Predicting Interest Rates Statistical Models Economic vs. Statistical Models Economic models are designed to match correlations between interest rates and other economic aggregate variables Pro: Economic (structural) models use all the latest information available to predict interest rate movements Con: They require a lot of data, the equation can be quite complex, and over longer time periods are very inaccurate Economic vs. Statistical Models Statistical models are designed to match the dynamics of interest rates and the yield curve using past behavior. Pro: Statistical Models require very little data and are generally easy to calculate Con: Statistical models rely entirely on the past. They don’t incorporate new information. The Yield Curve 6 4.3 4 2 5.07 2.12 2.61 2.94 1 yr 2 yr 5yr 10 yr 20yr S(1) S(2) S(5) S(10) S(20) 0 Recall that the yield curve is a collection of current spot rates Forward Rates Forward rates are interest rates for contracts to be written in the future. (F) F(1,1) = Interest rate on 1 year loans contracted 1 year from now F(1,2) = Interest rate on 2 yr loans contracted 2 years from now F(2,1) = interest rate on 1 year loans contracted 2 years from now S(1) = F(0,1) Spot/Forward Rates S(3) Spot Rates S(2) S(1) Now F(0,1) 1yr F(1,1) F(0,2) 2yrs F(1,3) 4yrs 5yrs F(2,1) F(2,2) F(1,2) 3yrs Forward Rates Calculating Forward Rates Forward rates are not observed, but are implied in the yield curve Suppose the current annual yield on a 2 yr Treasury is 2.61% while a 1 yr Treasury pays an annual rate of 2.12% 2.61%/yr S(2) S(1) Now 2.12%/yr 1yr F(1,1) 2yrs 3yrs 4yrs 5yrs Calculating Forward Rates 2.61%/yr S(2) S(1) Now 2.12%/yr 1yr 2yrs 3yrs 4yrs 5yrs F(1,1) Strategy #1: Invest $1 in a two year Treasury $1(1.0261)(1.0261) = 1.053 (5.3%) For these strategies, to pay the same return, the one year forward rate would need to be 3.1% Strategy #2: Invest $1 in a 1 year Treasury and then reinvest in 1 year $1(1.0212)(1 + F(1,1)) $1(1.0261)(1.0261) = $1(1.0261)(1+F(1,1) 1+F(1,1) = $1(1.0261)(1.0261) $1(1.0212) =1.031 Calculating Spot Rates We can also do this in reverse. If we knew the path for forward rates, we can calculate the spot rates: ??? S(3) ??? S(2) S(1) Now 2% 2% 1yr 3.3% 2yrs 2.9% 3yrs 4yrs 5yrs Calculating Spot Rates ??? S(2) Now 1yr 2% 2yrs 4yrs 5yrs 3.3% Strategy #1: Invest $1 in a two year Treasury $1(1+(S(2))(1+S(2)) For these strategies, to pay the same return, the two year spot rate would need to be 2.6% 3yrs Strategy #1: Invest $1 in a 1 year Treasury and then reinvest in 1 year $1(1.02)(1.033) = 1.054 (5.4%) 2 $1(1.02)(1.033) = $1(1+S(2)) 1/2 1+S(2) = ((1.02)(1.033)) =1.026 Arithmetic vs. Geometric Averages 2.6% S(2) Now 2% 1yr 2yrs 3yrs 4yrs 5yrs 3.3% In the previous example, we calculated the Geometric Average of expected forward rates to get the current spot rate 1/2 1+S(2) = ((1.02)(1.033)) =1.026 (2.6%) The Arithmetic Average is generally a good approximation S(2) = 2% + 3.3% 2 = 2.65% Spot rates are equal to the averages of the corresponding forward rates (expectations hypothesis) 2.73% S(3) 2.65% S(2) S(1) Now 2% S(2) = 2% 1yr 3.3% 2% + 3.3% 2 2yrs 3yrs 4yrs 5yrs 2.9% = 2.65% S(3) = 2% + 3.3% + 2.9% 3 = 2.73% However, the expectations hypothesis assumes that investing in long term bonds is an equivalent strategy to investing in short term bonds This rate is “locked in” at time 0 2.65% S(2) Now 2% 1yr 2yrs 3yrs 4yrs 5yrs 3.3% This rate is flexible at time 0 Long term bondholders should be compensated for inflexibility of their portfolios by adding a “liquidity premium” to longer term rates (preferred habitat hypothesis) Statistical Models Now F(0,1) 1yr F(1,1) 2yrs F(2,1) 3yrs F(3,1) 4yrs 5yrs 3.3% F(4,1) First, write down a model to explain movements in the forward rates Then, calculate the yield curve implied by the forward rates. Does it look like the actual yield curve? S(3) S(2) S(1) Now 1yr 2yrs 3yrs 4yrs 5yrs Lattice Methods (Discrete) Lattice models assume that the interest rate makes discrete jumps between time periods (usually calibrated monthly) Binomial: Two Possibilities each Period Trinomial: Three Possibilities each Period An Example At time zero, the interest rate 5%: F(0,1) = S(1) An Example In the first year, the interest rate has a 50% chance of rising to 5.7% or falling to 4.8%: F(1,1) An Example In the second year, there is also a 50% chance of rising or falling conditional on what happened the previous year: F(2,1) Calculating the Yield Curve S(1) 5.7% Path 1: (1.05)(1.057) = 1.10985 (10.985%) 5% 4.8% Path 2: (1.05)(1.048) = 1.10040 (10.04%) Expected two year = (.5)(1.10985) + (.5)(1.10040) = 1.105125 (10.5125%) cumulative return 1/2 Annualized Return = (1.105125) = 1.0512 (5.12%) = S(2) 6.4% Path 1: (1.05)(1.057)(1.064) = 1.181 (18.1%) 5.7% Path 2: (1.05)(1.057)(1.052) = 1.168 (16.8%) 5% 5.2% Path 3: (1.05)(1.048)(1.052) = 1.157 (15.7%) 4.8% 4.6% Path 4: (1.05)(1.048)(1.046) = 1.151 (15.1%) Expected three year = (.25)(1.181) + (.25)(1.168) + (.25)(1.157) +(.25)(1.151) = 1.164 cumulative return 1/3 Annualized Return = (1.164) = 1.0519 (5.19%) = S(3) Future Yield Curves 6.4% 5.7% 5% 5.2% Path 1: (1.048)(1.052) = 1.1025 (10.25%) 4.8% Suppose that next months interest rate turns out to be 4.8% = S(1)’ 4.6% Path 2: (1.048)(1.046) = 1.096 (9.6%) (.5)(1.1025) + (.5)(1.096) = 1.0993(9.3%) 1/2 S(2)’ = (1.099) = 1.049 (4.9%) Volatility & Term Structure A common form for a binomial tree is as follows: 1 it with probabilit y .5 it 1 it with probabilit y .5 (1 ) Sigma is measuring volatility Higher volatility raises the probability of very large or very small future interest rates. This will be reflected in a steeper yield curve 8.3 8.25 8.2 8.15 8.1 High Sigma Low Sigma 8.05 8 7.95 7.9 7.85 1 2 3 4 5 6 7 8 9 10 11 12 Continuous Time Models dit ait , t dt it , t dz Change in the interest rate at time ‘t’ Deterministic (Non-Random) component Random Error term with N(0,1) distribution Random component Vasicek The Vasicek model is a particularly simple form: dit it dt dz Controls Persistence Controls Variance Controls Mean Using the Vasicek Model Choose parameter values Choose a starting value Generate a set of random numbers with mean 0 and variance 1 dit .26 it dt 2dz i0 6% t=0 6% t=1 6.8% t=2 6.84% t=3 4.202% 0 -.16 -.168 .3596 dz .4 .2 -1.1 .5 di .8 .04 -2.368 1.3596 i .2(6-i) t=4 5.5616% -.9 Vasicek (sigma = 2, kappa = .17) 96 88 80 72 64 56 48 40 32 24 16 Path1 Path2 Path 3 Path 4 Path 5 Average 8 0 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 Vasicek (sigma = 4, kappa = .17 ) 0.3 0.25 0.2 Path1 Path2 Path 3 Path 4 Path 5 Average 0.15 0.1 0.05 -0.1 -0.15 96 88 80 72 64 56 48 40 32 24 0 -0.05 8 16 0 Vasicek (sigma = 2, kappa = .4) 0.16 0.14 0.12 Path1 Path2 Path 3 Path 4 Path 5 Average 0.1 0.08 0.06 0.04 0.02 96 88 80 72 64 56 48 40 32 24 16 8 0 0 Cox, Ingersoll, Ross (CIR) The CIR framework allows for volatility that depends on the current level of the interest rate (higher volatilities are associated with higher rates) drt r dt r dz dit it dt i dz Heath,Jarow,Morton (HJM) Vasicek and CIR assume a process for a single forward rate and then use that to construct the yield curve In this framework, the correlation between different interest rates of different maturities in automatically one (as is the case with any one factor model) HJM actually model the evolution of the entire array of forward rates df t, T at, T , f (t, T )dt t, T , f (t, T )dz Change it the forward rate of maturity T ant time t Table 1 Summary Statistics for Historical Rates Shape Inverted Humped 11.6% 13.4% Normal 68.8% Mean S.D. Skewness Exc. Kurtosis 1% 5% 50% 95% 99% Yield Statistics 1 yr. 3 yr. 6.08 6.47 3.01 2.88 0.97 0.84 1.10 0.69 1 yr. 1.07 2.05 5.61 12.08 15.17 Percent iles 3 yr. 1.59 2.52 6.20 12.48 14.69 Other 6.3% 5 yr. 6.64 2.84 0.77 0.48 5 yr. 1.94 2.72 6.44 12.59 14.59 10 yr. 6.81 2.81 0.68 0.16 10 yr. 2.38 2.90 6.68 12.56 14.29 Corr (1 yr,10 yr) = 0.944 Tables 1-4 from Ahlgrim, D’Arcy, and Gorvett, CAS 1999 DFA Call Paper Program Table 2 Summary Statistics for Vasicek Model Normal 41.6% Mean S.D. Skewness Exc. Kurtosis 1% 5% 50% 95% 99% Shape Inverted Humped 54.8% 3.6% Yield Statistics 1 yr. 3 yr. 8.81 8.75 3.83 3.24 -0.16 -0.16 -0.19 -0.19 1 yr. -0.38 2.33 8.94 14.69 17.22 Percent iles 3 yr. 0.97 3.27 8.86 13.73 15.87 Other 0.0% 5 yr. 8.68 2.77 -0.16 -0.19 5 yr. 2.04 4.00 8.77 12.94 14.76 10 yr. 8.52 1.95 -0.16 -0.19 10 yr. 3.84 5.22 8.59 11.53 12.82 Corr (1 yr,10 yr) = 1.000 Notes: Number of simulations = 10,000, = 0.1779, = 0.0866, = 0.0200 Table 3 Summary Statistics for CIR Model Shape Normal Inverted Humped Other 47.7% 47.6% 4.7% 0.0% Mean S.D. Skewness Exc. Kurtosis 1% 5% 50% 95% 99% Yield Statistics 1 yr. 3 yr. 8.08 8.04 2.89 2.31 0.92 0.92 1.49 1.49 5 yr. 7.98 1.88 0.92 1.49 Percentiles 3 yr. 3.90 4.73 7.73 12.31 15.33 5 yr. 4.62 5.29 7.73 11.45 13.90 1 yr. 2.92 3.95 7.71 13.42 17.19 10 yr. 7.86 1.20 0.92 1.49 10 yr. 5.71 6.14 7.70 10.09 11.66 Corr (1 yr,10 yr) = 1.000 Notes: Number of simulations = 10,000, = 0.2339, = 0.0808, = 0.0854 Table 4 Summary Statistics for HJM Model Mean S.D. Skewness Exc. Kurtosis 1% 5% 50% 95% 99% Yield Statistics 1 yr. 3 yr. 7.39 7.51 2.26 2.27 0.51 0.53 -0.88 -0.85 1 yr. 4.45 4.79 7.48 11.57 12.09 Percentiles 3 yr. 4.48 4.85 7.58 11.74 12.26 5 yr. 7.60 2.31 0.54 -0.85 5 yr. 4.52 4.90 7.65 11.92 12.44 Corr (1 yr,10 yr) = 0.999 Notes: Number of simulations = 100 10 yr. 7.80 2.44 0.54 -0.86 10 yr. 4.59 4.99 7.83 12.38 12.89