M11 Lognormal Models for Investment Rates
Topics in Insurance, Risk, and Finance
Learning outcomes
- Know and derive the properties of lognormal distributions.
- Derive explicit formulas for the distributions and related quantities of \(S_n\) and \(V_n\) in the lognormal model.
- Understand the unique role of the lognormal distribution for modelling rates of return in varying rate models.
- Define, calculate, and interpret Value-at-Risk.
Review: normal distribution
Lognormal model
- We have studied the moments of cash series.
- To have a more detailed analysis of investment activities (e.g., probability of default), we need to derive distribution functions.
- In particular, distributions can be used to determine the capital requirements for financial institutions (Value-at-Risk for insurance companies and Expected Shortfall for banks).
- The task of finding distributions is however very challenging.
- We will study the case when the accumulation factor \((1+i_t)\) follows a lognormal distribution, in which case we can derive the exact distribution of \(S_n\).
Review: normal distribution I
We say a random variable \(X\) follows a normal distribution with parameters \(\mu\) and \(\sigma^2\), denoted by \(X\sim N(\mu,\sigma^2)\), if the density function of \(X\) can be written as \[f(x)=\frac{1}{\sqrt{2\pi}\sigma}\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right).\]
- \(\mu\) and \(\sigma^2\) are the mean and variance of \(X\).
- If \(\mu=0\) and \(\sigma=1\), \(X\) is usually denoted by \(Z\) and is called a standard normal random variable.
- The distribution function and the density function of \(Z\) are usually denoted by \(\Phi\) and \(\phi\), and are called the standard normal distribution function/density function.
Review: normal distribution II
- Symmetry: \(\Phi(x)+\Phi(-x)=1\) for \(x\in\R\).
- Standardization: For \(X\sim N(\mu,\sigma^2)\), we have \[\frac{X-\mu}{\sigma}\sim N(0,1).\]
- Moment generating function: Let \(X\sim N(\mu,\sigma^2)\). For \(t\in \mathbb R\), \[\E[\exp(tX)]=\exp(\mu t +t^2\sigma^2/2).\]
- If \(X_1\sim N(\mu_1,\sigma_1^2)\) and \(X_2\sim N(\mu_2,\sigma_2^2)\) are independent, \[c_1X_1+c_2X_2\sim N(c_1\mu_1+c_2\mu_2,c_1^2\sigma_1^2+c_2^2\sigma_2^2).\]
Question: If \(X_1\sim N(\mu_1,\sigma_1^2)\) and \(X_2\sim N(\mu_2,\sigma_2^2)\), does \(X_1+X_2\) always follow the normal distribution?
Review: normal distribution III
Review: normal distribution example
For \(X\sim N(5,5^2)\), what is \(\p(X\le 6)\)?
Lognormal distribution
Lognormal distribution: definition
Let \(Z\) be a standard normal random variable, i.e., \(Z\sim N(0,1)\). Let \(\mu\in\mathbb{R}\) and \(\sigma>0\). A lognormal random variable can be written as \[X=\exp(\mu+\sigma Z).\]
- We write \(X\sim LN(\mu,\sigma^2)\).
- Note that \(\mu\) and \(\sigma^2\) here are not the mean and variance of \(X\).
- The logarithm of \(X\) is normally distributed (with mean \(\mu\) and variance \(\sigma^2\)), and hence the name.
- Equivalently, we can write \[X=\exp(Y)\] where \(Y\sim N(\mu,\sigma^2)\).
Lognormal distribution: density
Proof:
Lognormal distribution: moments
Proof:
Lognormal distribution: product
Proof:
Lognormal distribution: reciprocal
Proof:
Lognormal models: \(S_n\) and \(V_n\)
Lognormal model
- Due the properties of lognormal distributions, it is convenient to model the accumulation factors (i.e., \(1+i_t\)) as lognormal distributions.
- We can derive distributions of \(S_n\) and \(V_n\) using lognormal models, which turn out to be lognormal distributions.
- Even if the accumulation factor is not lognormally distributed, we can still use lognormal distributions to approximate the distribution of \(S_n\) and \(V_n\).
- However, distributions of \(A_n\) and \(P_n\) are still not available.
Lognormal model: \(S_n\) and \(V_n\) in a varying rate model
Question: What if the rates are not identically distributed? Any criticism on the lognormal assumption?
Proof:
Lognormal model: \(S_n\) example
In a varying rate model, suppose that the iid returns, \(i_k\), are such that \(1+i_k \sim LN(0.12,0.04^2)\). What is the probability that the accumulated value of 10000 at time 5 is greater than 21000?
Lognormal model: \(V_n\) example
In a varying rate model, suppose the iid returns, \(i_k\), are such that \(1+i_k \sim LN(0.08,0.04^2)\). What is the probability that the present value of a benefit of 100 at time 10 is less than 40?
Lognormal model: approximation I
In a varying model, even if accumulation factors do not follow a lognormal distribution, one can still use lognormal distributions to approximate \(S_n\) and \(V_n\).
Lognormal model: approximation II
- Since \(\log S_n=\sum_{t=1}^n\log(1+i_t)\), the summation of iid random variables, by the central limit theorem, \(\log S_n\) has a normal distribution as its limiting distribution, after normalization.
- Therefore, \(S_n\) can be approximated by a lognormal distribution.
- To apply the central limit theorem, technically, you need to verify the moment constraints/assumptions (why).
Lognormal model: \(S_n\) and \(V_n\) in a fixed rate model
Proof:
Lognormal model: a comparison
Let the accumulation factor for one period follow \(LN(\mu,\sigma^2)\). Calculate the coefficient of variation of \(S_n\) in a varying rate model and fixed rate model respectively.
We will see that \(S_n\) is more spread-out in the fixed rate model, which is intuitive (why?).
Value-at-Risk
With the distribution of \(S_n\), more risk metrics of \(S_n\) can be calculated. For a random variable \(X\), Value-at-Risk (VaR) at level \(p\in(0,1)\) is defined as \[\VaR_p(X)=F^{-1}(p)=\inf\{x: \p(X\le x)\ge p\}. \]
- Here, \(X\) is regarded as losses.
- In general, \(p\) is close to 1 (e.g., \(p=0.99\)).
- VaR is used as a regulatory risk measure in the realm of bank and insurance.
- VaR: the event that the loss is greater than this level has a probability less than \(1-p\).
- For a continuous random variable \(X\sim F\), \(\VaR_p(X)\) is the inverse of \(F\).
Value-at-Risk : example A
Suppose that a loss \(X\) has the distribution that \(P(X=-7)=0.04\) and \(P(X=5.5)=0.96\). What is \(\VaR_{0.95}(X)\)? Ans: \(\VaR_{0.95}(X)=5.5\).
Value-at-Risk : example B I
In a varying rate model, the annual accumulation factor follows a lognormal distribution with parameters \(\mu=0.075\) and \(\sigma^2=0.025^2\). Let the initial investment be 1000 and the accumulated value at the end of 5 years be \(X\). What are \(\VaR_{0.25}(X)\) and \(\VaR_{0.75}(X)\)?