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

1

Lecture 1 Introduction Risk-neutral pricing State price deflators Constructing a stochastic asset model for actuarial use

2

Lecture 2 The Wilkie model for price inflation Criticisms of the Wilkie model for price inflation

3

Lecture 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

Lecture 1 Lecture 2 Lecture 3 Introduction Risk-neutral pricing State price deflators 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

• n 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 Lecture 1 Lecture 2 Lecture 3 Introduction Risk-neutral pricing State price deflators 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

Lecture 1 Lecture 2 Lecture 3 Introduction Risk-neutral pricing State price deflators 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 Lecture 1 Lecture 2 Lecture 3 Introduction Risk-neutral pricing State price deflators 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

Lecture 1 Lecture 2 Lecture 3 Introduction Risk-neutral pricing State price deflators 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 Lecture 1 Lecture 2 Lecture 3 Introduction Risk-neutral pricing State price deflators 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 S0. Then, the Black-Scholes method of option pricing assumes that the price of the underlying asset at time t, St, is a geometric Brownian motion defined by the following stochastic differential equation: dSt = 휇Stdt + 휎StdWt where 휇 is the drift parameter (or expected annual rate of return), 휎 is the volatility parameter and Wt is a standard Brownian motion. And, the solution to the above SDE is: St = S0 × exp [( 휇 − 휎2 2 ) t + 휎Wt ]

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 Introduction Risk-neutral pricing State price deflators 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 = S0 × Φ (d1) − Ke−rt × Φ (d2) where d1 = ln [S0 K ] + [ r + 휎2 2 ] T 휎 √ T and d2 = d1 − 휎 √ T Suppose that we have S0 = 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

• ption with a term of T = 3 and an exercise price of K = 110 is:

c = 100 × Φ (0.3311) − 110e−0.05×3 × Φ (−0.0153) = 16.21

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 Introduction Risk-neutral pricing State price deflators 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: Wt has independent increments, with Wt − Ws ∼ N (0, t − s) Then, for t > 0, we have Wt ∼ 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, ST, is a log-Normal random variable with: mean of E (ST) = S0e휇T and variance of var (ST) = S2

0e2휇T (

e휎2T − 1 ) Then, we can use Monte-Carlo simulation to generate 10,000 different (but equally likely) realisations of Z (and, thus, of S3), thereby obtaining the empirical distribution function of S3 shown in Figure 1.

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 Introduction Risk-neutral pricing State price deflators Constructing a stochastic asset model for actuarial use

Risk-neutral pricing (5)

• Figure 1: Empirical distribution of price of underlying asset at time 3, S3

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 Introduction Risk-neutral pricing State price deflators Constructing a stochastic asset model for actuarial use

Risk-neutral pricing (6)

Then, for each realisation of S3, we can find the corresponding maturity pay-off from a call option with exercise price of K = 110, given by: C3 = max (0, S3 − K) = { S3 − K if S3 ≥ K if S3 < K Then, the corresponding empirical distribution function of C3, obtained from the same 10,000 realisations of Z, is shown in Figure 2. And, from this, the expected pay-off at time 3 is given by: EP (C3) = 23.5949 where we use the subscript P to denote that this expectation is based on projected future cash flows using the real-world probability measure, known as the P-measure.

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 Introduction Risk-neutral pricing State price deflators Constructing a stochastic asset model for actuarial use

Risk-neutral pricing (7)

• Figure 2: Empirical distribution of maturity pay-off from call option at time 3, C3

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 Introduction Risk-neutral pricing State price deflators Constructing a stochastic asset model for actuarial use

Risk-neutral pricing (8)

So, how do reconcile this value with the option price of c = 16.21, obtained previously using the Black-Scholes methodology? Traditional actuarial methodology would involve discounting the expected future cash flow at either: the expected return on the underlying asset, given by 휇 = 7% per annum, giving: c1 = e−3휇 × EP (C3) = e−3×0.07 × 23.5949 = 19.13

• r

the risk-free rate of return, given by r = 5% per annum, giving: c2 = e−3r × EP (C3) = e−3×0.05 × 23.5949 = 20.32 In fact, to recover the market-consistent value, we would need to discount the future cash flow at a rate of interest of j = 12.51% per annum such that e−3j × EP (C3) = c = 16.21.

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 Introduction Risk-neutral pricing State price deflators Constructing a stochastic asset model for actuarial use

Risk-neutral pricing (9)

Thus, whilst both approaches give a prudent estimate of the cost of the option (in this case, at least), neither is consistent with the “fair” value derived using the Black-Scholes approach. Is this a problem? Yes and no. If the aim is to estimate an appropriate market price for an embedded option (e.g. a minimum maturity guarantee on a unit-linked life insurance contract), then the prudent “price” obtained by this approach may prove to be uncompetitive. However, if the aim is to estimate a suitable reserve to be held in respect of such an embedded guarantee, then the implicit prudence may be appropriate (and any excess will be released when a claim is made). It is important to emphasise a crucial difference at this stage: purchasing such an option in the market (if available) will transfer the risk to a third-party, whereas simply estimating the likely cost of the option will not.

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 Introduction Risk-neutral pricing State price deflators Constructing a stochastic asset model for actuarial use

Risk-neutral pricing (10)

Before we consider this further, we return to the risk-neutral framework underlying the Black-Scholes methodology. In this artificial environment, the expected return on a risky asset is equal to the risk-free return. Now, from above, the price of the underlying asset at time t, St, is a random variable given by: St = S0 × exp [( 휇 − 휎2 2 ) t + 휎Wt ] where 휇 is the expected annual return (or drift parameter). Thus, in the risk-neutral world, we replace 휇 by r when projecting the future price of the underlying asset. Then, the expected annual return on the asset will now be equal to r, the risk-free return. Then, Figure 3 shows the empirical distribution of this “risk-neutral” price at time 3, which we denote S′

3, based on the same 10,000 realisations of Z.

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 Introduction Risk-neutral pricing State price deflators Constructing a stochastic asset model for actuarial use

Risk-neutral pricing (11)

• Figure 3: Empirical distribution of “risk-neutral” price of underlying asset at time 3,

S′

3 – green line

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 Introduction Risk-neutral pricing State price deflators Constructing a stochastic asset model for actuarial use

Risk-neutral pricing (12)

From Figure 3, we have: E ( S′

3

) ≡ EQ (S3) = 116.1873 ≈ S0erT = 100e0.05×3 = 116.1834 where the subscript Q denotes that this expectation is based on the projected future cash flows using the risk-neutral probability measure, Q. Then, as in the real-world framework, we can find the corresponding maturity pay-off for each realisation of S′

3 given by:

C ′

3 = max

( 0, S′

3 − K

) = { S′

3 − K

if S′

3 ≥ K

if S′

3 < K

Then, the corresponding empirical distribution function of C ′

3, obtained from

the same 10,000 realisations of Z, is shown in Figure 4. In this case, we have E (C ′

3) ≡ EQ (C3) = 18.8384.

However, crucially, when we discount this expected pay-off in the risk-neutral world using the risk-free interest rate, r, we get: e−3r × EQ (C3) = e−3×0.05 × 18.8384 = 16.21

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 Introduction Risk-neutral pricing State price deflators Constructing a stochastic asset model for actuarial use

Risk-neutral pricing (13)

• Figure 4: Empirical distribution of “risk-neutral” maturity pay-off from call option at

time 3, C ′

3

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 Introduction Risk-neutral pricing State price deflators Constructing a stochastic asset model for actuarial use

State price deflators (1)

So, what rate of interest should be used to discount the actual (i.e. real-world) future cash flows to give a market-consistent valuation? We can reconcile the risk-neutral approach and the actuarial approach using state-price deflators. A deflator is a stochastic discount factor that varies according to the actual economic conditions, in particular: a payment of 1 made at time t is worth more (or less) if investment returns have been low (or high) between time 0 and time t We define Dt to be the deflator applied to cash flows at time t (i.e. the value at time 0 of a payment of 1 made at time t). Then, if the risky asset is priced correctly, we have S0 = EP (DtSt). Also, e−rt is the price at time 0 of a risk-free asset paying 1 at time t. Thus, we have e−rt = EP (Dt × 1) = EP (Dt).

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 Introduction Risk-neutral pricing State price deflators Constructing a stochastic asset model for actuarial use

State price deflators (2)

These conditions allow us to find an explicit expression for the deflator at time t, Dt, given by: Dt = exp ( − [( r + 휆2 2 ) t + 휆Wt ]) where 휆 = 휇 − r 휎 See “State price deflator for Black-Scholes model” for a full derivation. Crucially, we note that Dt is a decreasing function of Wt (and, thus, of the price of the underlying asset at time t, St). So, we discount future cash flows at a higher rate of interest when the actual return on the risky asset increases. Hence, Dt is a stochastic, risk-adjusted discount rate, which assumes a different value for each scenario generated. Then, for each realisation, we discount the real-world payoff, C3, using the corresponding state-price deflator, D3. Figure 5 shows the empirical distribution function of the random variable D3C3, obtained from the same 10,000 realisations of Z.

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 Introduction Risk-neutral pricing State price deflators Constructing a stochastic asset model for actuarial use

State price deflators (3)

• Figure 5: Empirical distribution of PV of maturity pay-off from call option at time 3,

C3, discounted using state-price deflator, D3

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 Introduction Risk-neutral pricing State price deflators Constructing a stochastic asset model for actuarial use

State price deflators (4)

Crucially, from Figure 5, we see that: EP (D3C3) = [D3 × max (S3 − K, 0)] = 16.21 = c i.e. discounting the real-world cash flows using the corresponding state-price deflator allows us to recover the “fair” value of the future cash flows without the need to derive artificial risk-neutral cash-flows For future reference, we can note that a simpler stochastic discount factor would be represented by D′

t = S0

St – i.e. the actual return achieved on the underlying (risky) asset up to time t. However, whilst this would ensure that the risky asset was priced correctly (as, by definition, we would have S0 = EP (D′

tSt), it would NOT return the current

market price for the risk-free asset also. We will return to this in Lecture 6 below.

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 Introduction Risk-neutral pricing State price deflators Constructing a stochastic asset model for actuarial use

Constructing a stochastic asset model for actuarial use (1)

Actuaries model the future cash flows of financial institutions such as life insurance companies and pension funds. So, a stochastic asset model for actuarial use should generate future experience for the key economic variables affecting the amount (and, perhaps, timing) of future cash flows. Question: What are the main factors affecting the amounts of future cash flows for a final salary pension scheme? It is crucial that each scenario generated by the stochastic model is realistic. Thus, in particular, we need to ensure that we model the interactions between the key economic variables appropriately.

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 Introduction Risk-neutral pricing State price deflators Constructing a stochastic asset model for actuarial use

Constructing a stochastic asset model for actuarial use (2)

In particular, the current level of price inflation is likely to influence the current (and, perhaps even, future) value of many of the other key economic variables, such as: if the current rate of price inflation rises, what is likely to be the effect on the current rate of salary inflation? why? if the current rate of price inflation rises, what is likely to be the effect on the current level of the short-term interest rate (i.e. the return on cash deposits)? why? if the current rate of price inflation rises, what is likely to be effect on the current level of long-term interest rates (i.e. the GRY on long-dated fixed-interest bonds)? why? if the current rate of price inflation rises, what is likely to be the short-term effect on the level of the equity market? and, what is likely to be the longer-term effect on the equity market? why?

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 Introduction Risk-neutral pricing State price deflators Constructing a stochastic asset model for actuarial use

Constructing a stochastic asset model for actuarial use (3)

Thus, we begin by considering the construction of a suitable model for the rate of price inflation. Should the model be continuous time or discrete time? Continuous-time models are commonly used in financial economics (e.g. GBM model used for Black-Scholes option pricing, Vasicek and CIR interest rate models). These models tend to use simple structures for the underlying stochastic processes, giving closed-form analytical solutions. And, in the short-term, the resulting inaccuracies do not cause serious problems. However, actuaries tend to work with a much longer time horizon (which can compound these small inaccuracies), and are used to projecting cash flows in discrete time periods (e.g. every year). Thus, whilst use of a discrete-time approach is more complicated and time-consuming, it ultimately gives the freedom to model the individual processes (and the interactions between them) more accurately.

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 Introduction Risk-neutral pricing State price deflators Constructing a stochastic asset model for actuarial use

Constructing a stochastic asset model for actuarial use (4)

ARIMA(p, d, q) time series models are covered in detail in Subject CT4. Before fitting such models, we require that the series to be modelled is stationary (i.e. the mean and variance of the underlying stochastic process can be assumed to be constant). A non-stationary series can be transformed into a stationary series by: de-trending to remove a deterministic trend, differencing to remove a stochastic trend, variance stabilising transformations (e.g. log transform) For an AR(p) model, the value of the process at time t, Xt, is represented by a linear regression on the p previous values of the process. For an MA(q) model, the value of the process at time t, Xt, depends on the random components of the q previous values of the process. See “Introduction to Time Series Modelling” for more details.

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 The Wilkie model for price inflation Criticisms of the Wilkie model for price inflation

The Wilkie model for price inflation (1)

• Figure 6: Annual force of price inflation in UK, 1923-2009

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 The Wilkie model for price inflation Criticisms of the Wilkie model for price inflation

The Wilkie model for price inflation (2)

We propose that the annual price inflation process to be a stationary auto-regressive process. i.e. the level of price inflation in any given time period will be significantly correlated with the level in the previous time period In particular, we can see that periods of low (and, often, stable) inflation tend to bunch together (e.g. pre 1935, 1950-1970 and, although not shown, 1994

• nwards).

However, these periods appear to be interrupted by shorter periods of higher (and, often, more volatile) inflation (e.g. 1940s, mid 1970s, early 1990s). Thus, a simple model for price inflation may be expected to be an AR(p) model, given by: Xt = 휇 + 훼1 (Xt−1 − 휇) + . . . + 훼p (Xt−p − 휇) + et

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 The Wilkie model for price inflation Criticisms of the Wilkie model for price inflation

The Wilkie model for price inflation (3)

Let Q(t) denote the value of the RPI at time t. Then, we define: I(t) = ln [ Q(t) Q(t − 1) ] as the annual force of price inflation over the interval (t − 1, t). Then, the annual rate of price inflation over (t − 1, t) is ln [1 + I(t)]. For any model, it is crucial to choose a suitable subset of past data for the parameter estimation. In particular: too short ⇒ high standard error for parameter estimates, and too long ⇒ some of the data used to estimate the parameters may be of limited relevance for future projections Balancing these conflicting objectives, the Wilkie (2011) model uses UK price inflation data over the period 1923-2009.

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 The Wilkie model for price inflation Criticisms of the Wilkie model for price inflation

The Wilkie model for price inflation (4)

Then, the Wilkie model for the annual force of price inflation in year t, I(t), is an AR(1) model given by: I(t) = QMU + QA × [I(t − 1) − QMU] + QE(t) where QMU is the long-term mean annual force of price inflation QA is the auto-regression parameter

i.e. measures the strength of the effect of the current value of the process

• n the following value

QE(t) = QSD ×QZ(t) is the random component of the process at time t

QSD is the (constant) standard deviation parameter of the price inflation process QZ(t) is a standard N(0, 1) random variable

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 The Wilkie model for price inflation Criticisms of the Wilkie model for price inflation

The Wilkie model for price inflation (5)

Then, fitting to 1923-2009 data gives parameter estimates as follows: QMU = 0.043 QA = 0.58 QSD = 0.04 But, it is important to remember that we are trying to model future experience. In particular, an actuary’s views of the likely future experience should depend on more than just an analysis of the past data. Question: What other factors may affect our views on the likely long-term mean (or volatility) of future price inflation? Question: On average, might we expect price inflation in future to be higher than, lower than or the same as in the past? Why? What about the volatility of price inflation?

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 The Wilkie model for price inflation Criticisms of the Wilkie model for price inflation

The Wilkie model for price inflation (6)

Thus, if desired, we can adjust the parameters to reflect expectations of future experience, such as: use a lower value of QMU to reflect the belief that, on average, future price inflation will be lower than past price inflation, use a higher value of QSD to reflect greater economic uncertainty However, as with any subjective adjustment to past data, it is crucial that we do not give too much weight to short-term influences. e.g. we have had periods of low and stable inflation in the past without this resulting in a fundamental change in the underlying process (e.g. relative stability of 1960s followed by hyper-inflation in early 1970s) Before analysing the output, we need an initial starting value, I(0). For simplicity, we will use the value at the end of 2009. Thus, we have I(0) ≡ I(2009) = −0.0158.

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 The Wilkie model for price inflation Criticisms of the Wilkie model for price inflation

The Wilkie model for price inflation (7)

• Figure 7: Two possible simulations of force of price inflation over the period 2010-2029

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 The Wilkie model for price inflation Criticisms of the Wilkie model for price inflation

The Wilkie model for price inflation (8)

• Figure 8: Empirical distribution function for I(1) and I(20), based on 10,000

simulations

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 The Wilkie model for price inflation Criticisms of the Wilkie model for price inflation

The Wilkie model for price inflation (9)

Consider the distribution of I(1) – i.e. red line – based on 10,000 simulations. The distribution has a Normal shape with: E [I(1)] = (1 − QA)×QMU+QA×I(0) = 0.0090 and var [I(1)] = QSD2 = 0.042 Also, the 5th and 95th percentiles of the distribution are given by: Pr [I(1) < −0.0569] = 0.05 and Pr [I(1) > 0.0747] = 0.05 Now, consider the distribution of I(20) – i.e. blue line – based on 10,000

• simulations. Again, the distribution has a Normal shape with:

E [I(20)] = 0.0430 ≈ QMU and var [I(20)] = 0.04912 ≈ QSD2 1 − QA2 And, in this case, the 5th and 95th percentiles are given by: Pr [I(20) < −0.0389] = 0.05 and Pr [I(20) > 0.1236] = 0.05

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 The Wilkie model for price inflation Criticisms of the Wilkie model for price inflation

The Wilkie model for price inflation (10)

• Figure 9: Empirical distribution function of I(t), for t = 0, 1, 2, . . . , 20

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 The Wilkie model for price inflation Criticisms of the Wilkie model for price inflation

The Wilkie model for price inflation (11)

This is known as the “increasing funnel of doubt” – i.e. the volatility (or uncertainty) increases as we look further into the future. From Figure 9, we can see the mean-reversion of the price inflation process and that the distribution of I(t) reaches a stationary (or equilibrium) position very quickly (i.e. within 4 to 5 years). We can define GQ(t) as the mean annual rate of price inflation up to time t. Thus, we have: GQ(t) = 100 × [( Q(t) Q(0) )

1 t

− 1 ] Figure 10 shows one possible sample path for both I(t) and GQ(t) over the period 2010 to 2029. We can see that the mean rate of price inflation up to time t, GQ(t), becomes more stable as t increases. Why?

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 The Wilkie model for price inflation Criticisms of the Wilkie model for price inflation

The Wilkie model for price inflation (12)

• Figure 10: One possible simulation of the mean annualised force of price inflation up

to time t, GQ(t), over the period 2010-2029

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 The Wilkie model for price inflation Criticisms of the Wilkie model for price inflation

The Wilkie model for price inflation (13)

• Figure 11: Empirical distribution function for GQ(1) and GQ(20), based on 10,000

simulations

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 The Wilkie model for price inflation Criticisms of the Wilkie model for price inflation

The Wilkie model for price inflation (14)

• Figure 12: Empirical distribution function of GQ(t), for t = 0, 1, 2, . . . , 20

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 The Wilkie model for price inflation Criticisms of the Wilkie model for price inflation

The Wilkie model for price inflation (15)

From Figure 11 and Figure 12, we can see that the average rate of price inflation becomes more stable as we consider a longer time horizon: this long-term cross-sectional stability is the basis of the traditional deterministic approach used by actuaries In particular, by using a long time horizon, the effect of the inherent volatility in the economic variable is significantly reduced and, thus, can be reasonably represented by a constant ‘average’ rate. However, even over 20 years, the volatility in the average rate of price inflation is not insignificant, in particular: the 90% confidence interval for the average rate of price inflation is (0.6%, 7.5%) ... indictating a much greater degree of uncertainty that most actuaries would imagine!

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 The Wilkie model for price inflation Criticisms of the Wilkie model for price inflation

Criticisms of the Wilkie model for price inflation (1)

• Figure 13: Standardised residuals, QZ(t), after fitting basic Wilkie price inflation

model

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 The Wilkie model for price inflation Criticisms of the Wilkie model for price inflation

Criticisms of the Wilkie model for price inflation (2)

But, residuals from fitting the model do not have a Normal distribution. In particular, skewness parameter, √b1, is 1.25 ⇒ residuals are positively-skewed. Thus, in practice, the process more likely to make ‘large’ jump upwards than downwards. Could introduce asymmetry in process by transforming data: use log transform, but would exclude negative values – undesirable? use translated Gamma distribution for residuals A consequence of this enforced symmetry is the high probability of negative price inflation noted above. Also, kurtosis parameter, b2, is 5.97 ⇒ weight in tails of the distribution is greater than would be expected for a Normal distribution (with b2 = 3), particularly the right-hand tail (due to positive skewness). So, model may under-estimate probability of LARGE rises in inflation!

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 The Wilkie model for price inflation Criticisms of the Wilkie model for price inflation

Criticisms of the Wilkie model for price inflation (3)

However, more importantly, it appears from Figure 13 that the volatility of QZ(t) is not constant over the period 1923-2009. In particular, high volatility in early 1940s and in 1970s to early 1980s, with much lower volatility in 1950s to 1960s and since mid 1990s. Thus, main criticism of Wilkie model is that it does not allow for observed (short) bursts of high (and, usually, more volatile) price inflation. Instead, crucially, it imposes a constant volatility on the process. What is the effect of this? Volatility is too high during periods when inflation should be low and stable AND too low during periods when it should be high and volatile! We consider two methods of addressing this key problem: an auto-regressive conditional heteroskedastic (or ARCH) model, a regime-switching model

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

ARCH model for price inflation (1)

In an ARCH model, the variance is modelled as a stochastic process. Thus, we model the annual force of price inflation in year t, I(t), as: I(t) = QMU + QA × [I(t − 1) − QMU] + QE(t) where QMU is the long-term mean annual force of price inflation QA is the auto-regression parameter QE(t) = QSD(t) × QZ(t) is the random component of the process at time t

QSD(t) is the (non-constant) standard deviation of the price inflation process at time t, given by: QSD(t)2 = QSA2 + QSB × [I(t − 1) − QSC]2 QZ(t) is a standard N(0, 1) RV ⇒ QE(t) ∼ N ( 0, QSD(t)2)

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

ARCH model for price inflation (2)

Thus, assuming the parameter QSB > 0, the (one-step ahead) variance of the process, QSD(t)2, increases as the current value of the process, I(t), moves further away from the parameter QSC. In general, QSC = QMU (i.e. the long-term mean of the process), but it is possible to use some other measure. Thus, an ARCH process becomes more volatile as it moves further away from the mean. Question: Does this seem reasonable as a model for the underlying price inflation process? If so, why? Also, note that we must include the parameter QSA and require QSA ∕= 0. Otherwise, if the value of the process at any time t, I(t), was equal to the long-term mean, QMU, then the variance of the process would collapse to zero – and, thus, we would have I(t + k) = QMU for all k ≥ 0.

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

ARCH model for price inflation (3)

Then, fitting to 1923-2009 data gives parameter estimates as follows: QMU = 0.035 QA = 0.59 QSA = 0.023 QSB = 0.63 QSC = 0.035 Note that Wilkie (2011) constrained the model such that QSC = QMU. The values of the QMU and QA parameters are different to those suggested for the standard model. Why? For simulation purposes, we again need to set the initial value of the process, I(0). As for the standard model, we use the value of the process in 2009 giving I(0) ≡ I(2009) = −0.0158.

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

ARCH model for price inflation (4)

• Figure 14: Two possible simulations of force of price inflation over the period

2010-2029 using the ARCH model

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

ARCH model for price inflation (5)

From Figure 14, one of the sample paths exhibits “hyper-inflation” (i.e. force

• f price inflation in year 2021, I(12), is over 60%).

This can happen because, as the process gets further away from the mean, the variance increases exponentially (as a result of the [I(t − 1) − QSC]2 term). Then, a positive step (i.e. QZ(t) > 0) will lead to a further large rise in inflation (leading to hyper inflation) and a negative step (i.e. QZ(t) < 0) will lead to a sharp fall. Whilst this feature does not appear in every sample path generated by the ARCH model, it is reasonably common. Hyper-inflation has not (yet!) been experienced in the UK or US, but arose in Germany after World War I (as well as in many developing economies from time to time – e.g. Argentina in the late 1980s, Zimbabwe in the mid 2000s).

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

ARCH model for price inflation (6)

• Figure 15: Distribution function for I(1) using both standard Wilkie model and ARCH

model

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

ARCH model for price inflation (7)

Thus, for the distribution of I(1) for the ARCH model (in comparison with the standard model), we have: normal shape to distribution (as for standard model) mean = (1 − QA) × QMU + QA × I(0) = 0.0050 variance = 0.04642 (cf. 0.0402 for standard model)

this is because the process is starting relatively far from the long-term mean, QMU however, when the process is close to the mean, the one-step ahead variance is significantly lower using the ARCH model

Pr [I(1) < −7.14%] = 0.05

compared with 5th percentile of −5.69% for standard model

Pr [I(1) > 8.13%] = 0.05

compared with 95th percentile of 7.47% for standard model

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

ARCH model for price inflation (8)

• Figure 16: Distribution function for I(20) using both standard Wilkie model and

ARCH model

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

ARCH model for price inflation (9)

And, for the distribution of I(20) for the ARCH model (in comparison with the standard model), we have: distribution is no longer normal

higher peak AND fatter tails thus, in the long term, the ARCH model gives both a greater concentration

• f values around the mean AND a higher probability of hyper-inflation

(but, also of hyper-deflation!)

mean = 0.0345 ≈ QMU (as for standard model) higher variance = 0.08022 (cf. 0.04912 for standard model) Pr [I(20) < −4.73%] = 0.05

compared with 5th percentile of −3.89% for standard model

Pr [I(20) > 11.78%] = 0.05

compared with 95th percentile of 12.36% for standard model

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

ARCH model for price inflation (10)

However, if we look at more extreme tail values of the distribution of I(20), we can see more clearly the effect of the increase in kurtosis: Pr [I(20) < −14.91%] = 0.01

compared with 1st percentile of −7.42% for standard model

Pr [I(20) > 22.47%] = 0.01

compared with 99th percentile of 15.73% for standard model

Thus, the ARCH model produces the desired long periods of low and stable inflation (i.e. greater concentration around mean) punctuated by short periods of high and volatile inflation (i.e. fatter tails). However, the model is still symmetrical – so, the periods of high volatility can produce both extreme positive inflation (possible, if unlikely) and extreme negative inflation (highly unlikely!). Removing the constraint that QSC = QMU will create skewness in the model

• utput (at the expense of increasing the number of parameters).

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

ARCH model for price inflation (11)

As can be seen from Figure 17, the price inflation process generated by the ARCH model also displays the “increasing funnel of doubt”. However, the greater concentration around the mean can be seen from the lower inter-quartile range, particularly at later durations (in comparison to the standard model in Figure 9). Also, as for the standard model, we can define GQ(t), the mean annual rate

• f price inflation up to time t, as follows:

GQ(t) = 100 × [( Q(t) Q(0) )

1 t

− 1 ] And, from Figure 18, we can see that, as expected, the volatility of the average rate of price inflation up to time t reduces as t increases. However, as a result of the greater volatility, the averaging effect is more gradual here and there remains considerable uncertainty in the mean rate of price inflation, even over the very long term.

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

ARCH model for price inflation (12)

• Figure 17: Empirical distribution function of I(t), for t = 0, 1, 2, . . . , 20, using the

ARCH model for price inflation

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

ARCH model for price inflation (13)

• Figure 18: Empirical distribution function of GQ(t), for t = 0, 1, 2, . . . , 20, using the

ARCH model for price inflation

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

ARCH model for price inflation (14)

Then, comparing average price inflation up to time t, GQ(t), under the ARCH model with that under the standard Wilkie model, we have: time, t 1 2 5 10 20 50 ARCH model E [GQ(t)] 0.60% 1.23% 2.29% 2.88% 3.22% 3.42% sd [GQ(t)] 4.69% 4.42% 4.24% 3.95% 3.30% 2.35% standard Wilkie model E [GQ(t)] 0.96% 1.68% 2.87% 3.59% 3.99% 4.23% sd [GQ(t)] 4.04% 3.81% 3.40% 2.76% 2.08% 1.36%

Table 1: Distribution of average rate of price inflation up to time t, GQ(t), for Wilkie ARCH model

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

ARCH model for price inflation (15)

Thus, as we would expect, we have: higher mean annualised rate of price inflation at all durations under the standard Wilkie model

this is due to the higher long-term mean parameter in the standard model (i.e. QMU = 0.043 compared with QMU = 0.035)

higher variance in the mean annualised rate of price inflation at all durations under the ARCH model

as noted previously, in the short term, this is due to the process starting relatively far from the long-term mean (as a result of the negative inflation experienced in 2009) if the process starts closer to the long-term mean, then the variance (or uncertainty) in the mean annualised rate of price inflation will be lower using the ARCH model however, as the duration increases, it becomes more likely that the ARCH model will generate periods of high volatility, thereby significantly increasing the uncertainty of the mean rate of price inflation in the long term (regardless of the starting position)

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

Regime-switching model for price inflation (1)

An alternative approach to generating processes with non-constant volatility is to use a regime-switching model. We will consider a model with two regimes (or states), but the idea can be readily extended to include more regimes. Suppose that we have two regimes defined as: “normal” inflation

i.e. price inflation process is (relatively) low and stable

“high” inflation

i.e. price inflation process is high and volatile

Within each regime, the process is usually assumed to follow a simple ARIMA model (but with different parameters values in particular, higher mean and variance parameters in the “high” inflation regime).

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

Regime-switching model for price inflation (2)

How do we enable the process to move between the two regimes? could define process as a time-homogenous Markov model with transition matrix given by: P = [ p11 p12 p21 p22 ] where pij = Pr [in state j at time (t + 1)∣in state i at time t], with 1 ≡ “normal” inflation state and 2 ≡ “high” inflation state

p12 = 1 − p11 ⇒ and, as we expect the process to spend long periods in state 1, we would have p11 close to 1 p21 = 1 − p22 ⇒ and, as we expect the process to spend only short periods in state 2, we would have p22 close to 0

can use a threshold approach

we define a threshold parameter and, if the process crosses this threshold, then it moves from state 1 to state 2 (or vice versa) this is the approach used by Whitten & Thomas (1999)

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

Regime-switching model for price inflation (3)

In the model proposed by Whitten & Thomas (1999), the price inflation process is assumed to follow a simple AR(1) model in each regime, and, the previous value of the process, I(t − 1), determines which regime the process currently occupies. Thus, we model the annual force of price inflation in year t, I(t), as: I(t) = ⎧     ⎨     ⎩ QMU1 + QA1 × [I(t − 1) − QMU1] +QSD1 × QZ(t) if I(t − 1) ≤ QR QMU2 + QA2 × [I(t − 1) − QMU2] +QSD2 × QZ(t) if I(t − 1) > QR where QR is the threshold parameter. However, a lack of data means that it is very difficult to fit even this very simple regime-switching model with much confidence – by definition, periods of high and unstable inflation should be few and far between!

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

Regime-switching model for price inflation (4)

• Figure 19: Annual force of price inflation in UK, 1923-2009

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

Regime-switching model for price inflation (5)

Looking again at UK price inflation data 1923-2009 in Figure 19, we can see that there only one or two short periods of high and volatile inflation (mid to late 1970s and, arguably, early 1940s). In total, this gives us, perhaps, 8-10 data points from which to estimate a credible model for the “high” inflation state. However, Whitten & Thomas (1999) gives the following parameter estimates: QR = 0.1 QMU1 = 0.04 QMU2 = 0.12 QA1 = 0.55 QA2 = 0.0 QSD1 = 0.0325 QSD2 = 0.05 Thus, we have a higher mean and (crucially) a higher variance in the “high” inflation regime.

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

Regime-switching model for price inflation (6)

• Figure 20: Two possible simulations of force of price inflation over the period

2010-2029 using the regime-switching model

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

Regime-switching model for price inflation (7)

Then, from Figure 20, we can see that, as the process has higher volatility in the “high” inflation state, it is likely that any sample path of future price inflation generated by such a model will remain in this state for only one or two time units before crossing the threshold again back to the “normal” inflation state (where the volatility is lower). Thus, as with the ARCH model, the price inflation process generated with the regime-switching model can be expected to show: long periods of stability (when process is in “normal” inflation state), and short periods of high and more volatile inflation (when process is in “high” inflation state).

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

Regime-switching model for price inflation (8)

• Figure 21: Distribution function for I(1) using both standard Wilkie model, ARCH

model and regime-switching model

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

Regime-switching model for price inflation (9)

• Figure 22: Distribution function for I(20) using both standard Wilkie model, ARCH

model and regime-switching model

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

Regime-switching model for price inflation (10)

• Figure 23: Empirical distribution function of I(t), for t = 0, 1, 2, ..., 20, using the

regime-switching model for price inflation

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

Regime-switching model for price inflation (11)

Then, as with the ARCH model, the price inflation process generated by the RS model exhibits heteroskedasticity (or fat-tailedness). However, for t > 1, the distribution of I(t) also exhibits the desired positive skewness i.e. a large jump upwards in the price inflation process is more likely than a corresponding large jump downwards Thus, for t > 1, the distribution of I(t) is no longer Normal e.g. for t = 20, we have skewness, √b1 ≈ 0.7 and kurtosis, b2 ≈ 4.2 As can be seen from Figure 23, the price inflation process generated by the RS model also displays an “increasing funnel of doubt”. The positive skewness of I(t) as t increases can also be clearly seen.

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

Regime-switching model for price inflation (12)

However, the main disadvantage of this model is the subjectivity of many of the parameter estimates: this reduces the confidence we can have in the model output Question: Which of the parameters in the regime-switching model are likely to be subject to the greatest uncertainty? Again, we can define GQ(t), the mean annual rate of price inflation up to time t, as follows: GQ(t) = 100 × [( Q(t) Q(0) )

1 t

− 1 ] And, from Figure 24, we can see that, as expected, the volatility of the average rate of price inflation up to time t reduces as t increases.

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

Regime-switching model for price inflation (13)

• Figure 24: Empirical distribution function of GQ(t), for t = 0, 1, 2, ..., 20, using the

regime-switching model for price inflation

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

Regime-switching model for price inflation (14)

Then, comparing average price inflation up to time t, GQ(t), under the regime-switching model with that under both the standard Wilkie model and the ARCH model, we have: time, t 1 2 5 10 20 50 regime-switching model E [GQ(t)] 1.27% 1.98% 3.13% 3.89% 4.35% 4.64% sd [GQ(t)] 3.29% 2.99% 2.75% 2.52% 2.05% 1.41% ARCH model E [GQ(t)] 0.60% 1.23% 2.29% 2.88% 3.22% 3.42% sd [GQ(t)] 4.69% 4.42% 4.24% 3.95% 3.30% 2.35% standard Wilkie model E [GQ(t)] 0.96% 1.68% 2.87% 3.59% 3.99% 4.23% sd [GQ(t)] 4.04% 3.81% 3.40% 2.76% 2.08% 1.36%

Table 2: Distribution of average rate of price inflation up to time t, GQ(t), for regime-switching model

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I

Lecture 1 Lecture 2 Lecture 3 ARCH model for price inflation Regime-switching model for price inflation

Regime-switching model for price inflation (15)

Thus, in comparison with the standard model, we have: lower variance (or uncertainty) for mean annualised rate of price inflation in the short term

this is because we assume that the process starts in the “normal” inflation regime with I(0) ≡ I(2009) = −0.0158

however, in the long term, the variance of the mean annualised rate of price inflation is similar to that for the standard model

as, in the long term, the process can be expected to exhibit short periods of more volatile price inflation

the mean annualised rate of price inflation will tend towards a long-term equilibrium QMU∗, such that QMU1 < QMU∗ < QMU2 Regime-switching model is intuitively appealing, but crucial to check robustness of results to changing assumed threshold, QR. Why?

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part I