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Investing for Retirement: A Downside Risk Approach


When saving for retirement how should an individual choose the allocation of ... Achieving a 70% probability of generating 80% of pre-retirement income at age 65. ... – PowerPoint PPT presentation

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Title: Investing for Retirement: A Downside Risk Approach

Investing for Retirement A Downside Risk Approach
  • Tom Root and Donald Lien

Motivating Questions
  • When saving for retirement how should an
    individual choose the allocation of funds between
    risky and risk free asset?
  • Can general guidelines be established to help in
    the allocation decision?
  • Can empirical estimates using downside risk
    improve out understanding of the allocation

Academic Literature
  • Samuelson (1969) and Merton (1969)
  • Expected utility maximization of the consumption
    saving decision.
  • Establish the end of investment period, then
    solve recursively for the allocation decision
    that maximizes the expected utility of
  • Allocation decision that is independent of the
    investment horizon.

Financial Planning Advice
  • Decreasing emphasis on risky assets through time.
  • The 100-age rule
  • The percentage of the portfolio placed in
    equities should be approximately equal to 100
    minus the age of the individual.
  • Retirement goal Generate a given percentage of
    pre retirement income for a given number of
    years. For example 80 or pre-retirement income
    at age 65 or 80 at 65

Bridging the Gap
  • Booth (2001)
  • A Value at Risk Approach
  • Individual attempts to contain the probability of
    failing to meet a given target wealth.
  • 70 of 80 at 65
  • Achieving a 70 probability of generating 80 of
    pre-retirement income at age 65.
  • The individual is concerned with the success or
    failure of meeting the target

Value at Risk (VaR)
  • An estimate of the amount of loss (or value) a
    portfolio is expected to equal or exceed at a
    given probability level.

A Simple Example
  • Assume a financial institution is facing the
    following three possible scenarios and associated
  • Scenario Probability Loss
  • 1 .97 0
  • 2 .015 100
  • 3 .015 0
  • The VaR at the 98 level would equal 0
  • This and subsequent examples are based on Meyers

VaR Problems
  • Artzner (1997), (1999) has shown that VaR is not
    a coherent measure of risk.
  • For Example it does not posses the property of
    subadditvity. In other words the combined
    portfolio VaR of two positions can be greater
    than the sum of the individual VaRs

A Simple Example
  • Assume you the previous financial institution and
    its competitor facing the same three possible
  • Scenario Probability Loss A Loss B Loss A B
  • 1 .97 0 0 0
  • 2 .015 100 0 100
  • 3 .015 0 100 100
  • The VaR at the 98 level for A or B alone is 0
  • The Sum of the individual VaRs VaRA VaRB 0
  • The VaR at the 98 level for A and B combined
  • VaR(AB)100

Coherent Measures of Risk
  • Artzner (1997, 1999) Acerbi and Tasche
    (2001a,2001b), Yamai and Yoshiba (2001a, 2001b)
    have pointed to Conditional Value at Risk or Tail
    Value at Risk as coherent measures.
  • CVaR and TVaR measure the expected loss
    conditioned upon the loss being above the VaR
  • Lien and Tse (2000, 2001) have adopted a more
    general method looking at the expected shortfall

The Original Financial Planning Model
  • Let end of period wealth be given by

Let G represent the target wealth then choose q
such that
VaR model
  • Booths (1999) model replaced the zero shortfall
    probability with a given level of probability, a.
  • The goal is then to choose q such that

Expected Shortfall Model
  • The individual should choose q to minimize the
    target expected shortfall such that the shortfall
    cannot be more than a given percentage (b) of
    target wealth.

A More Formal Treatment
  • The individual can satisfy both restrictions
  • The restrictions can be captured by the lower
    partial moment
  • LPM of random variable X is characterized by two
    parameters m, the target and n, the order of the
  • where f( ) is the probability density function of
    X. Then

Empirical Estimations
  • We attempt to use historical data to measure the
    past expected shortfalls across portfolio
    allocation, investment horizon, and target wealth

The Data
  • Return Data is the monthly return reported by
    Ibbotson Associates January 1926 to June 2002.
  • The risky return was proxied by the return on
    large company stocks and the risk free return by
    the return on long term government bonds.
  • The returns were adjusted by the inflation rate
    reported by the BLS.

Model Parameters
  • Assume that an individual is currently 35 years
    of age and has 100,000 in savings.
  • She is saving for the goal of reaching 70 of her
    pre-retirement real income of 50,000 per year or
    an annual annuity payment of 35,000 for 11 years
    (assuming retirement at age 65 and life
    expectance of 76).
  • The real return on the annuity is assumed to be
    either 1, 4, or 7 producing target wealth
    estimates of 362,866.99, 306,616.68, and
    262,453.60 respectively.

Portfolio Allocations and holding periods
  • 101 constant allocation portfolios beginning with
    100 in treasuries and decreasing the percentage
    in treasures by 1 until reaching 100 in
    equities were calculated.
  • The original investment period of 30 years was
    also deceased by one year until a holding period
    of one year was reached. Resulting in 30
    different holing periods.

Expected shortfall
  • The shortfall for each portfolio was calculated
  • The expected shortfall was generated by
    calculating the shortfall on successive
    portfolios of the holding period starting with
    each month in the sample (for those months with
    enough observations to satisfy the holding
  • The average of the shortfalls is then reported as
    the expected shortfall

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