly positive martingale ξ, which is a martingale if Novikov’s condition5 is satisfied and starts at ξ0 = 1. The differential equation is given by
dξt > = −σξt (xt) dt (3.8) ξt
∗ Again applying Girsanov’s theorem, we see that zt is a Brownian motion under Q, hence ∗ > dzt = dzt + σξ(xt) dt (3.9) 3Girsanov’s Theorem is stated in C.1 4Itˆo’slemma is stated in C.2 5 1 R T σ∗(x )σ∗(x )>du Novikov’s condition: E[e 2 0 ξ u ξ u ] < ∞, a more detailed overview is given by Duffie (2001).
21 ∗ Substituting this definition of zt 3.9 into 3.5 we find
∗ ∗ > ∗ dxt = (µx(xt) + σx(xt)σξ (xt))dt + σx(xt)dzt (3.10)
When looking at 3.10, we see that the volatility is unaffected and only the drift changes by the change in risk measure. This is known as the diffusion invariance principle. This section showed how to change the probability measure, which is needed for the AFNS model. Both the DNS- and the AFNS will be introduced in the remainder of this chapter.
3.2 Modelling yield curves and stylized facts
Portfolio selection with respect to interest rate risk involves measuring the expo- sure of one’s portfolio to adverse movement in the term structure of interest rates. Because the yield curves are not observed in practice, we have to estimate these from the (historical) bond prices. There are two different classes of term structure models to model these curves. The first class are affine term-structure models by building on the work of Vasicek (1977) and Cox et al. (1985). This class of mod- els works with the restriction that arbitrage opportunities are eliminated. These restrictions are appealing since bonds are traded in well-organized, highly liquid markets. These models have the advantage that the possess good tractability and a good economic foundation, however these models have difficulty capturing devia- tions from the expectation theory, see Bolder (2006). The second class, introduced by Diebold and Li (2003), works directly under the probability measure P. These models are basically a time-series description of the term structure and provides a better forecast than the affine models. A disadvantage of this approach is the lack of the theoretical model foundation. However recent work from Christensen et al. (2009) improved the theoretical foundation by imposing the arbitrage free restriction, which led to the AFNS model. A good model for the yield curve should be able to capture at least some of five stylized facts. First, the average yield curve is increasing and concave over
22 time. Secondly, the yield curve can take on a variety of shapes, for example up- and downward sloping, humped and S-shapes. Thirdly, yield dynamics are (very) persistent, which means that there are high correlations, in particular on short term. Further the short end of the curve is more volatile then the long end. And finally, yields for different maturities have high cross-correlations. The next section elaborates on the basis for the DNS and the AFNS model. Bolder (2006) gives a thorough derivation of the Nelson Siegel models, however for completeness this will partly be repeated in this thesis.
3.3 Nelson Siegel Term Structure Models
Recall that in the last section five stylized facts of the yield curve were stated. From these five stylized facts follows a typical yield curve, which Nelson and Siegel (1987) associated with solutions to differential or difference equations. This section introduces the Nelson-Siegel model and follows the work of Diebold and Li (2003) and Christensen et al. (2011) that result in the DNS - and AFNS model. The starting point in the Nelson Siegel models is the instantaneous forward rate, given as f(t, τ) = lim f(t, T, τ), (3.11) T →τ where f(t, T, τ) is the continuously compounded forward interest rate that is
1 P (t, τ) f(t, T, τ) = ln (3.12) T − τ P (t, T )
Substituting the continuous forward interest rate into 3.11 the instantaneous for-
23 ward rate can be obtained 1 P (t, τ) f(t, τ) = lim ln T →τ T − τ P (t, T ) ln P (t, τ) − ln P (t, T ) = lim T →τ T − τ ln P (t, τ) − ln P (t, T ) = lim T →τ T − τ ∂ (ln P (t, τ) − ln P (t, T )) (3.13) = lim ∂T T →τ ∂ ∂T (T − τ) P 0(t,T ) = lim P (t,T ) T →τ 1 P 0(t, τ) = − P (t, τ)
The fourth equation is obtained using L’Hˆopital’srule6. The instantaneous forward can be seen as the overnight interest rate, therefore it is possible to derive the yield curve as a function of the instantaneous forward curve P 0(t, τ) − = f(t, τ) P (t, τ) ∂ − (ln P (t, τ)) = f(t, τ) ∂τ ∂ − (ln e−y(t,τ)(τ−t)) = f(t, τ) ∂τ Z τ ∂ Z τ (3.14) (y(t, s)(s − t))ds = f(t, s)ds t ∂s t Z τ y(t, τ)(τ − t) − y(t, t)(t − t) = f(t, s)ds t 1 Z τ y(t, τ) = f(t, s)ds τ − t t It follows that the the zero-coupon yield is an equally-weighted average of forward rates. Nelson and Siegel (1987) proposed a functional form for f(t, τ) which results in a parsimonious representation of the yield curve. This form will be introduced in the next section.
0 6 f(x) f (x) 0 L’Hˆopital’srule: if limx→c g(x) = 0, +∞ or −∞, limx→c g0(x) exists and g (x) 6= 0 f(x) f 0(x) then limx→c g(x) = limx→c g0(x)
24 3.3.1 Nelson-Siegel model
The model originally suggested by Nelson and Siegel (1987) was a functional form for f(t, τ), which is given by
−λtτ −λtτ f(t, τ) = x0 + x1e + x2λtτe (3.15)
This is a parsimonious representation of a yield curve and does not depend on the expectations theory of term structure. Further it does not enforce the theoretically appealing condition of no arbitrage. From this functional form of the forward rate we can obtain a closed form solution of the corresponding yield curve by substituting this functional form into (3.14).
−λtτ −λtτ 1 − e 1 − e −λtτ y(t, τ) = x0,t + x1,t + − e x2,t (3.16) λtτ λtτ Where y(t, τ) is the zero coupon yield curve with τ denoting the time to maturity, and x0,t, x1,t, x2,t and λt are model parameters. Diebold and Li (2003) build on this model and made two important adjustments, which led to the DNS model. This model will be elaborated on in the next section.
3.3.2 Dynamic Nelson-Siegel model
The first adjustment that Diebold and Li (2003) made, was a clear interpretation of the factors. This interpretation can be derived from Figure 3.1. When investigating the zero-coupon yield curve in (3.16), one can see that the terms affect different tenors on the yield curve. The first loading on x0 is one, a constant, which does not decay to zero when t goes to τ. Diebold and Li (2006) interpret this loading as