# Data Summary – Scenario Generation

This blog  entry describes the model that was used in the generation of the various sets of scenarios from 1992 through 1994.

The generation of the yield curves used in the interest rate scenarios is not arbitrage free. This would require setting up a diffusion process of state variables and making sure that the various par bond prices are consistent with the resultant bond pricing partial differential equation. Instead, we used a two-factor model with a log-normal diffusion process on the short-rate (ninety-day) and a log-normal diffusion process on the long-rate (ten-year). This model does not have mean reversion and has fixed boundaries above and below. These fixed boundaries are not reflecting.

Below we use the notation $Y^m_t$, where $m$ denotes the maturity of the interest rate on the yield curve and $t$ denotes the time epoch. The only exception of this notation is that we use $Y^{90}_{t}$ to denote the value of the ninety-day rate instead of $Y^{.25}_{t}$.  Note that $m = \{1, \ldots , 20\}$.

First obtain the initial yield curve required, which will be from the last yield curve from the U. S. Treasury for the projection period.  These are from 1992, 1993 and 1994 . Set $Y^{5}_{0}$ to be U. S. constant  maturity Treasury five-year interest rate for the last day of the year and calculate the ninety-day rate to be $Y^{90}_{0} = Y^{5}_{0}\,\exp(\mu_{90})$ where $\mu_{90}$ and $\sigma_{90}$ and $\sigma_{10}$ are based on a historical log-normal analysis of the short and long rates.  In the following formulas, we assume below that $\mu_{10}$ is zero.

With maturity $m$, we use the log regression formula $N(m)=1.349\,\log(2m+1) + 1.051\,\log(m+1)$ to assure a “nice” positive or inverted yield curve. This formula precludes the possibility of humped yield curves.

Define the spread slope constant $C = (Y^{5}_{0}-Y^{90}_{0})/\,N(5).$ Letting $m$ range from one to twenty, we obtain the entire initial yield curve from $Y^m_{0} = Y^{90}_{0}+C\,N(m).$

For time $t>0$, the subsequent yield curves are based on a lognormal diffusion processes of the ten-year rate and the ninety-day rate as follows. The ten-year rate is projected with this formula $Y^{10}_{t+1} = Y^{10}_{t}\,\exp(\sigma_{10}\,Z_{10}).$

The ninety-day rate is projected as $Y^{90}_{t+1} = Y^{90}_{t}\,\exp({\mu_{90}\,+\sigma_{90}\,Z_{90}})$ where $Z_{90}$ and $Z_{10}$ are uncorrelated standard normal samples.

These values are then bracketed. The ninety-day brackets are 0.5% and 20%  and the brackets of the ten-rates are 1% and 25%.

However, in the belief that inverted yield curves are only observed in a rising interest rate environment, if the yield curve is inverted and the rates are falling (measured by the fact that $Y^{90}_{t+1}>Y^{10}_{t+1}$ and $Y^{10}_{t+1} < Y^{10}_{t}$) then the $Y^{90}_{t+1}$ is adjusted to be $Y^{90}_{t+1}=Y^{10}_{t+1}e^{\mu_{90}}$

This new value of $Y^{90}_{t+1}$ is then bracketed as before.

Now define the spread slope constant $C = (Y^{10}_{t+1}-Y^{90}_{t+1})/N(10)$  and obtain the entire yield curve by interpolating by this formula $Ym_{t+1} = Y^{90}_{t+1}+C\,N(m).$