Thou Shalt Still Buy and Hold - Osaka University...An investor might optimally never buy stock if...

Thou Shalt Still Buy and Hold

Xunyu Zhou/Oxford

3rd September 2008/Kyoto

Xunyu Zhou/Oxford Thou Shalt Still Buy and Hold

This Talk Based on ...

A. Shiryaev, Z. Xu, and X. Zhou, “Thou shalt buy and hold”,Quantitative Finance, 2008

M. Dai, H. Jin and X. Zhou, “Buy on the lows and sell on thehighs”, working paper, 2008

M. Dai, H. Jin and X. Zhou, “Thou shalt still buy and hold”,working paper, 2008.


Classical Continuous-Time Portfolio Selection Models

Merton’s model



Merton’s model

Markowitz’s model



Merton’s model

Markowitz’s model

Behavioral model (recently)



Merton’s model

Markowitz’s model


No transaction costs



Merton’s model

Markowitz’s model


No transaction costs

Optimal portfolios are to “trade all the time” so as to keepcertain “constant proportions”


Portfolio Selection with Transaction Costs

“Continuous Trading” clearly incurs infinite costs




In the presence of transaction costs: sell region, buy regionand no trade region





Trade only necessary






Liu and Loewenstein (2002), Rev Fin Studies, CRRA investor,transaction costs and finite horizon







An investor might optimally never buy stock if the horizon isshort







An investor might optimally never buy stock if the horizon isshortAn investor should largely buy and hold if the horizon is long


Conventional Wisdom: Buy and Hold

Buy and Hold: Buy a good stock and leave it alone for longtime




“Never sell a security unless you need money”(EfficientMarket.ca)





“Trading is hazardous to your wealth” (Barber and Odean,2000, J Fin)






Grounds






Grounds

Efficient market hypothesis (EMH): every security is fairlyvalued at all times, so there is no point to trade (you tradebecause something happens to you, not because somethinghappens to the market)






Grounds


Pure costs: bid/offer spread, brokerage, capital gain tax etc.(Warren Buffett is a buy-and-hold advocate rejecting EMH)






Grounds


Pure costs: bid/offer spread, brokerage, capital gain tax etc.(Warren Buffett is a buy-and-hold advocate rejecting EMH)

However, no well-established portfolio models produced purebuy-and-hold strategy


If I Were an Innocent Investor ...

Let’s say I just bought a stock, and must sell it in one year




Need to decide when to sell





Criterion?





Criterion?

Mean-Variance? Expected utility?





Criterion?

Mean-Variance? Expected utility? I don’t know what you aretalking about





Criterion?


Let’s sell higher ...





Criterion?


Let’s sell higher ... say, at the maximum price?





Criterion?



Selling at the maximum price is a “mission impossible”





Criterion?




How about selling at the price closest to the maximum?





Criterion?




How about selling at the price closest to the maximum?

...or sell at the time when the expected relative error betweenthe current price and the maximum price is minimised


The Model

A Black–Scholes market with a stock and a saving account


The Model


The discounted stock price follows, on (Ω,F , P ):

dPt = (a − r)Ptdt + σPtdBt, or Pt = eµt+σBt

where µ = a − r − 12σ2


The Model





Let Mt = max06s6t Ps, 0 6 t 6 T


The Model






Consider the following optimal stopping problem

min06τ6T

E

[

MT − Pτ

MT

]

where τ ∈ [0, T ] is a Bt-stopping time


The Model






Consider the following optimal stopping problem

min06τ6T

E

[

MT − Pτ

MT

]

where τ ∈ [0, T ] is a Bt-stopping time

... or equivalently

max06τ6T

E

[

Pτ

MT

]


Related (Probabilistic) Literature

Graversen, Peskir, Shiryaev (2000), Theory Prob Appl, studied

min0≤τ≤T

E(B0τ − S0

T )2

where S0t := max06s6t B0

s , and obtained optimal

τ∗ = inf

0 6 t 6 T∣

∣

∣

S0t − B0

t√T − t

> z∗

where z∗ = 1.12...


Related (Probabilistic) Literature

Graversen, Peskir, Shiryaev (2000), Theory Prob Appl, studied

min0≤τ≤T

E(B0τ − S0

T )2

where S0t := max06s6t B0

s , and obtained optimal

τ∗ = inf

0 6 t 6 T∣

∣

∣

S0t − B0

t√T − t

> z∗

where z∗ = 1.12...

Pedersen (2003), Stoch Stoch Rep, considered

min0≤τ≤T

E|B0τ − S0

T |p, 0 6 p < +∞


The Model Rewritten

Assume σ = 1 (otherwise rescale the time)


The Model Rewritten


Rewrite Bµt = µt + Bt, Sµ

t = max06s6t Bµs , 0 6 t 6 T


The Model Rewritten



t = max06s6t Bµs , 0 6 t 6 T

The problem is

max06τ6T

E

[

eBµτ

eSµT

]


The Model Rewritten



t = max06s6t Bµs , 0 6 t 6 T

The problem is

max06τ6T

E

[

eBµτ

eSµT

]

Not a standard optimal stopping problem: SµT is not

Bt-adapted!


A Few Good Steps ... Towards a Standard Problem

For any Bt-stopping time 0 6 τ 6 T , we have

E

[

eBµτ

eSµT

]

=E

[

eBµτ

maxeSµτ , emaxτ6t6T B

µt

]




E

[

eBµτ

eSµT

]

=E

[

eBµτ


µt

]

=E

[

min

e−(Sµτ −B

µτ ), e

− maxτ6t6T

(Bµt −B

µτ )

]




E

[

eBµτ

eSµT

]

=E

[

eBµτ


µt

]

=E

[

min

e−(Sµτ −B

µτ ), e

− maxτ6t6T

(Bµt −B

µτ )

]

=E

[

E

[

min

e−(Sµτ −B

µτ ), e

− maxτ6t6T

(Bµt −B

µτ )

∣

∣

∣

∣

Fτ

]]




E

[

eBµτ

eSµT

]

=E

[

eBµτ


µt

]

=E

[

min

e−(Sµτ −B

µτ ), e

− maxτ6t6T

(Bµt −B

µτ )

]

=E

[

E

[

min

e−(Sµτ −B

µτ ), e

− maxτ6t6T

(Bµt −B

µτ )

∣

∣

∣

∣

Fτ

]]

=E

[

E[

min

e−x, e−SµT−τ

]∣

∣

∣

x=Sµτ −B

µτ

]




E

[

eBµτ

eSµT

]

=E

[

eBµτ


µt

]

=E

[

min

e−(Sµτ −B

µτ ), e

− maxτ6t6T

(Bµt −B

µτ )

]

=E

[

E

[

min

e−(Sµτ −B

µτ ), e

− maxτ6t6T

(Bµt −B

µτ )

∣

∣

∣

∣

Fτ

]]

=E

[

E[

min


]∣

∣

∣

x=Sµτ −B

µτ

]

=E [G(τ, Xτ )]

where G(t, x) = E[

min

e−x, e−SµT−t

]

, Xt = Sµt − Bµ

t




E

[

eBµτ

eSµT

]

=E

[

eBµτ


µt

]

=E

[

min

e−(Sµτ −B

µτ ), e

− maxτ6t6T

(Bµt −B

µτ )

]

=E

[

E

[

min

e−(Sµτ −B

µτ ), e

− maxτ6t6T

(Bµt −B

µτ )

∣

∣

∣

∣

Fτ

]]

=E

[

E[

min


]∣

∣

∣

x=Sµτ −B

µτ

]

=E [G(τ, Xτ )]

where G(t, x) = E[

min

e−x, e−SµT−t

]

, Xt = Sµt − Bµ

t

Xt – drawdown process – is Bt-adapted!


Function G

Assuming µ 6= 12 , then

G(t, x) =2(µ − 1)

2µ − 1e−(µ− 1

2)(T−t)Φ

(−x + (µ − 1)(T − t)√T − t

)

+1

2µ − 1e−(1−2µ)xΦ

(−x − µ(T − t)√T − t

)

+ e−xΦ

(

x − µ(T − t)√T − t

)

,

where Φ is the CDF of standard normal distribution


Function G

Assuming µ 6= 12 , then

G(t, x) =2(µ − 1)

2µ − 1e−(µ− 1

2)(T−t)Φ

(−x + (µ − 1)(T − t)√T − t

)

+1

2µ − 1e−(1−2µ)xΦ

(−x − µ(T − t)√T − t

)

+ e−xΦ

(

x − µ(T − t)√T − t

)

,

where Φ is the CDF of standard normal distribution

If µ = 12 , then G has a different (explicit) expression


Dynamic Programming

Define Xxt+s = x ∨ Sµ

s − Bµs , s ≥ 0


Dynamic Programming


s − Bµs , s ≥ 0

Xxt+s is Markovian under P , ∀(t, x)


Dynamic Programming


s − Bµs , s ≥ 0


Value function

V (t, x) = sup06τ6T−t

E[

G(t + τ, Xxt+τ )

]


Dynamic Programming


s − Bµs , s ≥ 0


Value function

V (t, x) = sup06τ6T−t

E[

G(t + τ, Xxt+τ )

]

In particular

V (0, 0) = sup06τ6T

E

[

eBµτ

eSµT

]


Variational Inequalities

Dynamic programming equation (Variational Inequalities)

min−LV, V − G = 0, (t, x) ∈ [0, T ) × (0,∞) (1)

V (T, x) = G(T, x), x ∈ (0,∞) (2)

Vx(t, 0+) = 0, t ∈ [0, T ) (normal reflection) (3)

where

LV = Vt − µVx +1

2Vxx (4)


Holding and Selling Region

Holding region

C = (t, x) ∈ [0, T ] × [0,∞) : V (t, x) > G(t, x)



Holding region

C = (t, x) ∈ [0, T ] × [0,∞) : V (t, x) > G(t, x)

Selling region

D = (t, x) ∈ [0, T ] × [0,∞) : V (t, x) = G(t, x)



Holding region

C = (t, x) ∈ [0, T ] × [0,∞) : V (t, x) > G(t, x)

Selling region

D = (t, x) ∈ [0, T ] × [0,∞) : V (t, x) = G(t, x)

An optimal selling time is

τ∗ = inft ∈ [0, T ] : (t, Sµt − Bµ

t ) ∈ D



Holding region

C = (t, x) ∈ [0, T ] × [0,∞) : V (t, x) > G(t, x)

Selling region

D = (t, x) ∈ [0, T ] × [0,∞) : V (t, x) = G(t, x)

An optimal selling time is

τ∗ = inft ∈ [0, T ] : (t, Sµt − Bµ

t ) ∈ D

So it boils down to finding V - which is hard


Optimal Stopping with Running Payoff

To get around: turn terminal payoff into running payoff




Xxt+· = |Y·| in law, where

dYt = −µsign(Yt)dt + dBt, Y0 = x




Xxt+· = |Y·| in law, where

dYt = −µsign(Yt)dt + dBt, Y0 = x

By Tanaka’s formula

|Ys| = x−µ

∫ s

0I(Yu 6= 0)du+

∫ s

0sign(Yu)I(Yu 6= 0)dBu+ℓ0

s(Y )


Optimal Stopping with Running Payoff (Cont’d)

Ito’s formula then implies

G(t + s, Xxt+s) = G(t, x) +

∫ s

0H(t + u, Xx

t+u)du + Ms

provided that Gx(t, 0+) = 0 (so as to kill the local timeterm), where M is a martingale and

H(t, x) = LG(t, x) = Gt(t, x) − µGx(t, x) +1

2Gxx(t, x)




G(t + s, Xxt+s) = G(t, x) +

∫ s

0H(t + u, Xx

t+u)du + Ms



2Gxx(t, x)

Hence

V (t, x) = G(t, x) + sup06τ6T−t

E

[∫ τ

0H(t + u, Xx

t+u)du

]




G(t + s, Xxt+s) = G(t, x) +

∫ s

0H(t + u, Xx

t+u)du + Ms



2Gxx(t, x)

Hence

V (t, x) = G(t, x) + sup06τ6T−t

E

[∫ τ

0H(t + u, Xx

t+u)du

]

Need to calculate H and show Gx(t, 0+) = 0


Function H

Write G(t, x) = 2(µ−1)2µ−1 A(t, x) + 1

2µ−1B(t, x) + C(t, x) where

A = e−(µ− 12)(T−t)Φ

(−x + (µ − 1)(T − t)√T − t

)

B = e−(1−2µ)xΦ

(−x − µ(T − t)√T − t

)

C = e−xΦ

(

x − µ(T − t)√T − t

)


Function H

Write G(t, x) = 2(µ−1)2µ−1 A(t, x) + 1


A = e−(µ− 12)(T−t)Φ

(−x + (µ − 1)(T − t)√T − t

)

B = e−(1−2µ)xΦ

(−x − µ(T − t)√T − t

)

C = e−xΦ

(

x − µ(T − t)√T − t

)

Lengthy calculations show for ϕ = A, B, C,

Lϕ = (µ − 12)ϕ − ϕx


Function H

Write G(t, x) = 2(µ−1)2µ−1 A(t, x) + 1


A = e−(µ− 12)(T−t)Φ

(−x + (µ − 1)(T − t)√T − t

)

B = e−(1−2µ)xΦ

(−x − µ(T − t)√T − t

)

C = e−xΦ

(

x − µ(T − t)√T − t

)


Lϕ = (µ − 12)ϕ − ϕx

Hence H = LG = (µ − 12)G − Gx


Function H

Write G(t, x) = 2(µ−1)2µ−1 A(t, x) + 1


A = e−(µ− 12)(T−t)Φ

(−x + (µ − 1)(T − t)√T − t

)

B = e−(1−2µ)xΦ

(−x − µ(T − t)√T − t

)

C = e−xΦ

(

x − µ(T − t)√T − t

)


Lϕ = (µ − 12)ϕ − ϕx


Moreover Gx = B − C


Function H

Write G(t, x) = 2(µ−1)2µ−1 A(t, x) + 1


A = e−(µ− 12)(T−t)Φ

(−x + (µ − 1)(T − t)√T − t

)

B = e−(1−2µ)xΦ

(−x − µ(T − t)√T − t

)

C = e−xΦ

(

x − µ(T − t)√T − t

)


Lϕ = (µ − 12)ϕ − ϕx


Moreover Gx = B − C

So Gx(t, 0+) = B(t, 0+) − C(t, 0+) = 0


The Good...

If µ ≥ 12 :


The Good...

If µ ≥ 12 :

Gx 6 0 by the definition G(t, x) = E[

min

e−x, e−Sµ

T−t

]


The Good...

If µ ≥ 12 :


min

e−x, e−Sµ

T−t

]

H = (µ − 12 )G − Gx > 0


The Good...

If µ ≥ 12 :


min

e−x, e−Sµ

T−t

]

H = (µ − 12 )G − Gx > 0

So the optimal τ∗ = T


The Good...

If µ ≥ 12 :


min

e−x, e−Sµ

T−t

]

H = (µ − 12 )G − Gx > 0


The stock is “good”


... and The Bad

If µ ≤ −12 :


... and The Bad

If µ ≤ −12 :

exG(t, x) = E[

min

1, e−Sµ

T−t+x

]


... and The Bad

If µ ≤ −12 :

exG(t, x) = E[

min

1, e−Sµ

T−t+x

]

∂(exG)∂x > 0 so ex(Gx + G) > 0 or Gx + G > 0


... and The Bad

If µ ≤ −12 :

exG(t, x) = E[

min

1, e−Sµ

T−t+x

]


H = (µ − 12 )G − Gx = (µ + 1

2 )G − (Gx + G) 6 0


... and The Bad

If µ ≤ −12 :

exG(t, x) = E[

min

1, e−Sµ

T−t+x

]


H = (µ − 12 )G − Gx = (µ + 1

2 )G − (Gx + G) 6 0

So the optimal τ∗ = 0


... and The Bad

If µ ≤ −12 :

exG(t, x) = E[

min

1, e−Sµ

T−t+x

]


H = (µ − 12 )G − Gx = (µ + 1

2 )G − (Gx + G) 6 0

So the optimal τ∗ = 0

The stock is “bad”


Finding More Good Guys

If µ = 0:

V (t, x) > Et,x[G(T, XxT )] > G(t, x), ∀ t ∈ [0, T ), x > 0



If µ = 0:


On the other hand ∀µ ∈ R:

∂

∂µ

e12µ2(T−t) (Et,x[G(T, Xx

T )] − G(t, x))

> 0



If µ = 0:



∂

∂µ


T )] − G(t, x))

> 0

Hence ∀µ > 0:



If µ = 0:



∂

∂µ


T )] − G(t, x))

> 0

Hence ∀µ > 0:

V (t, x) > Et,x[G(T,XxT )] > G(t, x), ∀ t ∈ [0, T ), x ≥ 0



If µ = 0:



∂

∂µ


T )] − G(t, x))

> 0

Hence ∀µ > 0:





If µ = 0:



∂

∂µ


T )] − G(t, x))

> 0

Hence ∀µ > 0:


So the optimal τ∗ = TThe stock is “good”


Optimal Seeling Times

Define a goodness index of the stock

α =a − r

σ2




α =a − r

σ2

Theorem

(Shiryaev, Xu and Zhou, Quantitative Finance 2008) Optimalselling time τ∗ = T if α ≥ 0.5, and τ∗ = 0 if α ≤ 0




α =a − r

σ2

Theorem

(Shiryaev, Xu and Zhou, Quantitative Finance 2008) Optimalselling time τ∗ = T if α ≥ 0.5, and τ∗ = 0 if α ≤ 0

Theorem

(Dai, Jin, and Zhou 2008) Optimal selling time τ∗ = T ifα ≥ 0.5, and τ∗ = 0 if α < 0.5.


Optimal Relative Error - Good Stock

Let Pt = eµt+σBt , Mt = max06s6t Ps

Assume α ≥ 0.5




Assume α ≥ 0.5

Optimal selling time τ∗ = T




Assume α ≥ 0.5


Need to determine E[

MT−PTMT

]

: the optimal relative error




Assume α ≥ 0.5


Need to determine E[

MT−PTMT

]

: the optimal relative error

The joint density function of (PT , MT ) is

fT (p, m) =2

σ3√

2πT 3

ln(m2/p)

pme−

ln2(m2/p)

2σ2T+ β

σln(p)− 1

2β2T

where 0 < p 6 m, m > 1, β = µσ≡

(

α − 12

)

σ


Optimal Relative Error - Good Stock (Cont’d)

We have

E

[

PT

MT

]

=

∫ 1

0

P

(

PT

MT> y

)

dy

=

∫ 1

0

∫

∞

y

∫ p/y

p∨1

fT (p,m)dmdpdy

= · · ·

=

(

1 − 1

2α

)

Φ

(

(α − 1

2)σ√

T

)

+

(

1 +1

2α

)

eασ2T Φ

(

−(α +1

2)σ√

T

)



We have

E

[

PT

MT

]

=

∫ 1

0

P

(

PT

MT> y

)

dy

=

∫ 1

0

∫

∞

y

∫ p/y

p∨1

fT (p,m)dmdpdy

= · · ·

=

(

1 − 1

2α

)

Φ

(

(α − 1

2)σ√

T

)

+

(

1 +1

2α

)

eασ2T Φ

(

−(α +1

2)σ√

T

)

Optimal relative error r∗(α, σ) decreases in α and increases inσ



We have

E

[

PT

MT

]

=

∫ 1

0

P

(

PT

MT> y

)

dy

=

∫ 1

0

∫

∞

y

∫ p/y

p∨1

fT (p,m)dmdpdy

= · · ·

=

(

1 − 1

2α

)

Φ

(

(α − 1

2)σ√

T

)

+

(

1 +1

2α

)

eασ2T Φ

(

−(α +1

2)σ√

T

)

Optimal relative error r∗(α, σ) decreases in α and increases inσ

0 ≤ r∗(α, σ) < 12α


Optimal Relative Error - Bad Stock

Assume α < 0.5



Assume α < 0.5

Optimal selling time τ∗ = 0



Assume α < 0.5

Optimal selling time τ∗ = 0

Optimal relative error

r∗(α, σ)

=1 − 2α − 1

2(α − 1)Φ

(

(1

2− α)σ

√T

)

− 2α − 3

2(α − 1)e(1−α)σ2T Φ

(

(α − 3

2)σ√

T

)


The Messages

If one accepts the investment criterion in our model, then

The stock is said to be “good” if α ≥ 0.5


The Messages



In this case one should apply a pure buy-and-hold policy


The Messages



In this case one should apply a pure buy-and-hold policyThe better the stock the smaller relative error; the latterdiminishes to zero as α goes to infinity


The Messages




The stock is said to be “bad” if α < 0.5


The Messages




The stock is said to be “bad” if α < 0.5

In this case one should not buy in the first place, or shouldshort if possible


S&P 500

Mehra & Prescott (1985): a − r = 6.18%, σ = 16.67% basedon S&P 500 (1889-1978)


S&P 500


α = 2.2239 > 0.5 (by large margin)!


S&P 500



Taking T = 1, r∗(α, σ) = 10.15%


S&P 500



Taking T = 1, r∗(α, σ) = 10.15%

Buy and hold an S&P 500 index fund for one year:Statistically expected to achieve almost 90% of the maximumpossible return!


Market Timing

Buy-and-hold rule believed to be the antithesis of markettiming


Market Timing


You can’t really enter on lows and sell on highs so you mightas well just buy and hold


Market Timing



Yet, our model indeed based on market timing (attempting tosell high)


Market Timing




It actually leads to buy and hold!


Market Timing




It actually leads to buy and hold!So the market timing is consistent with buy-and-hold


Insensitivity to Market Parameters

Our optimal solutions insensitive to the market parameters




Definition of good/bad stock involves a range of parameters,instead of specific values





As with S&P 500, α ≥ 0.5 satisfied by a large margin whichwould accommodate sufficient level of (estimation) errors






In statistical terms, verifying whether α ≥ 0.5 much easierthan estimating α itself







Notorious mean–blur problem hardly an issue in our model







Notorious mean–blur problem hardly an issue in our model

On the other hand: an interesting problem to test hypothesisα ≥ 0.5


How Essential is Bang–Bang Policy?

Consider

max06τ6T

E

[

U

(

Pτ

MT

)]

(5)

We have shown that if U is linear then optimal stopping isbang–bang (either τ∗ = T or τ∗ = 0)



Consider

max06τ6T

E

[

U

(

Pτ

MT

)]

(5)


If U is logarithm or power then optimal stopping times areexactly the same



Consider

max06τ6T

E

[

U

(

Pτ

MT

)]

(5)


If U is logarithm or power then optimal stopping times areexactly the same

What about a general U?



Let v(x) := U(ex). The problem is equivalent to

max06τ6T

E[

v(Bµτ − Sµ

T )]

Theorem

(Dai, Jin and Zhou 2008)

If v is increasing and convex, then optimal stopping isbang–bang (τ∗ = T if α ≥ 0.5, and τ∗ = 0 if α < 0.5).



Let v(x) := U(ex). The problem is equivalent to

max06τ6T

E[

v(Bµτ − Sµ

T )]

Theorem

(Dai, Jin and Zhou 2008)

If v is increasing and convex, then optimal stopping isbang–bang (τ∗ = T if α ≥ 0.5, and τ∗ = 0 if α < 0.5).

Assume v is increasing, C2, and |v′′(x)| ≤ ek(x2+1). If optimalstopping is bang–bang, then v must be convex.


Convexity and Behavioural Finance

Experiment 1: You have been given £1000. Now choosebetween




A: Win £1000 with 50% chance and £0 with 50% chance




A: Win £1000 with 50% chance and £0 with 50% chanceB: Win £500 with 100% chance




A: Win £1000 with 50% chance and £0 with 50% chanceB: Win £500 with 100% chanceB was more popular






A: Lose £1000 with 50% chance and £0 with 50% chance






A: Lose £1000 with 50% chance and £0 with 50% chanceB: Lose £500 with 100% chance






A: Lose £1000 with 50% chance and £0 with 50% chanceB: Lose £500 with 100% chanceThis time: A was more popular







Yet the two experiments have exactly the same final wealthpositions!








Reference point (Kahneman and Tversky 1979) or customarywealth (Markowizt 1952)








Reference point (Kahneman and Tversky 1979) or customarywealth (Markowizt 1952)

Risk-averse on gains, and risk-seeking on losses


Maximum as Reference Point

max06τ6T

E[

v(Bµτ − Sµ

T )]

In this model the maximum SµT implicitly taken as reference

point



max06τ6T

E[

v(Bµτ − Sµ

T )]


point

Always in a “loss” situation



max06τ6T

E[

v(Bµτ − Sµ

T )]


point


Hence risk-seeking or having convex v



max06τ6T

E[

v(Bµτ − Sµ

T )]


point


Hence risk-seeking or having convex v

Bang-bang (or buy-and-hold) behaviour consistent withbehavioural theory


Conclusions

It is only natural that an investor wishes to sell a stock closest

to the maximum price over a given planning horizon


Conclusions



The problem is formulated as an optimal stopping problem


Conclusions




We have shown that, even in the absence of transaction costs,

one should buy and hold, if the stock is good, that is


Conclusions






By a good stock we specify that its goodness index is

greater than 0.5


Conclusions







greater than 0.5

This rule is robust with respect to different “utilities” oneapplies


Conclusions







greater than 0.5


This, at least from one angle, reinforces the conventional

maxim of buy and hold


Conclusions







greater than 0.5


This, at least from one angle, reinforces the conventional

maxim of buy and hold

That our model produces buy-and-hold rule suggests thecriterion proposed sensible and warrants further investigations


A Serenity Prayer: Illusion of Control


A Serenity Prayer: Illusion of Control

God, grant me the serenity to accept the things I cannot

control, courage to control the things I can, and the wisdom to

know the difference.

—— Meir Statman


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