Stochastic asset models for actuarial use Part II Dr Douglas - - PDF document

Lecture 4 Lecture 5 Lecture 6

Stochastic asset models for actuarial use – Part II

Dr Douglas Wright, Cass Business School

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6

Table of contents

1

Lecture 4 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

2

Lecture 5 Dynamic ALM using the Wilkie model ARCH and regime-switching models for asset returns Economic-based models

3

Lecture 6 Using SAMs to price investment guarantees Using SAMs to determine strategic asset allocation

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

Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Modelling asset returns (1)

We now consider modelling asset returns, and will focus only on the two main asset classes for most institutional investors: domestic equities, and domestic long-dated fixed-interest government bonds. Because one of the main uses of stochastic asset models is to assist in financial planning, we are likely to want to project the cash flows arising from the assets (rather than just the total return achieved). Thus, we must consider income and capital gain components separately. Then, the total (gross) return over the year is given by: total return = asset price at end of year + income received over year asset price at start of year − 1 For fixed-interest bonds, the income component is fixed (and volatility in return comes from changes in capital value) – i.e. a change in GRY.

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Modelling asset returns (2)

We define the following key asset variables: Y (t) is the equity dividend yield at time t, K(t) is the force of equity dividend growth in year (t − 1, t), and C(t) is the (running) yield on undated fixed interest bonds at time t

we will also refer to this is the “long-term interest rate”

Note that, for simplicity, the Wilkie model assumes that fixed-interest bonds are undated (rather than redeemed at some specified future date). Then, level of dividend income on equities at time t, D(t), is given by: D(t) = D(t − 1) × exp [K(t)] Without loss of generality, we assume that D(0) = 1.0 – as D(t) is an index, the absolute value at time 0 is not important.

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

Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Modelling asset returns (3)

Define P(t) as the level of the share price index at time t. Then, by definition, we have: Y (t) = D(t) P(t) ⇒ P(t) = D(t) Y (t) Assuming equity dividends are payable in arrears and ignoring taxation of dividend income, the gross return on equities over the year (t − 1, t) is: L PR(t) = P(t) + D(t) P(t − 1) − 1 Define PR(t) to be the accumulation at time t of an initial investment of 1 made at time 0 in equities (assuming re-investment of dividend income). Then, we have: PR(t) = PR(t − 1) × [1 + L PR(t)] = PR(t − 1) × P(t) + D(t) P(t − 1)

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Modelling asset returns (4)

Without loss of generality, assume that the annual income from an undated fixed-interest bond is 1.0. Then, the (running) yield on this investment at time t is given by: C(t) = income at time t price at time t ⇒ price at time t = 1 C(t) Assuming bond income is payable in arrears and ignoring taxation of bond income, the gross return on fixed-interest bonds over the year (t − 1, t) is: L CR(t) =

1 C(t) + 1 1 C(t−1)

− 1 = C(t − 1) × [ 1 C(t) + 1 ] − 1 Define CR(t) to be the accumulation at time t of an initial investment of 1 made at time 0 in fixed-interest bonds. Then, we have: CR(t) = CR(t − 1) × [1 + L CR(t)] = CR(t) × C(t − 1) × [ 1 C(t) + 1 ]

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

Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for equity returns (1)

Wilkie model has a hierarchical (or cascade) structure.

force of price inflation equity dividend yield force of equity dividend growth yield on long-dated FIBs

• Figure 25: Cascade structure for Wilkie model

Then, price inflation is the primary process driving the others. Also, for reasons discussed below, equity dividend yield influences both force

• f equity dividend growth and long-term interest rate.

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for equity returns (2)

• Figure 26: Annual force of price inflation, I(t) and equity dividend yield, Y (t), in UK,

1923-2009

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

Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for equity returns (3)

Thus, equity dividend yield is much more stable than price inflation. There is some evidence of a positive correlation between the two variables (indeed, the correlation coefficient is c = 0.60). However, is is crucial to note that, in particular, it is a LARGE increase in price inflation (e.g. 1940 and 1974) that seems to lead to LARGE contemporaneous increases in the equity dividend yield. How can this relationship be explained? This illustrates one the key problems with time series modelling: there does not appear to be a simple linear relationship between these variables ideally, we want a system which allows for a positive correlation between the variables if there is a large change in I(t), but with little or no relationship otherwise but, such relationships cannot easily be modelled or parameterised

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for equity returns (4)

We require that Y (t) is positive ⇒ use log transform. Then, the model for the equity dividend yield at time t, Y (t), is: ln [Y (t)] = YW × I(t) + ln YMU + YN(t) where Y (t) is the equity dividend yield at time t I(t) is the force of price inflation over the year (t − 1, t) YN(t) = YA × YN(t − 1) + YE(t) is an AR(1) process independent of price inflation YE(t) = YSD × YZ(t) is the random component of the equity dividend yield process at time t

YSD is the (constant) standard deviation parameter of the equity dividend yield process YZ(t) is a standard N(0, 1) random variable

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

Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for equity returns (5)

Then, the (log) equity dividend yield is modelled as: a stationary AR(1) process (around a mean of ln YMU) plus a direct influence from the current level of price inflation Then, fitting to 1923-2009 data gives parameter estimates as follows: YW = 1.55 YMU = 0.0375 YA = 0.63 YSD = 0.155 As expected, we have YW > 0 – implying a positive correlation between the current levels of price inflation and equity dividend yield.

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for equity returns (6)

• Figure 27: Annual force of price inflation, I(t) and force of equity growth, K(t), in

UK, 1923-2009

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

Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for equity returns (7)

As might be expected, there is a positive correlation between the force of price inflation in any given year and the force of equity dividend growth in the same year – the correlation coefficient is c = 0.38. This seems reasonable, as equities are a real asset, such that: dividends are paid from company profits, which can be expected to grow in line with inflation (as well as real economic growth) However, in practice, it is unlikely that the full extent of any rise (or fall) in inflation would be reflected immediately in the level of dividends: growth in company profits may lag behind growth in inflation

thus, a weighted average of current and past inflation is used (so that the effect of a change in inflation feeds through more gradually)

however, in the textbflong term, we would expect a rise of x% in price inflation to feed through to give a rise of x% in share dividends

this is often referred to as the “principle of unit gain”

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for equity returns (8)

Also, in practice, companies often issue forecasts of the likely level of the next dividend payment: as a result, share prices often fall (and, hence, dividend yields rise) in advance of an actual fall in the level of dividends thus, we can expect the change in the level of equity dividend yield at time (t − 1), given by YE(t − 1), to have a negative correlation with the force of share dividend growth over the year (t − 1, t), given by K(t) And, it is common for companies not to pay out the full amount of the growth in earnings in any given year: rather, it is common to defer some of this profit distribution to use for dividends in future years (when earnings may be lower) thus, to reflect this inherent smoothing, we may expect a moving-average structure in the force of share dividend growth

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

Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for equity returns (9)

Thus, the force of equity dividend growth in year t, K(t), is given by: K(t) = DW × DM(t) + (1 − DW ) × I(t) + DMU + DY × YE(t − 1) + DB × DE(t − 1) + DE(t) where DM(t) = DD × I(t) + (1 − DD) × DM(t − 1) is an exponentially weighted average of current and past inflation DMU is the long-term mean force of real share dividend growth DY × YE(t − 1) is the effect from the change in the previous year’s share dividend yield DE(t) = DSD × DZ(t) is the random component of the process at time t

DZ(t) is a standard N(0, 1) random variable, and the additional term, DB × DE(t − 1), is the moving average component reflecting the smoothing of dividend payout

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for equity returns (10)

Then, fitting to 1923-2009 data gives parameter estimates as follows: DW = 0.43 DD = 0.16 DMU = 0.011 DY = −0.22 DB = 0.43 DSD = 0.07 As expected, the parameter DY is negative ⇒ a rise in the equity dividend yield may indicate an expected fall in future share dividends (resulting in an immediate fall in share prices). Also, some evidence that the use of a Normal distribution slightly under-estimates the likelihood of a large fall in share dividends and, hence, in share prices (e.g. a market crash due to fears of a recession).

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

Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for equity returns (11)

Standard error of DMU is large – 95% CI is (−1.1%, 3.3%). This parameter is crucial in determining the size of the textbfequity risk premium, such that: increasing the value of DMU will increase the ERP

• ver-estimating this parameter will bias the model towards equities

One of the key uses of stochastic ALMs is to assist with the framing of long-term investment strategy for a particular set of liabilities, but: if the model is biased towards equities, then the optimal portfolio will be

• ver-weight in equities

we consider the effect of this parameter (and others) on the results of simple stochastic ALM exercises in Lecture 6

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for equity returns (12)

The annual return on equities in year t, L PR(t), is given by: L PR(t) = PR(t) PR(t − 1) − 1 = P(t) + D(t) P(t − 1) − 1 Figure 28 shows two possible sample paths of L PR(t) for the period 2010-2029 (based on the real-word starting values). We can see that: the annual return on equities is very volatile from year to year

in the two simulations shown, the annual return varies between about −30% and 70% over the 20-year period

there is little evidence of any serial correlation

this can be confirmed from the auto-correlation function (ACF) why might this be as expected?

Also, from Figure 29, the distribution of L PR(t) is positively-skewed. Note also that, due to the sharp fall in dividends in 2009, expected return on equities in 2010 is somewhat lower than in other future years.

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

Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for equity returns (13)

• Figure 28: Two possible simulations of annual equity return, L

PR(t), over the period 2010-2029

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for equity returns (14)

• Figure 29: Distribution function for L

PR(1) – red, L PR(5) – green and L PR(20) – blue

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

Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for equity returns (15)

Similarly, the mean annual return on equities up to time t, GPR(t), is given by: GPR(t) = 100 × [( PR(t) PR(0) )

1 t

− 1 ] From Figures 30 and 31, the distribution of GPR(t) becomes significantly more stable as the time horizon increases, such that: we can be 90% sure that the equity return in the next year will be between −20.3% and 38.2%, and we can be 90% sure that the average equity return over the next 20 years will be between 4.6% and 15.8%. This reduction in long-term volatility is the rationale behind the use of deterministic assumptions in many areas of actuarial science. However, clearly, even in the long term, there is significant uncertainty in the average return on equities.

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for equity returns (16)

• Figure 30: Empirical distribution function of mean annual equity return up to time t,

GPR(t), for t = 0, 1, 2, . . . , 20

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

Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for equity returns (17)

• Figure 31: Distribution function for GPR(1) – red, GPR(5) – green and GPR(20) – blue

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for equity returns (18)

Then, for the mean annualised return on equities, GPR(t), we have:

time, t 1 2 5 10 20 50 mean rate of price inflation, GQ(t) 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% mean rate of equity return, GPR(t) E [GPR(t)] 6.40% 10.14% 10.89% 10.40% 10.08% 9.95% sd [GPR(t)] 17.98% 12.62% 7.17% 4.82% 3.42% 2.21% corr [GPR(t), GQ(t)] −0.20 −0.03 0.27 0.46 0.58 0.66

Table 3: Distribution of average rate of equity return up to time t, GPR(t), for basic Wilkie model

Long-term real return on equities of about 5.5% per annum. Equity return positively correlated with inflation in the long term (as equities are a real asset), but negatively correlated in the short term (as markets tend to dislike high inflation).

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

Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for bond returns (1)

• Figure 32: Annual force of price inflation, I(t) and long-term interest rate, C(t), in

UK, 1923-2009

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for bond returns (2)

As expected, long-term interest rate tends to rise when inflation rises. The long-term interest rate (or yield on undated fixed-interest bonds) at time t, C(t), depends on: market’s required real long-term yield

this is modelled as an AR(1) process, as it is likely that there will be a significant level of auto-correlation between interest rates in successive time periods

expected long-term future inflation

this is modelled using an exponential weighted-average of current and past inflation it is reasonable to expect the market’s expectations of future inflation to rise when current inflation rises

Again, as for the equity dividend yield, a log transform is used to ensure that the value of C(t) is positive.

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

Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for bond returns (3)

Then, the model for the long-term interest rate at time t, C(t), is: C(t) = CW × CM(t) + CR(t) where: CM(t) = CD × I(t) + (1 − CD) × CM(t − 1) is the exponentially weighted-average of current and past inflation at time t

reflecting the market expectation of long-term future inflation

CR(t) = CMU × exp [CN(t)]

as CW is usually taken to be equal to 1, CR(t) represents the “real” component of the long-term interest rate, C(t)

CMU represents the mean long-term “real” interest rate CN(t) = CA × CN(t − 1) + CY × YE(t − 1) + CE(t) is a zero-mean AR(1) process determining the long-term “real” interest rate at time t

the term CY × YE(t − 1) allows for a direct link between contemporaneous changes in the levels of the equity and bond markets

CE(t) = CSD × CZ(t) is the random component of the process at time t

CZ(t) is a standard N(0, 1) random variable

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for bond returns (4)

Then, fitting to 1923-2009 data gives parameter estimates as follows: CW = 1.0 CD = 0.045 CMU = 0.0223 CA = 0.92 CY = 0.37 CSD = 0.255 Low value of CD indicates that market expectation of long-term future inflation changes gradually over time, as we have: CM(t) = CD × I(t) + (1 − CD) × CM(t − 1) = CD ×

∞

∑

k=0

(1 − CD)k × I(t − k)

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

Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for bond returns (5)

Also, CA = 0.92 implies that there is very high auto-correlation in the long-term “real” interest rate process i.e. the long-term “real“ interest rate also changes gradually over time thus, long-term interest rate, C(t), will change gradually over time And, the parameter CY is positive - why? an unexpected rise in the share dividend yield at time t (i.e. YE(t) > 0) will, all other things being equal, lead to a rise in the long-term interest rate at time t

i.e. fall in equity prices can be expected to coincide with fall in bond prices

however, this relationship depends crucially on the fact that LARGE rises in the share dividend yield (e.g. in 1952, 1968 and 1974) lead to LARGE rises in the long-term interest rate

i.e. in extreme circumstances, a crash can be expected in both markets

thus, in practice, we may choose to set this parameter to zero

• r, even use a negative value – why?

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for bond returns (6)

• Figure 33: One possible simulation of force of price inflation, I(t), and long-term

interest rate, C(t), over the period 2010-2029

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

Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for bond returns (7)

• Figure 34: Distribution function for C(1) – red, C(5) – green and C(20) – blue

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for bond returns (8)

• Figure 35: Empirical distribution function of long-term interest rate at time t, C(t),

for t = 0, 1, 2, . . . , 20

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

Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for bond returns (9)

From Figure 33, we can see that: long-term interest rate, C(t), is generally higher and more stable than force of price inflation, I(t), and long-term interest rate, C(t), tends to move in line with the current value

• f price inflation, I(t)
• r, more correctly, expectations of future price inflation (which are strongly

influenced by current and recent values of price inflation)

From Figures 34 and 35, we can see that: distribution of C(t) is positively skewed – as a result of log transformation C(t) can be expected to increase over time (from C(2009) = 4.51%) to the long-term mean (approximately equal to QMU + CMU = 6.53%)

however, as CA is close to one, mean-reversion happens very slowly (in fact, it takes around 60 years to reach a stable long-term mean)

the volatility (or uncertainty) in the long-term interest rate, C(t), increases as the time horizon, t, increases (i.e. the “increasing funnel of doubt”)

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for bond returns (10)

The annual return on fixed-interest bonds in year t, L CR(t), is given by: L CR(t) = CR(t) CR(t − 1) − 1 = C(t − 1) × [ 1 C(t) + 1 ] − 1 Figure 36 shows one possible sample path of both L CR(t) and L PR(t) for t = 1, 2, . . . , 20. We can see that: the annual return on fixed-interest bonds is significantly less volatile than that on equities, and as for equities, the annual return on fixed-interest bonds shows little evidence of any serial correlation

⇒ as with equity market, fixed-interest bond market is broadly efficient

From Figure 37, we can see that, as long-term interest rates can be expected to rise in the near future - why? - the return on fixed-interest bonds can be expected to fall in the next few years (before rising gradually to reach the stable long-term mean position).

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

Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for bond returns (11)

• Figure 36: One possible simulation of annual equity return, L

PR(t), and annual return

• n fixed-interest bonds, L

CR(t), over the period 2010-2029

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for bond returns (12)

• Figure 37: Empirical distribution function of annual return on fixed-interest bonds in

year t, L CR(t), for t = 0, 1, 2, . . . , 20

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

Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for bond returns (13)

Also, the mean annual return on fixed-interest bonds up to time t, GCR(t), is given by: GCR(t) = 100 × [( CR(t) CR(0) )

1 t

− 1 ] From Figure 38, as with GPR(t), the distribution of GCR(t) becomes significantly more stable as the time horizon increases. Also, from Figures 39 and 40, we can see that, even in the long term, mean annual returns on equities are significantly more volatile than mean annual returns on fixed-interest bonds: we can be 95% sure that the return on fixed-interest bonds in the next year will be between −6.8% and 17.6%, and

corresponding values for equities were −20.3% and 38.2%

we can be 95% sure that the average return on fixed-interest bonds over the next 20 years will be between 2.2% and 6.1%

corresponding values for equities were 4.6% and 15.8%

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for bond returns (14)

• Figure 38: Empirical distribution function of mean annual gilt return up to time t,

GCR(t), for t = 0, 1, 2, . . . , 20

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

Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for bond returns (15)

• Figure 39: Distribution function for GCR(1) – red and GPR(1) – blue

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for bond returns (16)

• Figure 40: Distribution function for GCR(20) – red and GPR(20) – blue

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

Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for bond returns (17)

Then, for the mean annualised return on fixed-interest bonds, GCR(t), we have:

time, t 1 2 5 10 20 50 mean rate of price inflation, GQ(t) 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% mean rate of equity return, GPR(t) E [GPR(t)] 6.40% 10.14% 10.89% 10.40% 10.08% 9.95% sd [GPR(t)] 17.98% 12.62% 7.17% 4.82% 3.42% 2.21% corr [GPR(t), GQ(t)] −0.20 −0.03 0.27 0.46 0.58 0.66 mean rate of return on long-dated gilts, GCR(t) E [GCR(t)] 5.45% 4.64% 3.63% 3.45% 4.04% 5.38% sd [GCR(t)] 7.44% 5.77% 3.85% 2.31% 1.22% 1.12% corr [GCR(t), GQ(t)] −0.56 −0.64 −0.69 −0.64 −0.37 0.36 corr [GCR(t), GPR(t)] 0.26 0.15 −0.09 −0.20 −0.11 0.27

Table 4: Distribution of average rate of return on fixed-interest bonds up to time t, GCR(t), for basic Wilkie model

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Modelling asset returns Wilkie model for equity returns Wilkie model for bond returns

Wilkie model for bond returns (18)

Thus, the long-term real return on fixed-interest bonds is just over 1% per annum (which increases to about 2.5% per annum if we extend the time horizon to over 100 years)

• cf. long-term real return of equities of about 5.5% per annum

i.e. equities can be expected to produce a return of about 3% per annum higher than fixed-interest bonds in the long term However, return on equities in considerably more volatile (or risky) than return

• n fixed-interest bonds (even over the very long term).

Also, as would be expected, there is a negative correlation between return on fixed-interest bonds and inflation in the short and medium term high inflation means that interest rates will rise (and, hence, price of fixed-interest bonds will fall ⇒ return on bond portfolio will be lower) however, in very long term, high inflation ⇒ future income is reinvested at higher yield (or lower price) ⇒ return on fixed-interest bond portfolio rises

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

Lecture 4 Lecture 5 Lecture 6 Dynamic ALM using the Wilkie model ARCH and regime-switching models for asset returns Economic-based models

Dynamic ALM using the Wilkie model (1)

In Lecture 6, we will use the model for simple stochastic asset-liability modelling exercises. However, if we use a dynamic investment approach in this case, then we can

• btain some very misleading results

a “dynamic investment approach” means allowing for the ability to change the investment strategy adopted at each future date in light of actual experience up to that point why might this be a desirable feature of an actuarial model? However, the Wilkie model does not follow the efficient market hypothesis – because it is fitted to real data that does not comply with EMH! Then, as a result of the mean-reversion identified previously, it is relatively straightforward to construct rules governing the dynamic asset allocation that can be expected to generate significantly higher returns than alternative static asset allocations.

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Dynamic ALM using the Wilkie model ARCH and regime-switching models for asset returns Economic-based models

Dynamic ALM using the Wilkie model (2)

For example, suppose that the current long-term interest rate, C(t), is above the historic long-term mean (of, according to our stochastic asset model, approximately QMU + CMU = 6.53%). Then, as a result of the mean-reversion implicit in the Wilkie model, we can expect long-term interest rates to fall in future ⇒ we can expect long-dated bond prices to rise. Thus, we can construct a dynamic investment strategy to buy fixed-interest bonds in this case (and benefit from the expected future price rise). However, an efficient market would NOT allow this! If it was as easy as this to make profits, all investors would by fixed-interest bonds now (and the price would rise immediately, thereby wiping out the future gains!). Thus, whilst such a strategy would actually have proved successful over the period 1923-2009, it is generally considered imprudent to assume that such excess returns could continue to be achieved in future and, as a result, we have to careful when implementing dynamic asset allocations in conjunction with a stochastic asset model.

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

Lecture 4 Lecture 5 Lecture 6 Dynamic ALM using the Wilkie model ARCH and regime-switching models for asset returns Economic-based models

ARCH and regime-switching models for asset returns (1)

When carrying out stochastic modelling exercises, it is absolutely crucial to investigate the sensitivity of the final results (e.g. the cost of a guarantee or the optimal investment strategy) to changes in the structure and/or parameters of the stochastic asset model. Thus, we should also consider alternative models for the asset returns based

• n the ARCH and regime-switching methodology discussed above:

ARCH processes are not required for the share dividend yield, force of share dividend growth and long-term interest rate series

thus, the ARCH model uses the ARCH price inflation process with the simple linear models for Y (t), K(t) and C(t) above

regime-switching model has two separate models for each process

however, the threshold determining which set of parameters to use is based

• n the current value of the price inflation process

so, each process being modelled is in either the “normal” volatility or “high” volatility state at the same time

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Dynamic ALM using the Wilkie model ARCH and regime-switching models for asset returns Economic-based models

ARCH and regime-switching models for asset returns (2)

Then, for the ARCH model, we have:

time, t 1 2 5 10 20 50 mean rate of price inflation, GQ(t) 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% mean rate of equity return, GPR(t) E [GPR(t)] 6.78% 10.26% 10.65% 9.94% 9.44% 9.18% sd [GPR(t)] 18.20% 12.70% 7.39% 5.87% 4.31% 4.14% corr [GPR(t), GQ(t)] −0.24 −0.06 0.27 0.58 0.73 0.86 mean rate of return on long-dated gilts, GCR(t) E [GCR(t)] 5.89% 5.21% 4.36% 4.03% 4.32% 5.25% sd [GCR(t)] 7.86% 6.52% 5.73% 3.45% 1.76% 1.37% corr [GCR(t), GQ(t)] −0.61 −0.70 −0.75 −0.68 −0.45 0.35 corr [GCR(t), GPR(t)] 0.30 0.18 −0.12 −0.29 −0.22 0.38

Table 5: Distribution of average rate of return on equities and fixed-interest bonds for ARCH model

Then, we can compare these results with Table 4 above for the basic Wilkie model.

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

Lecture 4 Lecture 5 Lecture 6 Dynamic ALM using the Wilkie model ARCH and regime-switching models for asset returns Economic-based models

ARCH and regime-switching models for asset returns (3)

Similarly, for the regime-switching model, we have:

time, t 1 2 5 10 20 50 mean rate of price inflation, GQ(t) 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% mean rate of equity return, GPR(t) E [GPR(t)] 2.28% 8.55% 12.13% 13.49% 14.15% 14.48% sd [GPR(t)] 15.50% 11.55% 6.52% 4.45% 3.06% 1.94% corr [GPR(t), GQ(t)] −0.05 −0.01 0.13 0.29 0.42 0.53 mean rate of return on long-dated gilts, GCR(t) E [GCR(t)] −0.63% 0.76% 1.41% 2.41% 4.01% 6.03% sd [GCR(t)] 6.45% 5.28% 3.67% 2.13% 1.18% 1.16% corr [GCR(t), GQ(t)] −0.49 −0.59 −0.65 −0.56 −0.06 0.62 corr [GCR(t), GPR(t)] 0.25 0.18 0.06 −0.01 0.14 0.40

Table 6: Distribution of average rate of return on equities and fixed-interest bonds for regime-switching model

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Dynamic ALM using the Wilkie model ARCH and regime-switching models for asset returns Economic-based models

ARCH and regime-switching models for asset returns (4)

Compared with the basic model, the key features of the output are: the ARCH model has a similar expected long-term real return on equities (of about 5.5% p.a.) and a slightly higher expected long-term real return

• n fixed-interest bonds (of about 1.8% p.a., increasing to about 2.5% p.a.

for a time horizon of over 100 years)

however, the higher long-term volatility in price inflation feeds through to give significantly higher investment risk (as measured by the variance of the return) for both fixed-interest bonds and (in particular) equities

the regime-switching model has a much higher expected long-term real return on equities (of almost 10% p.a.) and a similar expected long-term real return on fixed-interest bonds (of about 1.4% per annum, again increasing to about 2.5% p.a. for a time horizon of over 100 years)

however, in the long-term, the equity risk is considerably lower than for the standard model (i.e. less than 2.0% per annum) and the fixed-interest bond risk is similar (i.e. about 1.1% per annum) thus, in any asset-liability modelling exercise, the regime-switching model is likely to favour equities significantly more than the other two models

the variance-covariance structures are similar for each model

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

Lecture 4 Lecture 5 Lecture 6 Dynamic ALM using the Wilkie model ARCH and regime-switching models for asset returns Economic-based models

Economic-based models (1)

The Wilkie model uses a statistical approach to model the key economic variables. In theory, each variable could interact with all others. However, only those that were statistically significant (and could be rationalised by an economic justification) were included in the final model. This has lead to criticism that the model is over-parameterised and, more importantly, that it is inconsistent with established economic theories (e.g. efficient market hypothesis, rational expectations hypothesis). Also, it is difficult to use the model to produce market-consistent valuations (as the derivation of appropriate state-price deflators cannot be readily determined).

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Dynamic ALM using the Wilkie model ARCH and regime-switching models for asset returns Economic-based models

Economic-based models (2)

An alternative approach is to construct an economic model based on a hypothetical environment in which certain economic theories are assumed to hold e.g. markets can be assumed to be efficient and all investors assumed to behave rationallly Examples of such models include: the jump-equilibrium model proposed by Smith (1996), and the Timbuk model proposed by Smith and Southall (2001) the economic scenario generators (ESGs) used by Barrie and Hibbert (who are one of the leading providers of stochastic ALM software) These models are also constructed to allow simple market-consistent valuations of future cash flows (using readily-available risk-neutral probability measures and/or state price deflators).

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Economic-based models (3)

However, such models are likely to fit past data less well than statistical models. Model structure (and parameterisation) is, by design, a subset of all possible models (as we only consider structures which satisfy the desired theories). And, economic theories are just that ... they may not be borne out in practice! Indeed, the recent financial crisis has caused many observers to question the wisdom of the efficient markets hypothesis. The main advantage of such models is that they can be readily used to give market-consistent valuations for non-tradable cash flows. However, the main disadvantage is that they may not model the actual cash flows arising as well as a statistical model (as so may be less useful for assisting with financial planning decisions).

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Using SAMs to price investment guarantees Using SAMs to determine strategic asset allocation

Using SAMs to price investment guarantees (1)

We can use stochastic asset-liability modelling techniques to price investment guarantees embedded in many life insurance contracts. Consider a single premium unit-linked endowment assurance with a term of 10 years. Suppose that the single premium is P. Let UFt denote value of unit fund at time t. On death before maturity, benefit is the unit fund (at the end of the year of death) subject to a minimum of 1.5P. On survival to maturity, benefit is the unit fund (at maturity) subject to a minimum of 1.5P. Thus, the policyholder is guaranteed to make a return of at least 50% over the term of the contract.

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Using SAMs to price investment guarantees (2)

For simplicity, we ignore expenses and assume that the cost of the guarantee is covered by an additional single premium, G, payable at outset. However, this is unlikely to be the approach adopted in practice. Question: In practice, how is the cost of the guarantee more likely to be paid for? Then, the benefit received in year t is given by: Bt = max (UFt, 1.5P) = UFt + max (0, 1.5P − UFt) Then, the cost of guarantee on death in year t is: G d

t = max (0, 1.5P − UFt)

And, the cost of guarantee on survival to maturity at time 10 is: G s

10 = max (0, 1.5P − UF10)

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Using SAMs to price investment guarantees Using SAMs to determine strategic asset allocation

Using SAMs to price investment guarantees (3)

Then, the cost of the guarantee represents the pay-off from a put option with underlying asset equal to the unit fund. However, we cannot use standard

• ption pricing techniques to determine the price to be charged. Why not?

Instead, we will use a stochastic ALM approach. Assume mortality is deterministic and consider a life of exact age 50. Then, expected cost of guarantee on death in year t is:

t−1∣q50 × G d t

And, expected cost of guarantee on survival to maturity is:

10p50 × G s 10

Suppose that the unit fund is invested entirely in equities. Then, using the notation of the Wilkie model, we have: UFt+1 = UFt × PR(t + 1) PR(t) where UF0 = P = 1

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Using SAMs to price investment guarantees (4)

Then, one possible realisation of the investment experience over the next 10 years is:

time, t PR(t) UFt G d

t

G s

t t−1∣q50 tp50

Gt 1.0000 1.0000 1 0.9740 0.9740 0.5260 0.00251 0.00132 2 0.8937 0.8937 0.6063 0.00280 0.00170 3 1.4415 1.4415 0.0586 0.00313 0.00018 4 2.4283 2.4283 0.0000 0.00351 0.00000 5 2.6098 2.6098 0.0000 0.00393 0.00000 6 3.6256 3.6256 0.0000 0.00440 0.00000 7 4.1657 4.1657 0.0000 0.00492 0.00000 8 4.6153 4.6153 0.0000 0.00551 0.00000 9 3.8941 3.8941 0.0000 0.00616 0.00000 10 4.1985 4.1985 0.0000 0.0000 0.00688 0.95626 0.00000

where Gt = t−1∣q50 × G d

t + tp50 × G s t represents the expected cash flow in

respect of the guarantee in year t.

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Using SAMs to price investment guarantees Using SAMs to determine strategic asset allocation

Using SAMs to price investment guarantees (5)

Then, discounting the total cash flows in respect of the guarantee gives us the cost of the guarantee (for this particular scenario). But, what rate of interest should we use to discount the cash flows? A traditional actuarial approach would suggest using the expected return on the underlying assets. However, from Lecture 1, we should use the state-price deflator approach to give a “market-consistent” cost. But, whilst we can derive an explicit expression for the deflator using the log-Normal model, this is much more difficult using a more realistic stochastic asset (such as the Wilkie model). Instead, we use an appropriate 10-year risk-free rate of interest – i.e. current yield on long-dated fixed-interest bonds of C(0) ≡ I(2009) = 4.51% p.a. Then, in this case, the cost of the guarantee is: G =

10

∑

t=1

Gtv t

4.51% = 0.002979

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

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Using SAMs to price investment guarantees (6)

However, in the above scenario, the unit fund performs well over the period (and the guarantee only bites on death in the first three years). Consider an alternative scenario, where the unit fund performs much less well, given by:

time, t PR(t) UFt G d

t

G s

t t−1∣q50 tp50

Gt 1.0000 1.0000 1 0.8958 0.8958 0.6042 0.00251 0.00151 2 1.2398 1.2398 0.2602 0.00280 0.00073 3 1.1022 1.1022 0.3978 0.00313 0.00125 4 1.6109 1.6109 0.0000 0.00351 0.00000 5 1.2809 1.2809 0.2191 0.00393 0.00086 6 1.2007 1.2007 0.2993 0.00440 0.00132 7 1.1415 1.1415 0.3585 0.00492 0.00176 8 0.9865 0.9865 0.5135 0.00551 0.00283 9 0.8890 0.8890 0.6110 0.00616 0.00376 10 0.9421 0.9421 0.5579 0.5579 0.00688 0.95625 0.53736

In this case, we have G = 0.35641 (i.e. over 35% of the initial premium).

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Using SAMs to price investment guarantees Using SAMs to determine strategic asset allocation

Using SAMs to price investment guarantees (7)

• Figure 41: Empirical distribution function for cost of guarantee at time 0, G

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Using SAMs to price investment guarantees (8)

Repeating this process a large number of times, we can obtain an empirical distribution for the random variable, G, as shown in Figure 41. Then, expected cost of guarantee is E(G) = 0.0200 (per unit premium). However, clearly, the distribution of G is very skewed – indeed, the mean of E(G) = 0.0200 corresponds to approximately the 91st percentile of the distribution. Thus, charging an additional single premium of E(G) = 0.0200 will ensure that we have enough to cover the cost of the guarantee in approximately 91%

• f future outcomes.

However, this leaves a probability of insolvency of 100 − 91 = 9%. We may consider this to be unacceptably high – particularly as, unlike the mortality risk in a portfolio of life insurance business, this risk cannot be reduced by selling a large volume of business! Why not?

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Using SAMs to price investment guarantees Using SAMs to determine strategic asset allocation

Using SAMs to price investment guarantees (9)

Alternatively, we could use a higher percentile of the distribution of G: G(0.95) = 0.1420 (per unit premium) would give a probability of insolvency of 5%, and G(0.99) = 0.3370 (per unit premium) would give a probability of insolvency of 1% The disadvantage of this approach is that it gives the same weight to each scenario in which insolvency occurs (i.e. it does not take account of the length

• f the tail beyond the chosen percentile).

Thus, it may be better to use more sophisticated risk measure, such as the expected shortfall (see Subject CT8): e.g. with an expected shortfall of 0.5% (per unit premium), the cost of the guarantee is X such that: E [max (0, G − X)] = 0.005 ⇒ X = 0.1737 (per unit premium)

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Using SAMs to price investment guarantees (10)

Question Given the uncertainty resulting from the recent financial crisis, what happens if we assume a higher (and more volatile) inflation environment in future? Thus, we repeat the above using QMU = 0.06 (rather than 0.043) and QSD = 0.05 (rather than 0.04). Then, we have: E(G) = 0.0163 (0.0200) G(0.95) = 0.0915 (0.1420) E [max (0, G − X)] = 0.005 ⇒ X = 0.1446 (0.1737) For comparison, the values for the original model given in brackets. Question: Why do we get a lower cost for the guarantee (under all risk measures) in a high and more volatile inflation environment?

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Using SAMs to price investment guarantees Using SAMs to determine strategic asset allocation

Using SAMs to price investment guarantees (11)

Question What happens if we change the equity risk premium (ERP)? The equity risk premium is the additional return that can be expected on equities to compensate for the additional risk faced (as measured by the volatility of the return). From Lecture 4, the parameter DMU is crucial in defining the level of the ERP increasing (or decreasing) DMU ⇒ higher (or lower) ERP ⇒ equities become a more (or less) attractive investment Also, as discussed in Lecture 4, this parameter has a high stand error. Thus, we consider the effect of both increasing the value from 0.011 to 0.033 and decreasing it to −0.011. Question: What is likely to be the effect on the cost of the guarantee of increasing (or decreasing) the equity risk premium?

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Using SAMs to price investment guarantees (12)

Then, for DMU = 0.033, we have: E(G) = 0.0080 (0.0200) G(0.95) = 0.0098 (0.1420) E [max (0, G − X)] = 0.005 ⇒ X = 0.0088 (0.1737) And, for DMU = −0.011, we have: E(G) = 0.0455 (0.0200) G(0.95) = 0.2954 (0.1420) E [max (0, G − X)] = 0.005 ⇒ X = 0.2974 (0.1737) As expected, a higher equity risk premium means that the guarantee is less likely to “bite” ⇒ lower cost of guarantee (under all risk measures). However, it is worth noting that the results are very sensitive to the value chosen for DMU and, as noted above, there is a considerable degree of uncertainty as to the ‘true’ value of this parameter.

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Using SAMs to price investment guarantees Using SAMs to determine strategic asset allocation

Using SAMs to price investment guarantees (13)

Question Given the current economic uncertainty, what happens if we increase the volatility of equity returns? The parameter DSD controls the volatility of equity returns. What is likely to be effect on the cost of the guarantee of increasing DSD from 0.07 to 0.105 (i.e. an increase of 50%)? Then, we have: E(G) = 0.0371 (0.0200) G(0.95) = 0.2804 (0.1420) E [max (0, G − X)] = 0.005 ⇒ X = 0.3115 (0.1737) Thus, as would be expected, increasing the volatility of equity returns increases the cost of the guarantee (under all risk measures), as the guarantee is more likely to bite.

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Using SAMs to price investment guarantees (14)

Question What happens if we change the investment strategy of the unit fund? Suppose that the unit fund is invested 50% in equities and 50% in fixed-interest bonds (rather than 100% in equities). Then, we have: E(G) = 0.0138 (0.0200) G(0.95) = 0.0809 (0.1420) E [max (0, G − X)] = 0.005 ⇒ X = 0.0655 (0.1737) Thus, as would be expected, if the fund is invested 50% in equities and 50% in fixed-interest bonds, then the cost of the guarantee is considerably reduced (as the volatility of the unit fund is much lower). However, from the point of view of the policyholder, the expected amount of the maturity benefit will also be reduced.

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Using SAMs to price investment guarantees Using SAMs to determine strategic asset allocation

Using SAMs to price investment guarantees (15)

We can also explore the effect on the cost of the guarantee of: changing the term of the contract

if the term is reduced to 5 years, what is likely to be the effect on the cost of the guarantee?

changing the level of the guarantee

if the guaranteed death and maturity benefit is reduced to P, what is likely to be the effect on the cost of the guarantee?

Also, using the same techniques, we can easily explore the cost of more complicated structures for the guaranteed benefit, such as: embedded Asian options, where the benefit on death (or maturity) is based on the average of the unit fund over a specified period (e.g. the last two years or, even, the whole term) embedded look-back options, where the benefit at maturity is based on the maximum fund value over the term of the investment

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Using SAMs to determine strategic asset allocation (1)

We can also use stochastic asset-liability modelling techniques to assist in reserving and strategic (or long-term) asset allocation decisions. Consider a defined-benefit pension scheme that provides a lump sum benefit

• n age retirement at age 65 of 1

4× final salary for each year of service.

Consider a current active member of exact age 45 with current salary of £25,000 per annum and current past service of 20 years. Then, the actuary carrying out the statutory (triennial) valuation of this scheme has to answer two crucial (and related) questions: (1) How much money should the scheme hold today in respect of the accrued (or past service) liabilities of this member? (2) How should these funds be invested in the period prior to retirement (to minimise the cost of the promised benefits subject to maintaining an acceptable level of risk)?

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Using SAMs to price investment guarantees Using SAMs to determine strategic asset allocation

Using SAMs to determine strategic asset allocation (2)

In a deterministic environment, ignoring salary scale and other decrements prior to retirement, the present value of the accrued liability is: V = 1 4 × 20 × 25, 000 × (1 + j)65−45 × l45 l65 × v 65−45

i

where i is the assumed rate of investment return, and j is the assumed rate of salary growth Now, based on 1923-2009 data, we have: the average rate of price inflation is about 4% per annum, the average rate of salary growth is about 5.5% per annum, the average rate of return on equities is about 9.5% per annum, and the average rate of return on fixed-interest bonds is about 6.5% per annum

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Using SAMs to determine strategic asset allocation (3)

Then, suppose the actuary decides on the following prudent valuation basis: assumed rate of investment return, i, of 8% per annum assumed rate of salary growth, j, of 6.5% per annum mortality prior to retirement of AM92 ultimate Then, the present value of the accrued liabilities is V = 85, 050. And, given the nature of the liability, the actuary is likely suggest an asset allocation strategy that will involve mainly real assets (e.g. equities) with some diversification into nominal assets (e.g. fixed-interest bonds). Note that, in practice, a defined-benefit pension scheme will usually provide a whole life annuity on retirement (rather than a lump sum) and this annuity will usually be valued using the expected yield on long-dated fixed-interest bonds at retirement. How is this likely to change the suggested asset allocation strategy?

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Using SAMs to determine strategic asset allocation (4)

In practice, as the assumptions made by the actuary will be prudent, the actual cost of providing the benefits is expected to be lower than the reserve calculated above. However, even if we hold a reserve of V = 85, 050, there is still a possibility that this will prove to be insufficient to meet the promised liabilities. But, how high is this probability? Less than 50% ... but is it 25%, 10% or even 1%??? Also, how can we determine an optimal investment strategy to be adopted? A stochastic asset-liability modelling approach can help to answer these questions. We consider the current required reserve, V , to be a random variable. For simplicity, suppose that the fund can invest only in equities or fixed-interest bonds. Let z be the proportion of the fund invested in equities, with (1 − z) invested in fixed-interest bonds.

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Using SAMs to determine strategic asset allocation (5)

Then, using the Wilkie model notation, we can define the accumulated value

• f time t of an initial fund of 1, denoted by MR(t), as:

MR(t) = MR(t − 1) × [ z × PR(t) PR(t − 1) + (1 − z) × CR(t) CR(t − 1) ] Note that this assumes that the fund is re-balanced annually in the defined proportions. The Wilkie model for the force of salary inflation in year t, J(t), is: J(t) = WW 1 × I(t) + WW 2 × I(t − 1) + WMU + WSD × WZ(t) i.e. current force of salary inflation based on force of price inflation in current and previous year plus real salary inflation of WMU plus a noise term Then, we can define an earnings index at time t, W (t), as: W (t) = W (t − 1) × exp [J(t)] with W (0) = 1.0

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Using SAMs to price investment guarantees Using SAMs to determine strategic asset allocation

Using SAMs to determine strategic asset allocation (6)

Then, for each simulation, the actual amount required to be held now to meet the liability at retirement is a random variable given by: V = 1 4 × 20 × 25, 000 × W (20) W (0) × l65 l45 × MR(0) MR(20) i.e. actual benefit on retirement (based on actual salary growth up to retirement) discounted at actual rate of return on assets prior to retirement Note that, if the scheme provides an annuity on retirement (rather than a lump sum), we can value this at retirement using the random variable C(20) – i.e. the then current yield on long-dated fixed-interest bonds. Then, we are using a state-price deflator (or “stochastic discount factor”) equal to the actual return on the fund over the period up to retirement. However, as seen in Lecture 1, this approach will not give a “market-consistent” value. But, as no such market exists, what does “market-consistent” mean here?

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Using SAMs to determine strategic asset allocation (7)

• Figure 42: Empirical distribution function of required reserve, V , based on asset

allocation of 50% in equities and 50% in fixed-interest bonds

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Using SAMs to determine strategic asset allocation (8)

Then, for any chosen asset allocation strategy, repeating this a large number of times gives an empirical distribution for the random variable, V – the current reserve required – as shown in Figure 42. Then, assuming that 50% of the fund is invested in equities (with the remainder in fixed-interest bonds), we have: E(V ) = 81, 386 However, to ensure a reasonable level of security, we could consider holding a reserve equal to a higher percentile of the distribution. For example, holding a reserve equal to the 95th percentile of the distribution

• f V , denoted by V (0.95), will ensure that the fund can meet the liability at

retirement in 95% of all possible future outcomes. Then, we have: V (0.95) = 139, 865 ⇒ Pr (insolvency) = 5%

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Lecture 4 Lecture 5 Lecture 6 Using SAMs to price investment guarantees Using SAMs to determine strategic asset allocation

Using SAMs to determine strategic asset allocation (9)

Note that, the deterministic reserve calculated above (i.e. V = 85, 050) corresponds to approximately the 61st percentile of the distribution of V ⇒ probability that, after investing for 20 years, this amount is insufficient to meet the liability as it falls due is approximately 39%! Or, we could again use a more sophisticated risk measure such as the expected shortfall: with a reserve of V = 85, 050, the expected shortfall is: E [max (0, V − 85, 050)] = 10, 390 with a reserve of V (0.95) = 139, 865, the expected shortfall is: E [max (0, V − 139, 865)] = 1, 084 We can also explore the effect of changing the asset allocation strategy. Given the nature of the liability, it is likely that more of the fund should be invested in equities. Why?

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Using SAMs to price investment guarantees Using SAMs to determine strategic asset allocation

Using SAMs to determine strategic asset allocation (10)

Then, we can find the optimal asset allocation strategy such that: for a given level of risk (as measured by the chosen risk measure), this strategy allows us to hold the lowest reserve

z E (V ) sd (V ) V (0.95) 0% 164, 938.90 77, 695.00 310, 144.80 10% 139, 669.70 61, 329.50 253, 937.90 20% 119, 500.50 47, 948.96 207, 211.30 30% 103, 744.50 39, 354.70 175, 179.80 40% 91, 235.84 33, 508.21 151, 068.00 50% 81, 385.84 30, 459.88 139, 864.80 60% 73, 411.79 28, 663.63 127, 249.90 70% 67, 056.17 28, 216.68 119, 067.80 80% 62, 032.29 28, 145.77 115, 198.50 90% 57, 926.00 28, 707.29 113, 279.70 100% 55, 059.73 30, 439.85 114, 263.60

E(V ) increases as proportion of fund invested in equities, z, decreases. Why? But, for V (0.95), we can represent the change graphically (see Figure 43).

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

Lecture 4 Lecture 5 Lecture 6 Using SAMs to price investment guarantees Using SAMs to determine strategic asset allocation

Using SAMs to determine strategic asset allocation (11)

• Figure 43: Effect of investment strategy on required reserve, using

Pr (insolvency) = 5%

Dr Douglas Wright, Cass Business School Stochastic asset models for actuarial use – Part II Lecture 4 Lecture 5 Lecture 6 Using SAMs to price investment guarantees Using SAMs to determine strategic asset allocation

Using SAMs to determine strategic asset allocation (12)

We can draw a best-fitting curve through the points (with each point on the curve having the same level of risk – i.e. probability of insolvency of 5% – for the corresponding reserve, V (0.95), and asset allocation). Then, the optimal asset allocation is given by the minimum of this curve: approximately 90-95% of fund should be invested in equities (with the balance in fixed-interest bonds), and the corresponding reserve required to be held in respect of this liability is V ≈ 112, 000 Alternatively, we could repeat the above exercise using a risk measure such as the expected shortfall – e.g. for each asset allocation, we determine the reserve, X, such that: E [max (0, V − X)] = 1, 084 where, from above, with an asset allocation of 50% in equities and 50% in fixed-interest bonds, we have X = 139, 865.

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

Lecture 4 Lecture 5 Lecture 6 Using SAMs to price investment guarantees Using SAMs to determine strategic asset allocation

Using SAMs to determine strategic asset allocation (13)

Or, we may decide that a risk level of 5% (as measured by the probability of insolvency) is too high – then, we could repeat the above exercise with a risk level of 1% – i.e. use V (0.99) rather than V (0.95). If we reduce the level of risk faced, what effect is this likely to have on size of the reserve required? And what is likely to be the effect on the

• ptimal asset allocation?

As for the life insurance investment guarantee above, it is important to assess the sensitivity of the results to changes in some of the key parameters in the model (e.g. DMU) or to using a different stochastic asset model such as: ARCH model has a similar long-term real return on both equities and fixed-interest bonds to those for the standard model (but with a significantly higher long-term variance of return in both cases)

what effect is this likely to have on the level of the required reserve and

• n the optimal asset allocation?

regime-switching model has a significantly higher long-term real return

• n equities (and a lower variance of return)

what effect is this likely to have on the level of the required reserve and

• n the optimal asset allocation?

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