Stochastic asset models for actuarial use Part I Dr Douglas Wright, - PDF document

Lecture 1 Lecture 2 Lecture 3 Stochastic asset models for actuarial use – Part I Dr Douglas Wright, Cass Business School Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 Table of contents Lecture 1 1 Introduction Risk-neutral pricing State price deflators Constructing a stochastic asset model for actuarial use Lecture 2 2 The Wilkie model for price inflation Criticisms of the Wilkie model for price inflation Lecture 3 3 ARCH model for price inflation Regime-switching model for price inflation Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Introduction Lecture 1 Risk-neutral pricing Lecture 2 State price deflators Lecture 3 Constructing a stochastic asset model for actuarial use Introduction (1) Actuaries are responsible for managing the risks faced by financial institutions. This involves modelling uncertain future cash flows, requiring the actuary to make assumptions about factors affecting the amount and/or timing of the future cash flows. This includes both economic factors such as price inflation and investment return, and demographic factors such as mortality and withdrawal rates. Each of these factors can be thought of as a random variable . Traditionally, actuaries make deterministic assumptions for each factor, based on the expected value of the underlying random variables. Then, the risk (or uncertainty) is allowed for by including margins in the assumptions made and/or discounting future cash flows using an appropriate risk discount rate . Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Introduction Lecture 1 Risk-neutral pricing Lecture 2 State price deflators Lecture 3 Constructing a stochastic asset model for actuarial use Introduction (2) Differences in the actual future experience (compared to that assumed) will give rise to profits/losses . Often, the actuary will use scenario testing to investigate the risks associated with the actual future experience departing from the assumptions made. However, there are a number of problems with this approach, in particular: it is difficult to determine the cumulative effect of including margins in more than one variable, as many of the variables are correlated it does not give an indication of the level of the uncertainty associated with the various cash flow projections it cannot be used to price options or guarantees – as the pay-off will always be either in-the-money or out-of-the-money, depending on the assumptions made Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Introduction Lecture 1 Risk-neutral pricing Lecture 2 State price deflators Lecture 3 Constructing a stochastic asset model for actuarial use Introduction (3) We consider the use of stochastic modelling techniques to more satisfactorily model the uncertainties in the future cash flows as a result of random fluctuations in future experience. We will concentrate mainly on random fluctuations in the economic variables (e.g. price inflation and investment return), as these as non-diversifiable risks. Risks associated with random fluctuations in mortality experience reduce as the size of the population increases (due to the Law of Large Numbers). We use a stochastic asset model to generate a large number of simulations of the future economic experience and, for each simulation, we then model the future cash flows of the financial institution. This allows to explore the sensitivity of the financial position at various future dates to fluctuations in the economic experience. Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Introduction Lecture 1 Risk-neutral pricing Lecture 2 State price deflators Lecture 3 Constructing a stochastic asset model for actuarial use Introduction (4) Actuarial models have two crucial (and related) steps: projecting the cash flows in each future time period, and discounting these future cash flows to give an appropriate present (or capital) value Traditional actuarial techniques involve discounting the expected future cash flows at the expected rate of return on the assets held to back the liabilities (or, more prudently, at the rate of return on a risk-free asset). However, it can easily be shown that both approaches gives a capital value that is inconsistent with current market conditions. Some actuaries argue that this is not a valid concern, as the objective is not to put a market value on the cash flows (rather, it is to value assets and liabilities consistently). However, there is increasing pressure to use market-consistent valuations for insurance liabilities. Why? Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Introduction Lecture 1 Risk-neutral pricing Lecture 2 State price deflators Lecture 3 Constructing a stochastic asset model for actuarial use Risk-neutral pricing (1) Financial economics makes considerable use of risk-neutral pricing , which involves projecting future cash flows using an artificial (or risk-neutral) probability measure, known as the Q -measure. The methodology is known as “risk-neutral” because, in this artificial environment, the return on every asset is equal to the risk-free return. Then, these (uncertain) future cash flows are valued by discounting at the risk-free interest rate . The expected present value of the future (artificial) cash flows in the risk-neutral framework gives a market-consistent price for the (real-world) future cash flows. However, the projected cash flows using the risk-neutral probability measure are not the real-world cash flows and, thus, cannot be used to assist with on-going risk management and decision making (e.g. choosing asset allocation, reinsurance, bonus distribution strategies etc). Why? Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Introduction Lecture 1 Risk-neutral pricing Lecture 2 State price deflators Lecture 3 Constructing a stochastic asset model for actuarial use Risk-neutral pricing (2) We now consider the two different approaches when pricing a call option. A call option gives the holder the right, but not the obligation, to buy a specified asset at specified price on a specified future date. Consider a call option with a term of T years and an exercise price of K , written on an underlying asset with a current price of S 0 . Then, the Black-Scholes method of option pricing assumes that the price of the underlying asset at time t , S t , is a geometric Brownian motion defined by the following stochastic differential equation: dS t = 휇 S t dt + 휎 S t dW t where 휇 is the drift parameter (or expected annual rate of return), 휎 is the volatility parameter and W t is a standard Brownian motion . And, the solution to the above SDE is: 휇 − 휎 2 [( ) ] S t = S 0 × exp t + 휎 W t 2 Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Introduction Lecture 1 Risk-neutral pricing Lecture 2 State price deflators Lecture 3 Constructing a stochastic asset model for actuarial use Risk-neutral pricing (3) Then, assuming a risk-free rate of interest of r per annum and using the Black-Scholes methodology, the current price of the call option is given by: c = S 0 × Φ ( d 1 ) − Ke − rt × Φ ( d 2 ) where r + 휎 2 [ S 0 ] [ ] ln + T √ K 2 d 1 = √ and d 2 = d 1 − 휎 T 휎 T Suppose that we have S 0 = 100, a drift parameter of 휇 = 7% per annum and a volatility parameter of 휎 = 20% per annum. Then, using a risk-free interest rate of r = 5% per annum, the price of a call option with a term of T = 3 and an exercise price of K = 110 is: c = 100 × Φ (0 . 3311) − 110 e − 0 . 05 × 3 × Φ ( − 0 . 0153) = 16 . 21 Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Introduction Lecture 1 Risk-neutral pricing Lecture 2 State price deflators Lecture 3 Constructing a stochastic asset model for actuarial use Risk-neutral pricing (4) However, what does this “market-consistent” value represent? And, how can it be replicated by considering the real-world cash flows required by an actuary to manage risks that cannot be hedged using such derivatives? From the basic properties of a Brownian motion , we know that: W t has independent increments , with W t − W s ∼ N (0 , t − s ) Then, for t > 0, we have W t ∼ N (0 , t ) ≡ √ tZ , where Z ∼ N (0 , 1) is a standard Normal random variable. And, it is straightforward to show that the price of the underlying asset at time T > 0, S T , is a log-Normal random variable with: mean of E ( S T ) = S 0 e 휇 T and variance of var ( S T ) = S 2 0 e 2 휇 T ( e 휎 2 T − 1 ) Then, we can use Monte-Carlo simulation to generate 10,000 different (but equally likely) realisations of Z (and, thus, of S 3 ), thereby obtaining the empirical distribution function of S 3 shown in Figure 1. Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I