Investing for Retirement:
A Downside Risk Approach
Tom Root and Donald Lien
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Motivating Questions
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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
decision?
Academic Literature
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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
consumption.
Allocation decision that is independent of the
investment horizon.
Financial Planning Advice
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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
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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)
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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*
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Assume a financial institution is facing the following three
possible scenarios and associated losses
Scenario
1
2
3
Probability
.97
.015
.015
Loss
0
100
0
The VaR at the 98% level would equal = 0
*This and subsequent examples are based on Meyers 2002
VaR Problems
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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 VaR’s
A Simple Example
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Assume you the previous financial institution and its
competitor facing the same three possible scenarios
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 VaR’s = VaRA + VaRB = 0
The VaR at the 98% level for A and B combined
VaR(A+B)=100
Coherent Measures of Risk
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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
level.
Lien and Tse (2000, 2001) have adopted a more
general method looking at the expected shortfall
The Original Financial
Planning Model
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Let end of period wealth be given by:
~
WT   (1  rf 1 )(1  rf 2 )    (1  rfT )T W  (1   )(1  ~
r1 )(1  ~
r2 )    (1  ~
rT )W
where rfi is the less risky return in period i
~
r is a the return for a risky asset, a random variable
i
 is the % of the portfolio in the risk free asset
Let G represent the target wealth
then choose  such that
~
Prob(WT  G)  0
VaR model
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Booth’s (1999) model replaced the zero
shortfall probability with a given level of
probability, a.
The goal is then to choose  such that
~
Prob(WT  G)  a
Expected Shortfall Model
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The individual should choose  to minimize
the target expected shortfall such that the
shortfall cannot be more than a given
percentage (b) of target wealth.
~
~
E ( S )  E[max( 0, G  WT )]  bG
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A More Formal Treatment
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~
Prob(WT  G)  a
~
E[max( 0, G  WT )]  bG
The individual can satisfy both restrictions simultaneously
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
moment
where f( ) is the probability density function of X. Then
m
LPM ( X , m, n)   (max[ m  X ,0]) f ( x)dx
n

~
~
Pr ob(WT  G)  LPM (WT , G,0)
~
~
E[max( 0, G  WT )]  LPM (WT , G,1)
Empirical Estimations
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We attempt to use historical data to measure
the past expected shortfalls across portfolio
allocation, investment horizon, and target
wealth assumptions.
The Data
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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
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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
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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
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The shortfall for each portfolio was calculated as
~
max( 0, G  WT )
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 period).
The average of the shortfalls is then reported as
the expected shortfall
Graph 1 Expected Shortfall for a Target Wealth of $306,616.68
250000
Expected Shortfall ($)
200000
150000
100000
24
16
8
0
32
40
50000
48
56
64
72
0
1
4
80
7
10 13
16 19
22 25
28
Holding period (years)
88
96
% of Portfolio
in Equities
200000-250000
150000-200000
100000-150000
50000-100000
0-50000
Graph 2 Probability of Shortfall for a target Wealth of $306,616.68
1
0.9
Probability of Shortfall
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
4
85
7 10
13 16
90
19 22
95
25 28
100
Holding Period (Years)
80
75
70
65
60
55
50
45
40
35
30
25
20
10
15
% of Portfolio
in Equities
5 0
0.9-1
0.8-0.9
0.7-0.8
0.6-0.7
0.5-0.6
0.4-0.5
0.3-0.4
0.2-0.3
0.1-0.2
0-0.1