Real Rate & Term Premium
For a capital market simulation of the multi-factor term structure or explicit corporate yield curve types, ProVal uses the parameters of this topic to develop the real rate of return component of the yield (or total return) on one year government bonds. Like the inflation rate, the real return rate is described as being mean-reverting (trending toward the long term expected rate) and serially correlated (affected by the prior year's value).
A real return rate is developed only for the 1 year government bonds in order to generate the spot curve (i.e., the yields of zero coupon bonds for durations from 1 to 30 years), and from it, the prices of zero coupon bonds. The yield for 30 year government bonds is developed from the zero coupon bond prices, not by adding a real return rate to inflation.
Nominal (or total) return rates for government bonds (specifically, a portfolio of thirty year government bonds at par) are developed from a series of equations utilizing the change in the yield curve, where the par value has been determined by evaluating the zero coupon bond price equations at each of the 60 payment dates applicable to a 30 year bond that pays coupons twice a year. One year and 30 year government bond returns are the “interest rate-dependent” building blocks of the multi-factor simulator: i.e., they are asset classes whose total return rates depend only on interest rates at bond coupon payment dates.
The parameters of this topic directly affect nominal returns for 1 and 30 year government bonds only. Nominal rates of return for the other asset classes are determined, in part from government bond nominal rates, by means of a regression equation.
The mathematical equation used to develop the current year 1-year government bond real rate, RR(t), is the following:
RR(t) = [ (RR(t-1) * w ] + [ LTRR * (1-w) ] + e
Where:
RR(t-1): Previous year real rate is the one year government bond annual real rate of return (excess of nominal rate over inflation rate) in year t-1. It is generally an empirical value, assuming continuous interest payments, but you can specify any rate between -0.02 and 0.2, inclusive, for the prior year’s real rate.
w: Lagged real rate coefficient indicates the relationship between the prior year’s real rate and the current year’s real rate. The value 1 – w is sometimes referred to as the speed of mean reversion. The expected real rate at time t is represented by w times the prior year’s real rate plus (1 - w) times the long-term expected real rate, i.e., the sum of the first two terms of the equation. You may specify any number from 0 to 0.9999, inclusive. Be mindful, however, that the lower the lag coefficient (and the higher the mean reversion speed) is, the lower the standard deviation of the real interest rate and the lower the yield curve and standard deviation of government bond returns.
LTRR: Long term expected real rate is your expected long-term, or equilibrium, one year government bond real rate of return. Enter the rate as a number between 0 and 0.2, inclusive.
sd(e): Standard deviation of real interest rate is the standard deviation of the one year government bond real rate of return and is used to generate the error term. This error term, e, is normally distributed. You may enter any number between 0 and 0.2, inclusive. Note that if a high standard deviation is chosen, negative one year government bond real rates may be produced for many points in time, t (although ProVal will limit the nominal return rates to zero), as may an inverted yield curve. Also, a higher standard deviation may result than is desirable for the government bond nominal return rates.
The other two parameters in this topic are:
c(e, ui): Correlation of unexpected real rate and unexpected inflation indicates the relationship between unexpected inflation and the unexpected real rate of return of one year government bond. This parameter is used in the simulator’s zero coupon bond price equations. To specify that the values move together in lockstep, enter the parameter as 1; to specify a total lack of correlation, code it as 0. Negative values indicate an inverse relationship. Any decimal number between +1 and -1, inclusive, may be used. Note that a small correlation produces a low yield curve and low standard deviation of 30 year government bond returns.
Expected term premium of 30-year government bonds over 1-year government bonds is the assumed excess of the long term yield of 30-year government bonds, for the entire time period of the simulation, over the yield of 1-year government bonds. This is a long term average, consistent with the liquidity theory—that a higher return is required for investments with a longer term. However, note that, while your simulated bond yields will exceed 1 year government bond rates on average by this amount, simulated yield curves will have many possible shapes including inversions. This parameter affects the simulation of yields and returns on 30-year government bonds. You may enter any number between 0 and 0.2, inclusive. Note that a high premium produces a high yield curve and high 30-year government bond returns. Decreasing the value of this parameter will decrease the value of the output variable “calculated term premium parameter” (also called “market price of risk”), whose value is displayed when Capital Market Simulation results are viewed by means of the View button of the Capital Market Simulation library entry (that is, this is available from the Execute menu, not from the Output menu or Output pane).