Actuarial Investment

1
Actuarial Investment

• Matching Immunisation
• (and other ideas)
• in developing an investment strategy

2
Setting a Strategy

• Investment strategy to be set with reference to
  the liabilities
• SYSTEM T
• What is best investment strategy when liability
  is
• to pay 100 in 20 years time?
• to pay 10 shares of AIB in 5 years time?
• to pay 100 increased by CPI over next 10 years?

3
Pure Matching

• Complete or pure matching is selecting a
  portfolio of assets so their income flow will
  coincide precisely (in timing, amount, and
  contingency) to the liabilities, i.e., the asset
  proceeds match the liability outgoes under all
  circumstances.
• Known monetary liabilities (up to say 30 years)
  can generally be matched by gilt strips of
  suitable currency, term and nominal amount.
• Such liabilities can be closely matched by
  traditional giltsgive example of how.
• But what if liability is contingent on survival
  and salary inflation and remaining in the same
  employment?
• Can two portfolios with different market value
  completely match the same liability stream?

4
Immunisation

• Immunisation is a limited form of matching. It is
  the investment of assets in such a manner that
  the value of the assets will equal the value of
  the liabilities even on a change in the general
  level of interest rates,
• i.e., it is a strategy to ensure that the value
  of the asset portfolio equals the value of the
  liability portfolio at all interest rates.
• Attempt to apply immunisation when complete
  matching not possible.

5
Motivating Immunisation

• Say liability of 100 due in 10 years time in an
  economy with an non-strippable gilt market.
• Say, at GRY, PV of liability is equal to the MV
  of assets.
• If we invest in 10 year bond then we are exposed
  (e.g. make a loss) on fall in interest rates
  (i.e., the ruling GRY).
• If we invest in, say, a 14 year stock, then on a
  fall of ruling interest rates we make on loss on
  reinvestment of coupons but a capital gain
  (relative to PV of liability)
• So select a bond of suitable maturity so that,
  for a change in interest rate, the relative
  capital gains offsets the relative reinvestment
  losses.
• This is the idea behind immunisation.

6
Formal Immunisation

• Due to Redington (JIA 1952).
• Let us say that, at the ruling rate of interest
  in the market, the PV of liabilities, VL(i)
  equals the value of assets, VA(i), with (i)
  indicating that they are both functions of the
  interest rate i.
• VA(i)VL(i) when i is GRY of suitable term.
• Consider f(i) VA(i)-VL(i)
• Expand f(i) about ruling i by Taylor
• f(i?)f(i) ?f(i) (?2/2)f(i)
• But f(i)0 so f(i?)?f(i) (?2/2)f(i)
• So arrange assets so that f(i)0 and f(i)gt0
  then above ensures a small profit when i moves to
  i?.

7
Redingtons immunisation

• So Redingtons immunisation assumes
• a level yield curve that moves continuously with
  t.
• that all liability outgoes are known in timing
  and amountsay Lt at time t.
• That the asset proceeds are known in timing and
  amountsay, At at time t.
• Now put VL?vtLt at ruling rate of interest on
  market.
• Similarly, VA ?vtAt

8
Condition for Immunisation

• The 3 conditions are
• I VLVA
• PV of liabilities assets at market rate of
  interest
• II (?tvtLt)/(?vtLt) (?tvtAt)/(?vtAt)
• Discounted mean term of assets equals the
  discounted mean term of liabilities.
• III(?t2vtLt)/(?vtLt) lt (?t2vtAt)/(?vtAt)
• The spread of the liability proceeds about its
  mean term is less than that of the asset
  proceeds.

9
Limitations of Redingtons Immunisation

• It does not work in theory
• It assume a flat term structure of interest rates
  when it is not
• It assumes only small changes in interest
  ratesbut sometimes they can gap
• Even if theory worked it might not be applicable
  in practice
• Liability proceeds might not be known with
  certainty in timing and amount.
• Asset proceeds might not be known with certainty
  in timing and amount - or severely limits the
  investing universe
• Assets of suitably long discounted mean term
  might not exist.

10
Limitations of Redingtons Immunisation

• Even if it worked in theory and could be applied
  in practice we might not want to
• it immunises against gain as well as loss.
• It requires a lot a management and dealing to
  maintain the conditions - every time the interest
  rate changes and as time passes.
• And dealing and management expenses ignored in
  the theory.
• Note that theory implies a small, second order,
  profit with small moves in interest rate level
• this implies the model of interest rates moving
  in parallel level shifts is not an arbitrage free
  model of the interest rate structurethis sort of
  model is frowned on by financial economists.

11
Historical Notes

• T.C. Koopmans (1942) wrote a monograph on pure
  matching for Penn Mutual Life Insurance. He was,
  with others, awarded Nobel Prize in 1975 for
  Economics for linear programing.
• Redingtons 1952 breakthrough was anticipated by
  F.R. Macaulay (1938), whose father,T.B. Macaulay,
  was a distinguished NA actuary.
• Some date immunisation from Lidstones JIA paper
  of 1893.
• Immunisation is still being developed see, for
  instance, E.S. Shius work.

12
MPT CAPM and selecting a portfolio

• In actuarial applications we always have
  liabilities for which the assets are accumulated
  to meet.
• Hence we wish to select investments which
  maximise the surplus (PV of assets less PV of
  liabilities) at each point in time, subject to an
  acceptable ruin probability.
• Note change in risk-free asset from CAPM - it is
  not cash but matching portfolio.
• This creates a more interesting problem than CAPM
  as liabilities must be projected alongside
  assets, in a consistent manner - taking account
  of how influences on the former will affect the
  latter.
• Hence scenario modelling or stochastic modelling,
  altering the input (investment strategy) to
  optimise output (surplus, subject to acceptable
  ruin probability).

13
Stochastic Modelling

• Generally use techniques in time series (e.g.
  ARIMA modelling process) to model has key
  determinants of assets and liabilities vary
  together, using historical data sets of
  inflation, wage inflation, interest rates,
  dividend yields, equity capital values, etc.
• This takes care of consistency between projected
  values of assets and liabilities.
• The models have, of course, a white-noise
  residual term so output is stochastic.
• Proposed models in actuarial literature are an
  early step (e.g., Wilkie, Exley Dyson, Smith) -
  but use extreme caution in using them.