#### Risk Premia And Volatilities In A Nonlinear Term Structure Model by SSRN

London Business School

London Business School – Department of Finance

University of Pennsylvania – Finance Department

March 24, 2015

**Abstract: **

We introduce a reduced form term structure model with closed form solutions for yields where the short rate and market prices of risk are nonlinear functions of Gaussian state variables. The nonlinear model with three Gaussian factors matches both the time-variation in expected excess returns and yield volatilities of U.S. Treasury bonds from 1961 to 2014. Yields depend on all three factors, yet the model exhibits features consistent with unspanned risk premia (URP) and unspanned stochastic volatility (USV). The probability of a high volatility scenario increases with the monetary experiment and remains high during the Greenspan area, even though volatilities came back down to normal levels.

### Risk Premia And Volatilities In A Nonlinear Term Structure Model – Introduction

Investments in U.S. Treasury bonds require an assessment of both risk and reward, but existing term structure models find it difficult to simultaneously capture the time series variation of conditional first and second moments of yields.1 We therefore introduce an arbitrage free dynamic term structure model where the short rate and market prices of risk are nonlinear functions of Gaussian state variables. We provide closed-form solutions for bond prices and since the factors are Gaussian our nonlinear model is as tractable as a standard Gaussian model.

We use a monthly panel of five zero-coupon Treasury bond yields and their realized volatilities from 1961 to 2014 to estimate the nonlinear model with three factors. To compare the implications of the nonlinear model with those from the standard class of affine models, we also estimate a three-factor affine model with one stochastic volatility factor, the essentially affine A1(3) model.

To assess the ability of the nonlinear model to predict excess bond returns, we regress realized excess returns on model-implied expected excess return. The average R2 across bond maturities and holding horizons is 34% for the nonlinear model and only 7% for the A1(3) model. Moreover, the average R2 from regressing realized excess returns onto the five observed yields and realized variances is 17% and, hence, no affine model in which expected excess returns are linear functions of factors driving yields and yield volatilities can generate R2’s as high as those in the nonlinear model.

The first three principal components (PCs) of U.S Treasury yields | the level, slope, and curvature | explain almost all the cross-sectional variation in yields and, therefore, three factors have traditionally been used in the reduced form term structure literature to model bond yields.2 However, Cochrane and Piazzesi (2005) show that the fourth and fifth PC help predict excess returns in a linear regression. Furthermore, there is evidence that a component of expected excess returns is not explained by any linear combinations of yields; a phenomenon we refer to as Unspanned Risk Premia (URP).3 We regress expected excess returns implied by the nonlinear model on its implied PCs of yields and find that the first three PCs explain 68% to 74% of the variation in expected excess returns. The explanatory power increases to approximately 90% when we include the fourth and fifth PC. Hence, our nonlinear model can retain the parsimonious three-factor structure to price bonds and yet generate predictions consistent with the literature on URP. Duffee (2011b) and Joslin, Priebsch, and Singleton (2014) use five-factor models and consider hidden” factors that do not appear when pricing bonds but help explain the time variation in expected excess returns. In contrast, Unspanned Risk Premia arises due to a nonlinear relation between expected excess returns and yields in our model.

The nonlinear and A1(3) model fit realized yield volatilities well and thus they can capture the persistent time variation in volatilities and the high volatility during the monetary experiment in the early eighties. However, the two models have different predictions for future volatility. Specifically, in the nonlinear model the probability of a high volatility scenario increases with the monetary experiment and remains high during the Greenspan area even though volatilities came down significantly. This finding resembles the appearance and persistence of the equity option smile since the crash of 1987. In contrast, the distribution of future volatility in the A1(3) model is similar before and after the monetary experiment.

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