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# Interest Rate Simulation Crash Course

#### About the course

The course begins with a description of what models are and then lists features of a good model in general as well as specifically for an interest rate model. It then presents the criteria used for selecting a given model. This is followed by brief discussions of the two broad categories of term structure models, equilibrium and no-arbitrage, and the difference between one-factor and multifactor models. Basic concepts and terminology are introduced including how cash flows are discounted and the differences between various types of interest rates (e.g. spot, forward, YTM, short). A step by step walkthrough of how to derive spot rates and forward rates from a given par term structure using bootstrapping methodologies follows.

Next the course moves to an introduction to Monte Carlo (MC) simulators, in particular is discusses the various elements of the simulator, e.g. drift, diffusion, and presents a guide on how to build a MC simulator for stock prices in EXCEL. This model is easily extended to commodities and currencies but there are limitations to its application for interest rates which are highlighted. This discussion then leads to models designed specifically for generating interest rates. In particular the Cox Ingersoll Ross (CIR) one factor equilibrium model, Black Derman Toy (BDT) one factor no arbitrage model and Heath Jarrow Merton (HJM) multifactor no arbitrage model are presented in detailed.

For the CIR model, the course discusses the continuous representation of the interest rate process and uses the least squares methodology with either a simple discretisation or covariance equivalent discretisation process to estimate and calibrate the parameters of the model. Future short term rates are simulated from the CIR model process and the estimated parameters. Longer term rates are modelled based on the perfect correlation assumption of the one-factor model.

The course then moves on to the BDT model. This includes an overview of the basic features of a Black-Derman-Toy (BDT) one factor interest rate model as well as some of the simplifying assumptions made in its construction. It presents a detail guide that on how to construct and use the model that includes:

- Defining the input cells (including the initial zero yield curve & volatility term structures)
- Defining the calculation cells (including the state price lattices, short rate tree, prices, yields and yield volatilities)
- Defining the output cells (including mediate rates and volatilities)
- Linking all the pieces together with the Solver Function of the EXCEL worksheet
- Running the Solver Function to get the results, i.e. median rates and their time varying volatilities (sigmas)
- Procedure for generating the complete binomial interest rate tree for short rates from the median rates and their volatilities.
- Procedure for calculating the price of bonds and options on bonds using the short rate tree for determining and discounting cash flows/ payoffs of these instruments.

The HJM multifactor no arbitrage model is discussed next. This includes:

- The use of Principal Component Analysis (PCA) to determine the number of workable factors to explain volatility for the HJM interest rate model. In the course we focus on a three factor model. The PCA process consists of:
- Data selection from a given dataset
- Construction of the Covariance, eigenvector and diagonal matrices
- Setting up and running of Solver functionality to obtain solutions for eigenvectors and eigenvalues
- Determination of number of component/ factors to be used in the HJM model
- Determination of functional forms for selected eigenvectors
- Determination of weights for functional forms through derived volatility calibration

- An example of the construction of a three factor HJM interest rate model in EXCEL that shows how the following are determined:
- Brownian Shocks using a normally scaled random number generator
- Drifts using the principal component factors for volatility of the forward rate term structure
- Spot & forward rates using the initial zero rate yield rates, principal component factors and total drift
- Price/ discount factors using the derived spot & forward rates
- True & Path bond prices

The course ends with a real world application of a Monte Carlo simulator used for forecasting a country’s monetary policy rate decision.

#### Learning Objectives

After taking this course you will be able to:

- List the features of a good model
- Distinguish between equilibrium and no arbitrage models
- Describe one factor and multifactor models
- Describe the different types of interest rates
- Bootstrap zero and forward term structures from par term structures
- Build a Monte Carlo simulator in EXCEL for stock prices, commodities and currrencies
- Explain why this this simulator cannot be extended to interest rates
- Use discrete interest rate data with simple or covariance equivalent discretisation representation to estimate the parameters of the continuous CIR model
- Apply the least squares method to calibrate the model to a given historical interest rates data set
- Simulate the future short term rates from the CIR model process and the estimated parameters
- Derive longer term rates from the simulated short term rates and the perfect correlation assumption of the one factor model
- Describe the basic features of the BDT interest rate model
- List the simplifying assumptions used in the construction of the model
- Build a BDT model in EXCEL
- Derive results for the model, i.e. the median rates and time varying volatilities by setting and running the Solver functionality in EXCEL
- Build the short rates tree from the outputs of the BDT model (median rates, volatilities and calculated up-movements)
- Calculate the price of fixed income bonds using the derived short rate tree
- Calculate the price of options using the derived short rate
- Compute the principal components (eigenvectors) and their relative importance factors (eigenvalues) for the underlying term structure using EXCEL’s matrix multiplication, inverse matrix & Solver functionalities
- Determine functions for the selected eigenvectors/ principal components
- Calculate the scaling factor for each component function’s so that the combined volatilities derived from these functions match the volatilities inherent in the underlying term structure
- Calculate Brownian shocks
- Calculate the principal component factor based drifts
- Construct the HJM interest rate model comprising of a spot and forwards rates tree
- Calculate the discount factor or one-period price tree from the HJM interest rate model
- Calculate the value of a bond using the price tree derived above
- Use a Monte Carlo simulator to forecast the monetary policy rate

#### Prerequisites

Familiarity with EXCEL, mathematics, probability, regression/ curve fitting analysis. Some knowledge of bond and derivative markets.

#### Target Audience

The course is aimed at individuals responsible for the pricing of money market, derivatives and structured products as well as those involved in asset liability management and risk management, including the simulation and stress testing of rate sensitive asset and liability portfolios within banks, insurance companies and mutual funds.