drake drake university universite d’auvergne investing for retirement: a downside risk approach...

DrakeDRAKE UNIVERSITY

UNIVERSITED’AUVERGNE

Investing for Retirement:A Downside Risk Approach

Tom Root and Donald Lien


DrakeDrake University

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 decision?



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 consumption.Allocation decision that is independent of the investment horizon.



Financial Planning Advice

Decreasing emphasis on risky assets through time.

The “100-age” ruleThe 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 losses

Scenario Probability Loss1 .97 02 .015 1003 .015 0

The VaR at the 98% level would equal = 0

*This and subsequent examples are based on Meyers 2002



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 VaR’s



A Simple Example

Assume you the previous financial institution and its competitor facing the same three possible scenarios

Scenario ProbabilityLoss ALoss B Loss A & B1 .97 0 0 02 .015 100 0 1003 .015 0 100 100

The VaR at the 98% level for A or B alone is 0The 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

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

Let end of period wealth be given by:

asset freerisk in the portfolio theof % theis

variablerandom a asset,risky afor return thea is ~

i periodin return risky less theis where

)~1()~1)(~1)(1()1()1)(1(~

2121

i

fi

TT

fTffT

r

r

WrrrWrrrW

0)~

(Prob GWT

Let G represent the target wealth then choose such that



VaR model

Booth’s (1999) model replaced the zero shortfall probability with a given level of probability, The goal is then to choose such that

)~

(Prob GWT



Expected Shortfall Model

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

G)]~

,0[max()~

( TWGESE



A More Formal Treatment

The individual can satisfy both restrictions simultaneouslyThe restrictions can be captured by the lower partial momentLPM of random variable X is characterized by two parameters: m, the target and n, the order of the momentwhere f( ) is the probability density function of X. Then

)~

(Prob GWT GWGE T )]~

,0[max(

m

n dxxfXmnmXLPM )(])0,(max[),,(

)0,,~

()~

(Pr GWLPMGWob TT )1,,~

()]~

,0[max( GWLPMWGE TT



Empirical Estimations

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



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 as

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

)~

,0max( TWG

08

1624

3240

4856

6472

8088

96

1 4 7 10 13 16 19 22 25 28

0

50000

100000

150000

200000

250000

Exp

ecte

d S

ho

rtfa

ll (

$)

% of Portfolio in Equities

Holding period (years)

Graph 1 Expected Shortfall for a Target Wealth of $306,616.68

200000-250000

150000-200000

100000-150000

50000-100000

0-50000

05101520

2530

3540

4550

5560

6570

7580

8590

95100

1 4 7 10 13 16 19 22 25 28

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

bab

ilit

y o

f S

ho

rtfa

ll

% of Portfolioin Equities

Holding Period (Years)

Graph 2 Probability of Shortfall for a target Wealth of $306,616.68

0.9-10.8-0.90.7-0.80.6-0.70.5-0.60.4-0.50.3-0.40.2-0.30.1-0.20-0.1

drake drake university universite d’auvergne investing for retirement: a downside risk approach...

Documents