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Institute
=
+
0 5 10 15 20 25 30−5
0
5
10
15
20
25
Expected Single Period Returns
Period
r t+τ(%
)
Mean Return
1−std Bounds
0 5 10 15 20 25 300
5
10
15
20
Expected Multi Period Returns (Discount Rates)
Period
µt+τ(%
)
BC IRR
Mean
1−std Bounds
+
0 5 10 15 20 25 300.0
0.2
0.4
0.6
0.8
1.0
Stochastic Discount Factor
Period
m t+τ
Mean
1−std Bounds
0 5 10 15 20 25 300
5
10
15
Equity Price
Period
Pric
e (k
$)
Mean
1−std Bounds
0 50 100 150 2008
10
12
14
16
Evolution of IRR
Deals
IRR
(%)
BC IRR
Filtered IRR
1−std bounds
0 50 100 150 2006
7
8
9
10
11
12
Evolution of Prices
Deals
Pric
e (k
$)
Exact Mean
Observations
Filtered Mean
1−std Bounds
= , . . .
+
+ + = , . . .
+ = +
+
+ =
+
+
+ =
+
( + )
+
+
+ +
+ |
+ /
+
+
+
+
+ +
+
+
+
+
+ + + +
+
( + )
+
= /
+
=
∈] ,+∞[
+
+
+
=
=
( =
) = ( = ) = = −
= ( > )
= (
[=
]> )
= − , = ,
= ( + = | + = )
P + =
( )
+
+
+
+
+
+
+ = + + ( − + )( − )
[+
+
]
= P + .
[+
+
]
+
+
+
( + ) =
(+
+
)
.
([ + | + = ]
)
= + [ + | + = ]
+ P +
= , . . . ( − )
= =
= =
= , . . .
<
P + | =
(
≥ ≤
)
=
P + P + +
P +
P +
+ =
P =
(−−
)
P +
+
+
( , )
L( ; , ) = ( | )
=
( )
( − ) −
( ) = ( , )
( | , ) = ( + , + − )
∼ ( , ), ∼( , ) ⇒ ˆ| , ∼
( + , + − )
( )
∼ ∼
∼ ( , )
∼ ( , )
∼ ( , )
= %
= %
.
.
( − )
0.80 0.85 0.90 0.95 1.00
010
2030
4050
60
true value
round 1
0.80 0.85 0.90 0.95 1.00
010
2030
4050
60
true value
round 2
0.80 0.85 0.90 0.95 1.00
010
2030
4050
60
true value
round 3
0.80 0.85 0.90 0.95 1.00
010
2030
4050
60
true value
round 4
0.80 0.85 0.90 0.95 1.00
010
2030
4050
60
true value
round 5
0.80 0.85 0.90 0.95 1.00
010
2030
4050
60
true value
round 6
0.80 0.85 0.90 0.95 1.00
010
2030
4050
60
true value
round 7
0.80 0.85 0.90 0.95 1.00
010
2030
4050
60
true value
round 8
0.80 0.85 0.90 0.95 1.00
010
2030
4050
60
true value
round 9
0.80 0.85 0.90 0.95 1.00
010
2030
4050
60
true value
round 10
0.80 0.85 0.90 0.95 1.00
010
2030
4050
60
true value
round 11
0.80 0.85 0.90 0.95 1.00
010
2030
4050
60
true value
round 12
●
●●
● ● ●● ●
● ● ● ●
2 4 6 8 10 12
0.75
0.80
0.85
0.90
0.95
1.00
Posterior estimate of π11
observation rounds
prob
abili
ty o
f st
ate
tran
siti
on
posterior estimates
prior estimates
true value
●
●
●
●
●●
●●
● ● ● ●
2 4 6 8 10 12
0.01
00.
015
0.02
00.
025
0.03
0
Change in standard deviation (learning) of π11
observation rounds
Stan
dard
dev
iati
on
0.80 0.85 0.90 0.95 1.00
010
2030
4050
60
true value
round 1
0.80 0.85 0.90 0.95 1.00
010
2030
4050
60
true value
round 2
0.80 0.85 0.90 0.95 1.00
010
2030
4050
60
true value
round 3
0.80 0.85 0.90 0.95 1.00
010
2030
4050
60
true value
round 4
0.80 0.85 0.90 0.95 1.00
010
2030
4050
60
true value
round 5
0.80 0.85 0.90 0.95 1.00
010
2030
4050
60
true value
round 6
0.80 0.85 0.90 0.95 1.00
010
2030
4050
60
true value
round 7
0.80 0.85 0.90 0.95 1.00
010
2030
4050
60
true value
round 8
0.80 0.85 0.90 0.95 1.00
010
2030
4050
60
true value
round 9
0.80 0.85 0.90 0.95 1.00
010
2030
4050
60
true value
round 10
0.80 0.85 0.90 0.95 1.00
010
2030
4050
60
true value
round 11
0.80 0.85 0.90 0.95 1.00
010
2030
4050
60
true value
round 12
●●
●● ●
●● ● ●
● ● ●
2 4 6 8 10 12
0.80
0.85
0.90
0.95
1.00
Posterior estimate of π01
observation rounds
prob
abili
ty o
f st
ate
tran
siti
on
posterior estimatesprior estimates
true value
●
●
●
●
●● ●
●●
●● ●
2 4 6 8 10 12
0.01
00.
015
0.02
0
Change in standard deviation (learning) of π01
observation rounds
Stan
dard
dev
iati
on
=
−
=
+
+
●
●●
● ●●
●●
●● ●
●
2 4 6 8 10 12
0.75
0.80
0.85
0.90
0.95
1.00
Mean estimate of the probability of observing a positive dividend, pt
observation rounds
prob
abili
ty
posterior estimatesprior estimates
true value
+ ∈] ,+∞[
+
+ | + =
= , =
, . . .
L( , | ) ∝
/
(
−∑
=
( ( )− )
)
>
( , | , , , ) =
− (− )
( )
( ) (− ( − )
)
¯ =!
= ( )
(ˆ , ˆ)
ˆ = +
ˆ =
(+ +
(¯− ))
( + )
)−
ˆ =+ ¯
+ˆ = +
¯
ˆ ˆ
ˆ ˆ ˆ ˆ
. .
.
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.5
1.0
1.5
2.0
prior estimate
true density
posterior estimate
round 1
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.5
1.0
1.5
2.0
prior estimate
true density
posterior estimate
round 2
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.5
1.0
1.5
2.0
prior estimate
true density
posterior estimate
round 3
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.5
1.0
1.5
2.0
prior estimate
true density
posterior estimate
round 4
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.5
1.0
1.5
2.0
prior estimate
true density
posterior estimate
round 5
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.5
1.0
1.5
2.0
prior estimate
true density
posterior estimate
round 6
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.5
1.0
1.5
2.0
prior estimate
true density
posterior estimate
round 7
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.5
1.0
1.5
2.0
prior estimate
true density
posterior estimate
round 8
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.5
1.0
1.5
2.0
prior estimate
true density
posterior estimate
round 9
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.5
1.0
1.5
2.0
prior estimate
true density
posterior estimate
round 10
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.5
1.0
1.5
2.0
prior estimate
true density
posterior estimate
round 11
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.5
1.0
1.5
2.0
prior estimate
true density
posterior estimate
round 12
=
●
●
●● ● ● ● ●
● ● ● ●
2 4 6 8 10 12
−0.
20−
0.15
−0.
10−
0.05
0.00
Prior and posterior estimates of m
observation rounds
Expe
cted
val
ue
of t
he
mea
n o
f m
sample mean
●
●
●
●
●
●●
●●
●● ●
2 4 6 8 10 12
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
Change in standard deviation (learning) of m
observation rounds
Stan
dard
dev
iati
on o
f th
e es
tim
ate
of m
● ● ● ●●
● ● ● ● ●● ●
2 4 6 8 10 12
02
46
810
12
Posterior estimates of precision parameter p
observation rounds
expe
cted
val
ue
of p
posterior values
prior values
true precision
●
●
●
●
●
●●
●● ●
● ●
2 4 6 8 10 12
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Change in standard deviation (learning) of p
observation rounds
Stan
dard
dev
iati
on o
f th
e es
tim
ate
of p
−2 −1 0 1 2
0.0
0.1
0.2
0.3
Density of initial and final estimates of location m of ESCRt
N = 1000 Bandwidth = 0.2263
Dens
ity
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
Density of initial and final estimates of precision p of ESCRt
N = 10000 Bandwidth = 0.54
Dens
ity
posterior estimate
prior estimate
−2 −1 0 1 2
0.0
0.1
0.2
0.3
Density of initial and final estimates of location m of ESCRt
N = 1000 Bandwidth = 0.2263
Dens
ity
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
Density of initial and final estimates of precision p of ESCRt
N = 10000 Bandwidth = 0.54
Dens
ity
posterior estimate
prior estimate
, , , ,
P + =
! ",
+( + | + = )
+
+
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���
���
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��
�� �������
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���
���
���
��� ���������� �
��
�� �������
�����������
¯
+
¯
¯
+
∗ ∗∗⋆
∗ ∗∗ ⋆
+ +
+
●
●
●
●
+ = , . . .
+ |
+ |
+
+ | + + |
×=
×. −
×+
×
∼ N ( , )
×=
×.×
+× ×
+×
∼ N ( , )
(′) = ∀ ,
−
+
+
=
+
=
(+ + +
)−
= + + +
+
+
+ =
( )
+ −+
+
+
−
=
+
=
(+ + +
( )
)
=
((− )( + + + )
)
+
+ . . .
∞∏
=
(− + ) ∞ =
+∞
=
(∑
=
(( −∏
=
(− + )
)
+
))
=∑
=
(
(−−∑
=
+ ) +
)
=∑
=
( + + )
+ = { , , . . . , }
+
+
+
( + ) = + × ( + )
+
=∑
=
(+ + +
)
=∑
=+ ( + + )
+
=∑
=
( + + )
=
⎛
⎝−− −∑
=
+
⎞
⎠
= ,
∑
=+ ( + + )
,
=′
×.×
− ′
×.×.(
+)
×
× ×
×
×
×
′. − ′
. . − =′. .
′.( − . )− ( +
− −∑
=
+ ) =′. .
′.( − . )− ( +
′. ) =
′. .
′=[
· · · . . .]
+
×=
′. − ′
. . − ( − ′. )
×=
×
×=
′
×.×
×=
×.×
+×
∼N ( , )
+
=
=
( + ) =
( + ) −=
( + ) −=
+
= +
+
+
+ =+ + + · · ·+ + −
+ =−∑
=
+
+ = (− + )
+
+
= ( + ) −=
= + =
( + + + ) −=
=
=( , )
( )
=
= ( , )( )
=
=
= +
=
−
= − +
= ( − − − − ) +
= − + − − −
= − + ( − −
+ − ( − − )
= − + ˙ + − ( − − )
˙ = − −
−− ≈
= − + ˙
−
=
⎡
⎢⎢⎢⎢⎣+
+ −
⎤
⎥⎥⎥⎥⎦
= − + ˙
=
⎡
⎢⎢⎢⎢⎣
· · ·· · ·
· · ·
⎤
⎥⎥⎥⎥⎦
=
⎡
⎢⎢⎢⎢⎣
· · ·+ · · ·
· · · + −
⎤
⎥⎥⎥⎥⎦
˙ =
⎡
⎢⎢⎢⎢⎣
˙ +
˙ +
˙ +
⎤
⎥⎥⎥⎥⎦
+
+
( + )
˙+
+ = + +˙
+
+ ( +
)
+˙
+
+
= − + ˙ +
∼ ( , )
×=
×
×=
×
×=
×
˙×
=×
= − + +
−
=∑
=
[+ − (
∑= + ) +
∏ −= ( + + )
]
=
(−
− −∑
=
+
)
( + ) −=
( + ) −=
= + + + − −
− = − − −
=
+ = +
= − + +
∼ ( , )
∼ ( , )
= − + +
●
( + ) −=
● ×
● ××
●
( , )
= +
●
●
×
●
( , )
ˆ− = ˆ−− +
− = −− +
= − ˆ−
= − +
= − −
׈+ = ˆ− +
+ = ( − ) −
( + )=
= =
− ˆ×ˆ− +
×
ˆ+
ˆ−
ˆ−
ˆ
ˆ−
ˆ
− ˆ
ˆ+
( + ) −=
=
0 5 10 15 20 25 300
2
4
6
8
10
Base Case & Expected Dividend
Period
Divi
dend
(k$)
Base Case
Expected
0 5 10 15 20 25 300
20
40
60
80
100
Dividend Volatility
Period
Vola
tility
(%)
0 5 10 15 20 25 300
20
40
60
80
100
Transition Probabilities from Non−Payment State
Period
Prob
abili
ty (%
)
0 → 1
0 → 0
0 5 10 15 20 25 300
20
40
60
80
100
Transition Probabilities from Payment State
Period
Prob
abili
ty (%
)
1 → 1
1 → 0
+
+
0 5 10 15 20 25 300
20
40
60
80
100
Lockup and Payment Probabilities
Period
Prob
abili
ty (%
)
Lockup
Payment
0 5 10 15 20 25 300
5
10
15
20
Expected Period Losses
Period
Expe
cted
Los
s (%
)+ +
0 5 10 15 20 25 300
1
2
3
4
5
Risk Free Rate
Period
r t+τ
f(%
)
0 5 10 15 20 25 30−5
0
5
10
15
20
25
Expected Excess Returns
Period
λ t+τ
(%)
Prior
Filtered Mean
1−std Bounds
+
+
+
+ +
0 5 10 15 20 25 30−5
0
5
10
15
20
25
Expected Single Period Returns
Period
r t+τ(%
)
Mean Return
1−std Bounds
0 5 10 15 20 25 300
5
10
15
20
Expected Multi Period Returns (Discount Rates)
Period
µt+τ(%
)
BC IRR
Mean
1−std Bounds
+
0 5 10 15 20 25 300.0
0.2
0.4
0.6
0.8
1.0
Stochastic Discount Factor
Period
m t+τ
Mean
1−std Bounds
0 5 10 15 20 25 300
5
10
15
Equity Price
Period
Pric
e (k
$)
Mean
1−std Bounds
0 5 10 15 20 25 300
20
40
60
80
100
Cash Yield
Period
Cash
Yie
ld (%
)
0 5 10 15 20 25 300
5
10
15
20
Duration
Period
Dur
atio
n (y
ear)
20 40 60 80 1000
5
10
15
Price vs Volatility
Avg Dividend Volatility (%)
Pric
e (k
$)
20 40 60 80 1005
10
15
20
IRR vs Volatility
Avg Dividend Volatility (%)
IRR
(%)
0 50 100 150 2008
10
12
14
16
Evolution of IRR
Deals
IRR
(%)
BC IRR
Filtered IRR
1−std bounds
0 50 100 150 2006
7
8
9
10
11
12
Evolution of Prices
Deals
Pric
e (k
$)
Exact Mean
Observations
Filtered Mean
1−std Bounds
+
+
%
%
+ +
[ + ] +
+ +
5 10 15 208
9
10
11
12
Price after 3 Dividend Payments
Deals
Pric
e (k
$)
Ex−ante Expectation1−std BoundsObservedFiltered1−std Bounds
0 5 10 15 20 25 300
5
10
15
20
Term Structure after 3 Dividend Payments
Period
µt+τ(%
)
Ex−ante1−std BoundsAfter 3 Pyaments1−std Bounds
( )
+
=
=
=
+
( + ) =∑
=
+ × +
+
= . . .
= . . .
+ =
√√√√∑
=
+ ( + − ( + ))
+
+ |
+
+
+ = ( + )
= ( + )− ( )
( + ) = ( ( + )− ( ))
= ( ( + ))− ( ( ))
( ) ≈ ( ( )) +( )
× ( )
+
+
+ = + −
+ +
+
( + + ) = ( + )
+
+
+
+ = + + −
= ( + + ) + −
+
( + + )
+ = + + +
+
+ = ( + )
= ( + )− ( )
( + ) = + ( + )
( )−
= ( + + ) +( + ) −
[ + ] = [ ( + + )]
≈ ( [ + [ + ] +( +
[ + ]
≈ ( [ + [ + ]
( + ) ≈ + + +
= /
= /
= /
=− ∂
∂
=− ∂
∂
∑
= +
− (− )
⇒ =∑
= +
( − ) − (− )
=∑
= +
− (− )
=−
+
∈ [ ; ] P +
P +
L( | ) =
( )
( − ) −
( )= !
!( − )!
( | ) ∝ ( )L( | )
( )
( ; , ) =( + )
( ) ( )− ( − ) −
( | ) ∝ ( )L( | )
∝ ( − ) − × − ( − ) −
∝ ( + , + − )
= (( − ) − )×
= × ( − )
( )
+
( + )( + + )
L( , | ) ∝ /
(
−∑
=
( ( )− )
)
¯ =!
= ( )
( ; , ) =− (− )
( )
( ; , , , ) =
√( + )
( )×
(+ ( − )
)− −
(ˆ , ˆ)
ˆ = +
ˆ =
(+ +
(¯− ))
( + )
)−
ˆ =+ ¯
+ˆ = +
= ( +( )
)
= ( ( ))− ( +( )
)
= ( ( ))−
( + )
( ( )− ) ( + )
.
=
=
=
=
¯
ˆ ˆ ˆ ˆ
ˆ ˆ
[ ] ∗
∗
=∗
[ ]
∗∗
( + )
[ ]( + )
=∑
=
[ ]
( + )
=∑
=
[ ]
( + )
[ ]
( + )=
[ ]
( + )
⇒ =( + )
( + )
+ =( + )+
( + )+=
( + )
( + )
+ −= − ( + )
( + )
− ( + )( + )
[ + ]
[ + ]( + )
+ = +
( ) = + [ + ]( + )
( + )
= + [ + ]
( + )
= + [ + ]
( + )
= ( )
( ) =[ + ]( + )
( + )
=[ + ]
( + )
( + )
( + )
= ( )( + )
( + )
( ) < ( ) >
+ < +
+ = +
($ )
$
($ )
$
($ )
($ )
$
$
$
%
=
$
%
$
%
%
$ $
. %
%
%
% . %
+
′ ′
+
+ =∏
=
( + ) =
(∑
=
+
)
+
+
+
+
+ +
( + ) =!
= ( + + + )
=∑
=
(−! −
= +!
= ( + + + )
)
=∑
=
(!
= ( + + + − +− )
)
=
[∑
=
!= ( + + + − +− )
]
=∑
=
!= ( + + + − +− )
+
+
+
(∑
= ( + + + )−∑ −= + )
= ¯ = ¯
=
[∑
=
( ¯− )̄
]
=
[∑
=
( (¯− )̄)
]
¯ = ( + ¯) ¯= ( + ¯)
=
[∑
=
(( ( + ¯)− ( + ¯))
)]
=
[∑
=
( (+ ¯
+ ¯
) )]
=
[∑
=
(+ ¯
+ ¯
) ]
=
[ ∞∑
=
(+ ¯
+ ¯
)−
∞∑
= +
(+ ¯
+ ¯
) ]
=
[ ∞∑
=
( ) −∞∑
= +
( )
]
=
(+ ¯
+ ¯
)
∞∑
=
=∞∑
=
− =∞∑
=
= −
=+
−
∞∑
= +
= +∞∑
= +
− − = +∞∑
=
= +
− =+
−
[+
+
]
=+
−
⎡
⎣ −(
+
+
) ⎤
⎦
++ ∞
(¯− ∞)( +¯)
= [ + ] − + [ + ] − − + + ...
+ [ + ] − − + ... − + −
= ( [ + ] + · · ·+ [ + ])
− ( + ) ( [ + ] + · · ·+ [ + ])− . . .
− ( + − + + − ) [ + ]
=[
[ + ] · · · [ + ]]
⎡
⎢⎢⎢⎢⎣
⎤
⎥⎥⎥⎥⎦
−[
[ + ] · · · [ + ]].
⎡
⎢⎢⎢⎢⎣
· · ·· · ·
· · ·
⎤
⎥⎥⎥⎥⎦.
⎛
⎜⎝
⎡
⎢⎣
+ −
⎤
⎥⎦+
⎡
⎢⎣+ −
⎤
⎥⎦
⎞
⎟⎠
=′
×.×
− ′
×.×.(
+)
×
×=
⎡
⎢⎢⎢⎢⎣
⎤
⎥⎥⎥⎥⎦ ×=
⎡
⎢⎢⎢⎢⎣
[ + ]
[ + ]
[ + ]
⎤
⎥⎥⎥⎥⎦
×=
⎡
⎢⎢⎢⎢⎣
· · ·· · ·
· · ·
⎤
⎥⎥⎥⎥⎦
×=
⎡
⎢⎢⎢⎢⎣+
+ −
⎤
⎥⎥⎥⎥⎦
×=
⎡
⎢⎢⎢⎢⎣+
+ −
⎤
⎥⎥⎥⎥⎦
′. − ′
. . − =′. .
=
− =+
+ − + −
− = [ − ˙ ]
=[
− ( − − )]
− = −−
−
[+ − + −
−
]=
+ − + − =−
− =− −
−
−
[+ − + −
−
]=
−[
+ −]= − −
− =− −
+ −
−
−
− = − + −
+ − + −
− = − + −
+ − + −−
− =
[− + −
−
]
=−
[
− +− −
+ −
]
=−
[
− ++ −
]
| − − | −−
− ( + − + −− ) = − + −
−
[
+ − + −
−
[
− ++ −
]]
= − +− −
+ −
−[
+ −]+ −
− + −
+ −
= − +− −
+ −
− = −
+ −+
( + − )( + − )−
− −
( + − )− ( − + − )
( + − )( + − )
−
=∑
=
[+ − (
∑= + ) +
∏ −= ( + + )
]
=
(−
− −∑
=
+
)
= + + + − −
− = − − −
=
, . . . ,
( , . . . , ; )
= { , . . . , , , . . . , , , , . . . , }
, . . . ,
, . . . ,
, . . . ,
( | : ) ( | : ) . . .
( | : − )
( , . . . , ; )
( , . . . , ; ) =∏
=
( | : − ; )
( | : − ; )
( − )
( ) = −∑
=
| |−∑
=
( − )
ˆ
ˆ = ( )
( +
) ( + )++ ( +
)
+
+ ∝+ − ( +
)++ ∝ +
+ −( + ) +
+
++ −
+
= ( )×
×
( )
− , . . . ,
− , . . . ,
( + )=
( + )=( + )= = =
( + )= = . =
( + )=
( + )=
= . = = = . =
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