Title LM06 Simulation Methods IFT Notes.pdf ✅

LM06 Simulation Methods 2025 Level I Notes © IFT. All rights reserved 1 LM06 Simulation Methods 1. Introduction...........................................................................................................................................................2 2. Lognormal Distribution and Continuous Compounding......................................................................2 3. Monte Carlo Simulation.....................................................................................................................................4 4. Bootstrapping .......................................................................................................................................................4 Summary......................................................................................................................................................................6 Required disclaimer: IFT is a CFA Institute Prep Provider. Only CFA Institute Prep Providers are permitted to make use of CFA Institute copyrighted materials which are the building blocks of the exam. We are also required to create / use updated materials every year and this is validated by CFA Institute. Our products and services substantially cover the relevant curriculum and exam and this is validated by CFA Institute. In our advertising, any statement about the numbers of questions in our products and services relates to unique, original, proprietary questions. CFA Institute Prep Providers are forbidden from including CFA Institute official mock exam questions or any questions other than the end of reading questions within their products and services. CFA Institute does not endorse, promote, review or warrant the accuracy or quality of the product and services offered by IFT. CFA Institute®, CFA® and “Chartered Financial Analyst®” are trademarks owned by CFA Institute. © Copyright CFA Institute Version 1.0
LM06 Simulation Methods 2025 Level I Notes © IFT. All rights reserved 2 1. Introduction This learning module covers: • Lognormal distribution and continuously compounded asset return • Monte Carlo simulation • Bootstrapping 2. Lognormal Distribution and Continuous Compounding The Lognormal Distribution If x is a random variable that is normally distributed, then to create a lognormal distribution of x we take ex and plot the values on a graph. Normal Distribution Lognormal Distribution Instructor’s Note: It is called a ‘lognormal’ distribution because ‘the log is normal’. The properties of a lognormal distribution are: • It cannot be negative. It is bounded to the left by ‘0’. • The upper end of its range extends to infinity. • It is skewed to the right i.e. it has a long right tail. Both normal and lognormal distributions are important for finance professionals. A normal distribution is typically used to model asset returns, because returns generally vary about the mean, with a high probability of returns being close to the mean. Returns can also sometimes be negative. However, asset prices do not vary equally about a mean price, since the probability of extreme changes in price decreases as the price approaches zero. This means asset prices will not form a symmetrical graph like that of the normal distribution. Instead, asset prices follow a lognormal distribution, which is skewed to the right and cannot be negative. Continuously Compounded Rates of Return The continuously compounded rate of return can be calculated as the natural logarithm of the ending price over the beginning price.
LM06 Simulation Methods 2025 Level I Notes © IFT. All rights reserved 3 Rc = ln (Pt/P0) A property of the continuously compounded rate of return is that they are additive. The continuously compounded return from period 0 to period T is the sum of the incremental one-period returns between 0 and T. r0,T = rT−1,T + rT−2,T−1 + ⋯ + r0,1 This is illustrated in the below example. Example: (This is based on Practice Problem 1 from the curriculum.) The weekly closing prices of a share are as follows: Date Closing Price 1 August 112 8 August 160 15 August 120 Calculate the continuously compounded return of this share for the period August 1 to August 15. Solution: The continuously compounded return of an asset over a period is equal to the natural log of the asset’s price change during the period. In this case, ln(120/112) = 6.90%. The continuously compounded return from period 0 to period T is also equal to the sum of the incremental one-period continuously compounded returns, which in this case are weekly returns. Week 1 return: ln(160/112) = 35.67%. Week 2 return: ln(120/160) = –28.77%. Continuously compounded return = 35.67% + –28.77% = 6.90% Continuously compounded returns are used in many asset pricing models, as well as in risk management. Many investment applications make the assumption that returns are independently and identically distributed (i.i.d.). The concept of independence captures the idea that investors cannot predict future returns based on past returns. The concept of identical distribution captures the assumption of stationarity, which implies that the mean and variance of return do not change form period to period. Calculating Volatility The volatility (measured in terms of standard deviation) of the continuously compounded returns on an asset is typically stated as an annualized number. However, in practice volatility is often estimated using a historical series of continuously compounded daily
LM06 Simulation Methods 2025 Level I Notes © IFT. All rights reserved 4 return. To convert daily volatility into annual volatility, we annualize based on 250 days in a year (the approximate number of business days that financial markets are open for trading). For example, if daily volatility is 0.01, the annual volatility will be 0.01√250 = 0.1581 3. Monte Carlo Simulation Monte Carlo simulation is a computer simulation used to simulate possible security prices based on risk factors. As input, it uses randomly generated values for risk factors based on their assumed probability distributions. It processes this information as per the specified model and runs thousands of iterations. Then it gives the distribution of the expected value of the security as output. Monte Carlo simulation is widely used to estimate risk and return in investment applications. It is also used as a tool for valuing complex securities with embedded options where no analytic pricing formula is available (e.g. Mortgage backed securities). A limitation of Monte Carlos simulation is that it is fairly complex and will provide answers that are no better than the assumptions. Also, simulation is not an analytical method but a statistical one. It cannot provide more insight into cause-and-effect relationships like analytical methods. Exhibit 5 from the curriculum shows the steps for implementing a Monte Carlo simulation for valuing a contingent claim (an option). 4. Bootstrapping In an earlier lesson, we learnt how to find the standard error of the sample mean based on the central limit theorem. We will now cover ‘resampling’: a computational tool in which we