The Problem to be Solved - The Inflation Risk Model

The Inflation Risk Model

3.1 The Problem to be Solved

In this chapter, we model two assets world with the presence of inflation. We are interested in the post–retirement period of the member’s life and we investigate financial gains/losses due to optimal/suboptimal behaviour in postretirement period.

3.1.1 Economic Environment

Let us first explain the economic environment that is represented by the model.

We assume two sources of randomness in our model: one from the risky rate on equity investment and the other one from inflation. We assume a constant real interest rate. We have two assets: one risky asset (equities) and one risk free asset (cash).

We assume that the pensioner has income coming from social security, and that this income is constant in real terms. Regarding consumption, we assume that he consumes part of his available assets at the beginning of each year, and we assume that this amount is not subject to inflation during the period shorter than one year. The pensioner draws utility from consumed amounts and possibly from amounts bequeathed to his heirs. He draws utility from the amounts in real terms only.

After receiving income and consuming part of his pension wealth and income, the pensioner can invest his available assets into the risky and risk free asset at his own discretion.

We assume the presence of real and nominal annuities on the market. Real annuity provides constant income in real terms, while nominal annuity provides constant income in nominal terms. Having access to each type of annuity at each age is the most general case and obviously, the pensioner can act optimally in this environment and obtain the biggest gains. However, we will investigate this market with different limitations in order to investigate the importance of these limitations. Depending on constraints, the pensioner can be in the market where only real or only nominal annuities are available. It can also be that only real annuities are available and only at certain ages. There are many combinations and we will choose what we think are appropriate to shed light on the significance of having access to each of them.

Under our assumption about the market, the pensioner who takes nominal annuities has no protection against the risk of inflation. His income from a nominal annuity will be subject to inflation risk and will diminish in time if inflation is positive. However, nominal annuities provide better income in both real and nominal terms in the early years after purchasing nominal annuities. As we will see later, the pensioner will still choose optimally some nominal annuities and expose himself to the risk of inflation.

We assume that the member can annuitise his pension wealth only, and that the very first income from an annuity is receivable after one year time. So, at the beginning of each year of life, he annuitises the available pension wealth, receives income from social security and annuities bought in earlier years, consumes part of the remaining amount and invests the rest.

We will present our model and results in real terms. However, one should be aware that this is just for presentational reasons, and we will actually convert from nominal to real values in order to present the results more clearly.

We work in discrete time. We assume that postretirement decumulation process starts at age t=65, and finishes at age t=100. We assume that the maximum member’s age is 99, i.e. no member will be alive at age 100. The decumulation process lasts for 35 years. If the bequest motive exists, the pensioner aged 99 will consume part of his assets and the rest will be invested and bequeathed when he is going to die during that year. Otherwise, he will consume everything at age 99 and nothing will be left for investing. In the earlier periods, the pensioner consumes part of his available assets, uses one part for purchasing real annuities, one part for purchasing nominal annuities and invests the rest. We take the duration of one period to be one year. A slight modification would allow other lengths of each period. As we will see, the solution to

the problem follows the same pattern for different periods. That is why it is useful to investigate one representative period and then the solution for the whole problem can be derived from the solution of one representative period. We will always denote random variables with a ∼ sign above the name of variable. Graphical presentation of the most important variables appearing in our model is given as follows

State (information) variables Wt is pension wealth, Y_t is income,

dt is percentage of the real income received from nominal annuity, I_t₋₁ is inflation

W65 W₆₆ … W_t W_t₊₁ … W₁₀₀

Y65 Y₆₆ … Y_t Y_t₊₁ … Y₁₀₀ = 0

dNA d₆₆^NA … d_t^NA d_t^NA₊₁ … d₁₀₀^NA

I64 I₆₅ … It₋1 I_t … I99

Inflation

I is random inflation rate t

I 65 I ₆₆ … I _t I_t₊₁ … I₁₀₀ Returns

r is constant interest rate, r_t is random rate on risky asset

r r … r r … –––

r65 r₆₆ … r_t r_t₊₁ … –––

Control (decision) variables

Ct is consumption, α is proportion invested into equities, _t

mt is proportion used for purchasing nominal annuities,

mt is proportion used for purchasing real annuities

C65 C₆₆ … C_t C_t₊₁ … –––

α 65 α ₆₆ … α _t α_t₊₁ … –––

mNA m₆₆^NA … m_t^NA m_t^NA₊₁ … –––

mRA m₆₆^RA … m_t^RA m_t^RA₊₁ … –––

Age during the decumulation process

65 66 … t t+1 … 100

We assume that the maximum pensioner’s age is 100 years. However, we witness constantly increasing longevity in recent years and it is not unusual any more that the pensioner’s age is more than 100 years. We recognise that this assumption in the thesis is at the variance with the empirical evidence. However, as we will see in the later text in the thesis, we investigate a number of numerical examples. Producing the

numerical results is time consuming and increasing the maximum pensioner’s age to, for example, 115 years would require more time for calculation. Some other authors who investigate the problem of the pensioner’s optimal annuitisation and asset allocation use the maximum pensioner’s age of 100 years (Horneff, Maurer, Mitchell and Stamos (2009), Chai, Horneff, Maurer and Mitchell (2009), Horneff, Maurer and Stamos (2008)). When investigating consumption and portfolio choice over life cycle, but with no annuities, Cocco et al (2005) assume that the investor dies with probability 1 at age 100.

3.1.2 The Types of the Problem to be Investigated

We assume that the member can annuitise any part of the available pension wealth.

We will assume that the member never annuitises any part of his income, only pension wealth available at the beginning of the year can be used for purchasing annuities.

The pensioner aims to maximise the expected discounted utility derived from consumption and possibly from bequeathing wealth by choosing the optimal consumption, asset allocation and annuitisation. Regarding annuitisation, we distinguish the strategies for the proportions of the pension wealth m_t^RA and m_t^NA to be annuitised. We group these assumptions into six types of problems to be investigated as follows:

3.1 Annuitising mt^NA and m_t^RA parts of a pension wealth exogenously. In this type of the problem, the pensioner chooses a predetermined amount for purchasing real and nominal annuities and for given m_t^NA and m_t^RA the pensioner invests and consumes optimally. The control variables are

{

Ct^,αt

}

, and m_t^NA and m_t^RA are determined exogenously and are usually suboptimal. The model can handle any assumption about predetermined values of m_t^NA and m_t^RA for 65≤ ≤t 99. We will investigate in more details the results with no annuitisation which is the special case of this type of problem.

3.2 m_t^NA is chosen optimally for some ages and exogenously for others, and m_t^RA is chosen exogenously for all ages 65≤ ≤t 99. For ages where m_t^NA is chosen endogenously, the member chooses a predetermined amount for purchasing real annuities and for given m_t^RA the member maximises the value function with respect to three control variables

{

^C^t^,^α^t^,^m^t^NA

}

Otherwise, the control variables are

{

Ct^,αt

}

. The model allows us to

calculate the results for any combination of exogenous/endogenous nominal annuitisation. All we need to know is for which age nominal annuitisation is endogenous, and for which it is exogenous, and for exogenous annuitisation ages we need to know m_t^NA. We will thoroughly investigate the results under the assumption that the pensioner purchases nominal annuitises optimally at age 65 and no nominal annuities is available afterwards, and no real annuities.

3.3 m_t^NA is chosen exogenously for all ages, and m_t^RA is chosen optimally for some ages and exogenously for others. For ages where m_t^RA is chosen endogenously, the member maximises the value function with respect to three control variables

{

^C^t^,^α^t^,^m^t^RA

}

, and otherwise the control variables are

{

Ct^,αt

}

. Similarly to the type of problem 2, we will thoroughly investigate the results under the assumption that the pensioner purchases real annuitises optimally at age 65 and no real annuities is available afterwards, and no nominal annuities is bought at any age.

3.4 m_t^NA chosen endogenously and m_t^RA exogenously for all ages 65≤ ≤t 99. In this type of problem, the member chooses a predetermined amount for purchasing real annuities and for given m_t^RA the member maximises the value function with respect to three control variables

{

^C^t^,^α^t^,^m^t^NA

}

^{at all}

ages.

3.5 m_t^NA chosen exogenously and m_t^RA endogenously for 65≤ ≤t 99. In this type of problem, the member chooses a predetermined amount for purchasing nominal annuities and for given m_t^NA the member maximises the value function with respect to the three control variables

{

^C^t^,^α^t^,^m^t^RA

}

^{at all}

ages.

3.6 m_t^NA and m_t^RA are optimally chosen proportions for 65≤ ≤t 99. In this case, the member maximises the value function with respect to the four control variables, and control variables are

{

^C^t^,^α^t^,^m^t^NA^,^m^t^RA

}

for all ages.

We have six groups of problems to be investigated, and these groups are differentiated by the assumption regarding exogenous/endogenous nominal/real annuitisation. When we have a particular assumption about the values of m_t^NA and m_t^RA for ages when m_t^NA and/or m_t^RA are exogenous we will refer to this assumption as a case. We can think of different cases as being different markets which are comparable and which differ in offering annuities only. Actually, market and case are equivalent expressions in this thesis. That is why we sometimes referred to cases as markets.

Although we will not investigate the results for many other combinations of optimal/suboptimal annuitisation strategies, we want to emphasize that the model in this chapter can be used for any exogenous/endogenous nominal/real annuitisation strategies. For example, if we assume that full compulsory annuitisation is imposed at a certain age then annuitisation occurs at the pensioner’s discretion after retirement and before full compulsory annuitisation. Full compulsory annuitisation at a certain age can be deemed to be exogenous annuitisation with a proportion 100% at the age of compulsory annuitisation, and exogenous annuitisation with proportion 0%

afterwards. We have witnessed this example in UK (Blake (1999)). Many countries do not impose compulsory annuitisation at any age.

We allow that the pensioner has a certain utility from the bequest. If 100%

compulsory annuitisation happens, we exclude the bequest after that age since no pension wealth is left for bequeathing in the case of full annuitisation. In this context, it is sensible to assume that the bequest motive exists until the time of full annuitisation and not after that.

Regarding the amount to be annuitised at each age t , if exogenous annuitisation happens then it means that the member purchases real annuities for the amount of

t t

m W, or nominal ones for the amount of m W_t^NA _t, and these annuitisation choices are suboptimal. Endogenous annuitisation happens if m W_t^RA _t and/or m W_t^NA _t are chosen optimally from the model.

We will write {cv_t} to denote the {control variables_t} at age t , such that we have the general notation for any type of problem. As we will see later, we work with control variables for values in money units and with control variables for scaled down values suitable for the calculations. In order to differentiate between the two we will denote with {CV_t} the control variables for values in money units and with {cv_t} the control variables for scaled down values.

In document Optimal asset allocation and annuitisation in a defined contribution pension scheme (Page 62-67)