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![Page 1: Expected Utility and Post-Retirement Investment and Spending Strategies William F. Sharpe STANCO 25 Professor of Finance Stanford University .](https://reader035.fdocuments.in/reader035/viewer/2022062619/55166c8f5503469d698b560e/html5/thumbnails/1.jpg)
Expected Utility andPost-Retirement Investment and
Spending Strategies
William F. Sharpe
STANCO 25 Professor of FinanceStanford University
www.wsharpe.com William F. Sharpe
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Choosing a Post-retirement Financial Plan
Wealth 300
Prices 1.000 0.333 0.667 0.111 0.222 0.222 0.444Year 1 Year 2 Year 3
HH $103.37 $115.77 $124.04Now H T HH HT TH TT HT $103.37 $115.77 $103.37
TH $103.37 $93.03 $103.37TT $103.37 $93.03 $81.66
Spent Spent Spent Spent Spent Spent Spent1.00 1.12 0.90 1.20 1.00 1.00 0.79
50
60
70
80
90
100
110
120
130
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Von Neumann-Morgenstern (1)
Z
0
Z
0
Z
0
W1h1
h2
h3
2
3
1
W2
W3
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Von Neumann-Morgenstern (2)
Z
0
3*32*21*1 hhh
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Utility
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Expected Utility
)(
),...,;,...,( 11
sss
nn
XuEU
XXfEU
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First-order Conditions for Maximizing Expected Utility
sss
sss
ss
sss
PPCXmu
pXu
WXpts
XuMax
)(
)('
..
)(
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Marginal Utility
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Single-period Utility functions
• Quadratic (Mean/Variance)
• Constant Relative Risk Aversion (CRRA)
• Hyperbolic Absolute Risk Aversion (HARA)
• Prospect theory
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A Quadratic Utility Marginal Utility Function
cXbm
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A Quadratic Utility Marginal Utility Function (log/log scale)
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A CRRA Marginal Utility Function
bXm
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A CRRA Marginal Utility Function:(log/log scale)
Xbm lnln
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A HARA Marginal Utility Function(log/log scale)
)ln(ln MXbm
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A Kinked Marginal Utility Function
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A Kinked Marginal Utility Function(log/log scale)
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Minimum level
Typical level of retirement income(Perceived loss point)
Income levels (% of pre-retirement income)
100 moveable people, one of which represents the user(experienced frequency representation of probability)
Cost
The Distribution Builder
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Average Choices
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Do preferences conform with maximization of a CRRA utility function?
Or do preferences exhibit loss aversion?
Types of Choices
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Testing for CRRA Preferences
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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0R-Squared
Distribution of R-squared Valuesfor CRRA Utility
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Distribution Builder ResultsSplit at R2=0.90 (approx. median)
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Multi-period Financial Plans
• Multiple time periods• For each time period, multiple possible
states of the world– mutually exclusive– exhaustive
• Objective:– Select consumption for each time and state to
maximize expected utility, subject to a budget constraint
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The Simplest Possible Risky Capital Market
• Two periods– Now– Next year
• Two future states of the world– The market is up– The market is down
• Two securities– A riskless real bond– A portfolio of risky securities in market proportions
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Capital Market Characteristics
Bond 1.02 u
prob = 0.50 0 1.00
prob = 0.50
1.02 d
Market Portfolio 1.18 u
prob = 0.50 0 1.00
prob = 0.50
0.94 d
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Desired Spending
Spending 55.80 u
prob = 0.50 0 50.00
prob = 0.50
48.60 d
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Wealth, Financial Strategy and Desired Spending
W B0 M0 x 100 20 30
s C 1 -1 -1 50.00 S0
0 1.02 1.18 55.80 Su0 1.02 0.94 48.60 Sd
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Initial Wealth
W B0 M0 x 100 20 30
s C 1 -1 -1 50.00 S0
0 1.02 1.18 55.80 Su0 1.02 0.94 48.60 Sd
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Bond Investment
W B0 M0 x 100 20 30
s C 1 -1 -1 50.00 S0
0 1.02 1.18 55.80 Su0 1.02 0.94 48.60 Sd
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Market Portfolio Investment
W B0 M0 x 100 20 30
s C 1 -1 -1 50.00 S0
0 1.02 1.18 55.80 Su0 1.02 0.94 48.60 Sd
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Wealth, Financial Strategy, Capital Markets and Spending
Initial Wealth Financial Strategy
W B0 M0 x 100 20 30
s C 1 -1 -1 50.00 S0
0 1.02 1.18 55.80 Su0 1.02 0.94 48.60 Sd
Capital Market Characteristics Spending
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Decisions SpendingCx’= s
W B0 M0 x 100 20 30
s C 1 -1 -1 50.00 S0
0 1.02 1.18 55.80 Su0 1.02 0.94 48.60 Sd
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Decisions Spending x’ = C-1s
W B0 M0 x 50.000 20.000 30.000
sC-1 1.000 0.327 0.654 0.00 S0
0.000 -3.840 4.820 55.80 Su0.000 4.167 -4.167 48.60 Sd
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Arrow-Debreu Prices
W B0 M0 x 0.327 -3.840 4.167
sC-1 1.000 0.327 0.654 0.00 S0
0.000 -3.840 4.820 1.00 Su0.000 4.167 -4.167 0.00 Sd
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Lockbox Strategies
21
21
11
211
1
'
'
)('
'
xxx
sCsCx
ssCx
sCx
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Lockbox, Period 1
W B0 M0 x 50.000 20.000 30.000
sC-1 1.000 0.327 0.654 0.00 S0
0.000 -3.840 4.820 55.80 Su0.000 4.167 -4.167 48.60 Sd
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Desired Spending: Multiple Periods
63.84uu
55.60u
53.76ud
0 50.00
52.80du
49.20 d
47.04dd
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Dynamic Strategies
W B0 M0 Bu Mu Bd Md x 150.00 40.00 60.00 14.00 42.00 24.00 24.00
s C 1 -1 -1 0 0 0 0 50.00 S0
0 1.02 1.18 -1 -1 0 0 55.60 Su0 1.02 0.94 0 0 -1 -1 49.20 Sd0 0 0 1.02 1.18 0 0 63.84 Suu0 0 0 1.02 0.94 0 0 53.76 Sud0 0 0 0 0 1.02 1.18 52.80 Sdu0 0 0 0 0 1.02 0.94 47.04 Sdd
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Contingent Bond Purchases
W B0 M0 Bu Mu Bd Md x 150.00 40.00 60.00 14.00 42.00 24.00 24.00
s C 1 -1 -1 0 0 0 0 50.00 S0
0 1.02 1.18 -1 -1 0 0 55.60 Su0 1.02 0.94 0 0 -1 -1 49.20 Sd0 0 0 1.02 1.18 0 0 63.84 Suu0 0 0 1.02 0.94 0 0 53.76 Sud0 0 0 0 0 1.02 1.18 52.80 Sdu0 0 0 0 0 1.02 0.94 47.04 Sdd
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Contingent Market Portfolio Purchases
W B0 M0 Bu Mu Bd Md x 150.00 40.00 60.00 14.00 42.00 24.00 24.00
s C 1 -1 -1 0 0 0 0 50.00 S0
0 1.02 1.18 -1 -1 0 0 55.60 Su0 1.02 0.94 0 0 -1 -1 49.20 Sd0 0 0 1.02 1.18 0 0 63.84 Suu0 0 0 1.02 0.94 0 0 53.76 Sud0 0 0 0 0 1.02 1.18 52.80 Sdu0 0 0 0 0 1.02 0.94 47.04 Sdd
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Lockbox, Period 2
% Bonds 32.89% 25.00% 50.00%W B0 M0 Bu Mu Bd Md
x 49.67 16.34 33.33 14.00 42.00 24.00 24.00
sC-1 1.000 0.327 0.654 0.107 0.214 0.214 0.427 0.00 S0
0.000 -3.840 4.820 -1.255 -2.510 1.575 3.150 0.00 Su0.000 4.167 -4.167 1.362 2.723 -1.362 -2.723 0.00 Sd0.000 0.000 0.000 -3.840 4.820 0.000 0.000 63.84 Suu0.000 0.000 0.000 4.167 -4.167 0.000 0.000 53.76 Sud0.000 0.000 0.000 0.000 0.000 -3.840 4.820 52.80 Sdu0.000 0.000 0.000 0.000 0.000 4.167 -4.167 47.04 Sdd
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Lockbox Separation
• A retirement financial strategy is fully specified if spending in each year can be determined for any scenario of market returns
• A market is complete if any desired spending plan can be implemented with a retirement financial strategy
• If the market is complete, any fully specified retirement financial strategy can be implemented with a lockbox strategy
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Time-separableMulti-period Utility Functions
)()()()(),,,(
)()(),(
)()(
),,,(),()(
2222
11
000
0
ddddduduududuuuuddduuduu
dduudu
ddduuduudu
susususussssEU
susussEU
susEU
ssssEUssEUsEUEU
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Path-dependent Multi-period Utility Functions
)4()3()2()1(
,,:4
,,:3
,,:2
,,:1
4321
0
0
0
0
pupupupuEU
sssp
sssp
sssp
sssp
pppp
ddd
dud
udu
uuu
![Page 45: Expected Utility and Post-Retirement Investment and Spending Strategies William F. Sharpe STANCO 25 Professor of Finance Stanford University .](https://reader035.fdocuments.in/reader035/viewer/2022062619/55166c8f5503469d698b560e/html5/thumbnails/45.jpg)
A Habit Formation Utility Function
)1/()()( 11 gdCXaCu g
ttt
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Issac Gonzales Survey, 2009
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Survey Details
Investment Market ConditionThis investment represents the case where the market goes up in year 1.
The investment provides year-2 income if the market goes up in year 1.
Total Investment Cost(current dollars, rounded to nearest
thousand)Total investment cost required to produce $81,000 of year-2 income if the market goes up in year 1.
$81,000 x $0.33 = $26,730~ $27,000
Income(current dollars, rounded to
nearest thousand)The total amount of year-2 income you will receive if the market goes up in year 1.
Investment CostFor every $0.33 that you invest today you will receive $1 in year 2 if the market goes up in year 1.
Solid vs. Dashed BordersSolid borders represents cases where the market goes up in year 1 while dashed borders represent cases where the market goes down in year 1.
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Survey Example
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Average Response
0.8
0.9
1
1.1
1.2
1.3
Case 0
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Required Financial Strategy
N d u dd du ud uu0
0.5
1
1.5
2
2.5
3
spending
bonds
stockslg stx shrt bds
lg bds shrt stx
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Implied Marginal Utility Functions
0 0.5 1 1.5 2
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
time
c
Consumption versus time
-0.2 -0.1 0 0.1 0.2 0.3-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
log(c)
loc(
ppc)
log(PPC) versus log(c)
gu=2.59
gd=2.79
g1=2.70
v1
v2
d=v1/v2 = -0.82
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d Values
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Unanswered Questions
• How can we determine an individual’s true preferences?
• Are individual choices consistent with axioms of “rational decisions”?
• How can the influence of framing be minimized?
• How can an optimal financial strategy for complex preferences be determined?
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The Fidelity Income Replacement Funds
• Horizon date– E.g. 2036
• Investment strategy– Time-dependent “glide path” asset allocation
• Spending Rule– Pre-specified time-dependent proportions of
asset value
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Spending Rule
Annual Target Payment Rates
0
10
20
30
40
50
60
70
80
90
100
30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
Years to Horizon
Per
ent
of
Fu
nd
Sp
ent
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Investment Strategy
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Lockbox Equivalence
• Any strategy with a time-dependent proportional spending rule and a time-dependent investment strategy is equivalent to a lockbox strategy
• Each lockbox will have the same investment strategy and
• The initial amounts to be invested in the lockboxes can be computed from the pre-specified spending rates
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Initial Lockbox Values (1)
• Let:kt = the proportion spent in year t
Rt = the total return on investment in year t (e.g. 1.02 for 2%)
• The amounts spent in the first three years will be:
Wk0
(1-k0)WR1k1
(1-k0)WR1(1-k1) R2k2
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Initial Lockbox Values (2)
• Re-arranging:{Wk0}
{W(1-k0)k1} R1
{W(1-k0)(1-k1)k2} R1R2
• But these are the ending values for lockboxes with the initial investments shown in the brackets { }– therefore, investing these amounts in lockboxes will give
the same spending plan as the original strategy
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Percentages of Initial Wealth in Lockboxes
0.0%
1.0%
2.0%
3.0%
4.0%
5.0%
6.0%
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Lockbox Year
Per
cen
t o
f In
itia
l W
ealt
h W
ealt
h
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A Simple Income Replacement Fund
• Two assets– A riskless real bond– A market portfolio
• (e.g. 60% Stocks, 40% Bonds)
• A glide path similar to that for equity funds in the Fidelity Income Replacement Funds
• A 30-year horizon• Annual payment rates equal to those of
the Fidelity Income Replacement Funds
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Capital Market Characteristics
• Riskless real return– 2 % per year
• Market portfolio real return– Lognormally distributed each year– Expected annual return
• 6 % per year
– Annual standard deviation of return• 12 % per year
– No serial correlation
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Monte Carlo Simulations
• 10,000 scenarios of 29 years each
• Returns for each lockbox are simulated
• State prices for payment in year 29 are assumed to be log-linearly related to cumulative market returns– Consistent with a CRRA pricing kernel– Consistent with limit of a binomial i.i.d. return-
generating process
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Year 29: State Prices and Spending
0 0.5 1 1.5 2 2.5 3 3.5 4-18
-16
-14
-12
-10
-8
-6
-4
-2
log(Spending)
log(
Sta
te P
rice)
log(State Prices) and log(Spending), Year 29
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Year 29: Cumulative Market Return and Spending
-2 -1 0 1 2 3 40
0.5
1
1.5
2
2.5
3
3.5
4
Log(CumRms)
Log(
Spe
ndin
g)
Log(Spending) and log(Cumulative Returns on Market)
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Real-world Challenges
• Determining each individual’s true preferences
• Determining the return generating process
• Representing capital market instruments
• Estimating the feasibility of dynamic strategies
• Incorporating annuities
• Insuring the macro-consistency of optimal strategies