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Managing the risk associated with bandwidth demand · PDF fileManaging the risk associated...
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Managing the risk associated with bandwidth demand uncertainty Sverrir Olafsson Mobility Research Centre
Transcript of Managing the risk associated with bandwidth demand · PDF fileManaging the risk associated...
Managing the risk associated with bandwidth demand uncertaintySverrir OlafssonMobility Research Centre
Content
• Uncertain bandwidth requirements– Quantification
– Risk management
• Implementation of capacity instalment process
• Estimating time to capacity expiry
• Optimal timing of capacity instalment
• Use of real options
Bandwidth demand risk
• “Every day I look at the decision: should we build or should we lease”?
• Unknown demand for bandwidth – Uncertain future applications
– Uncertain uptake of future applications
– Uncertain customer base
Bandwidth price risk
• Unknown price of bandwidth– Prices are on their way down
– The rate of decline is unknown ⇒ risk
What uncertainties?
• Possible scenarios
• Require different risk management
• Most risk management procedures assume– Known demand (currency, commodity,…)
– Uncertain price
Uncertainties• Required bandwidth• Price of bandwidth
Analogy to commodity market were both the magnitude and price of
commodity are unknown
Demand knownUncertainties• Price of bandwidth
Analogy to commodity market were magnitude is known but the price
of commodity is unknown
Aim of risk management
• Identify risk sources
• Quantify the risk caused
• Control the risk– Hedge
• only some risks can be hedged (commodities markets,….)
– More efficient decision making
• operational caution
• insurance
• real options
Bandwidth demand risk
• Required bandwidth can be acquired in different ways– Building networks
– Install additional capacity, lit fibre
– Lease
– Enter derivative contracts (futures, swaps, options)
• Before deciding on action– Model bandwidth demand evolution
– Model price evolution
Modelling bandwidth demand
• Evidence for “exponential growth”
• But, rate of growth is uncertain
• Model as geometric Brownian motion
• Therefore tttt dWDdtDdD σµ +=
+
−= tt WtDD σσµ 2
0 21
exp
[ ] ( ) [ ]== tt DtDDE var;exp0 µ
( )1,0NdtdW
t
tt
∈=
ηη
0 100 200 300 400 500 600 700 800 900 10000
2 0
4 0
6 0
8 0
100
120
0 100 200 300 400 500 600 700 800 900 100020
40
60
80
100
120
140
Time [days]
Val
ue
0 100 200 300 400 500 600 700 800 900 10000
2 0
4 0
6 0
8 0
100
120
Sta
nd
ard
dev
iati
on
x0 = 50, µ = 0.3, σ = 0.5
ProcessMeanStandard deviation
gbmo(50,0.3,0.50,1/360,1000,1)
Possible demand scenarios
• Monte Carlo simulations – iterate a large number of scenarios each compatible with the assumptions
0 100 200 300 400 500 600 700 800 900 10000
50
100
150
200
250
Time (Days)
Req
uire
d ca
paci
tyCapacity evolution, µ = 0.3, σ = 0.25, D
0 = 50
justgbm.m
Assumptions and questions
• Given – Present demand D0
– Presently available capacity C0, D0 < C0
• Questions– What is the probability of exceeding the installed capacity within a
given time?
– What is the proper capacity instalment rate?
• Contrast the expected life of presently installed capacity with expectations about price evolution
Probability of exceeding capacity
• Probability of exceeding installed capacity
• Probability density function
• Cumulative probability function
0 500 1000 1500 2000 25000
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Days
Pro
babi
lity
den
sity
Parameter estimates, a = 6.2112, b = 141.9538
Empirical dataGamma fit
0 500 1000 1500 2000 2500 30000
0.2
0.4
0.6
0.8
1
Days
Cum
ulat
ive
prob
abil
ity
µ = 0.35, σ = 0.25, Initial demand = 50, Capacity = 150
Log-normalEmpiricalGamma
gmbreach.mgmbreach.m
Impact of uncertainty
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
Days
Pro
babi
lity
den
sity
µ = 0.5, σ = 0.2, Initial demand = 50, Capacity = 150
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Days
Cum
ulat
ive
prob
abil
ity
µ = 0.5, σ = 0.2, Initial demand = 50, Capacity = 150
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
Days
Pro
babi
lity
den
sity
µ = 0.5, σ = 0.4, Initial demand = 50, Capacity = 150
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Days
Cum
ulat
ive
prob
abili
ty
µ = 0.5, σ = 0.4, Initial demand = 50, Capacity = 150
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Days
Pro
babi
lity
den
sity
µ = 0.5, σ = 0.8, Initial demand = 50, Capacity = 150
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Days
Cum
ulat
ive
prob
abil
ity
µ = 0.5, σ = 0.8, Initial demand = 50, Capacity = 150
µ = 0.5 , σ = 0.2 µ = 0.5 , σ = 0.4 µ = 0.5 , σ = 0.8
All scenarios have the samemean capacity life
=
0
0log1
DC
tµ
Justification for log-normal modelling
• The empirical cumulativeprobability is well approximated by the log-normal distribution
0 1000 2000 3000 4000 5000 60000
0.2
0.4
0.6
0.8
1
Days
Cum
ulat
ive
prob
abili
ty
µ = 0.25, σ = 0.4, Initial demand = 50, Capacity = 150
Log-normalEmpiricalGamma
( ) ( ) ( )
−−+=≤t
tDCerftCDCP t 22//log1
21,
20
σσµ
• Deviations are explained by demand exceeding installed capacity and then go down below installed capacity again
Criteria for delaying instalment
• There are benefits in delaying the acquirement of additional capacity– Cost
– Efficient usage
• Risks– QoS reduction
– Loss of customers
• Quantifying the criteria requires assumptions on the price evolution
Bandwidth demand risk
• Bandwidth evolution is a stochastic process D(t)
• Match installed capacity optimally to demand
• Upgrade sequence
• The process C(t) should stochastically dominate D(t)
• Approach– Dynamic programming, simulation, real options
;...,,...,, 11 kk CtCt ∆∆=Ω
( ) ( )( ) 1Pr →≥ tDtC
Controlled instalments
• Probability to exceed installed capacity
• The instalment process will depend on– Expected growth
– Expected volatility
– Required QoS
( ) ( ) ( )
−−+=≤
ttDC
erftCDCP t 22//log
121
,2
000 σ
σµ
][
],[
1
1
instalmentcontrolleddCCCC
eduncontrollGBMdDDDD
TTTT
tttt
+=→+=→
+
+
Instalment strategy
• Probability that demand does not exceed installed capacity for different instalment strategies
Program:tempcapincrease.m0 1000 2000 3000 4000 50000.5
0.6
0.7
0.8
0.9
1Initial demand = 50, Initial capacity = 100, µ = 0.2, σ = 0.50
DaysPro
babi
lity
tha
t de
man
d is
bel
ow t
he in
stal
led
capa
city
µinc=0.05
µinc=0.10
µ inc=0.15
µinc=0.20
µinc=0.25
µ inc=0.30
0 200 400 600 800 10000.65
0.7
0.75
0.8
0.85
0.9
0.95
1Initial demand = 50, Initial capacity = 100, µ = 0.2, σ = 0.50
DaysPro
babi
lity
tha
t de
man
d is
bel
ow t
he in
stal
led
capa
city
µinc=0.05
µinc=0.10
µinc=0.15
µinc=0.20
µinc=0.25
µinc=0.30
Time to capacity exhaustion
• Assume additional capacity is installed at the average rate µi
• Time to exhaustion
( ) ( )tCttWtDD int µσσ
µ exp2
exp 0
2
0 =
+
−=
( )( )
( )
2
log
2
log
20
0
020
0
σµµσ
σµµ −−
=⇒+−−
==
i
WE
i
DC
tEtW
DC
t
Different instalment strategies
• Installing capacity at different– Rates
– Time intervals
010
2030
4050
050
100150
20090
92
94
96
98
100
Excess instalment [%]
Initial demand =50, Initial capacity = 75, µ = 0.35, σ = 0.25
Days between instalments
Cap
acit
y co
vera
ge [
%]
010
2030
4050
0
200
400
60075
80
85
90
95
100
Excess instalment [%]
Initial demand =50, Initial capacity = 75, µ = 0.35, σ = 0.25
Days between instalments
Cap
acit
y co
vera
ge [
%]
capacityplot([0:0.05:0.45],[1:50:500],50,75,0.35,0.25,1/360,1000,1000,1.5)capacityplot([0:0.05:0.45],[1:50:200],50,75,0.35,0.25,1/360,1000,1000,1.5);
Simple model for price of bandwidth
• Price of bandwidth has been going down– [P] = $/year/mile/megabit
• The real uncertainty is regarding the rate of decline in price
( ) ( ) 11 ++=+ ttaStS η
( ) ( ) 221 var,0,,0 ηη σηηηση ===∈ + ttttt EEN
( ) ( ) ∑=
+−+=+
k
iit
ikk atSaktS1
η
Simple model for price of bandwidth
• Price expectations and variance
• Therefore - even if the future expected spot price decreases its variance increases as long as a < 1
( ) ( )tSaktSE kt =+
( ) ( )( )σ σ ση ηSn
n
k k
k S t k aaa
2 2 2
0
12
2
2
11
= + = = −−
=
−
∑var
( ) ( ) ( ) E S t k E S t k E S t+ > + + > > + ∞1 ...
( ) ( ) ( )σ σ σS S Sk k2 2 21< + < < ∞...
0 5 0 100 150 200 250 300 3 5 0 4000
2
4
6
0 5 0 100 150 200 250 300 3 5 0 4004 0
5 0
6 0
7 0
8 0
9 0
100
Time
Val
ue
0 5 0 100 150 200 250 300 3 5 0 4000
1
2
3
4
5
6
Stan
dard
dev
iati
on
s0 = 100, a = 0.99858, σ = 0.35, Expected annual price change [%] = -0.4
When to install additional capacity
• Take into account the “damage” of capacity exhaustion
• Develop analogies to efficient frontier in portfolio management
• Optimal decisions» Attitudes to risk
» Utility function
» etc
When to install additional capacity?
• Delaying capacity instalment– Provides monetary benefits
– Incurs risk
0 500 1000 1500 20000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Days
Cum
ulat
ive
prob
abil
ity
µ = 0.3, σ = 0.25, Initial demand = 50, Capacity = 100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Probability of exceeding capacity
Gai
ns f
rom
dec
line
in p
rice
µ = 0.3, σ = 0.25, In dem = 50, Cap = 100 , Pr decline = 0.3
When to install additional capacity?
• The impact of volatility on expected benefits
gmbreach.m0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Probability of exceeding capacity
Gai
ns f
rom
dec
line
in p
rice
µ = 0.5, In dem = 50, Cap = 150 , Pr decline = 0.3
σ = 0.20σ = 0.40σ = 0.60σ = 0.80σ = 0.90
10000 iterations - days1500 experiments
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Probability of exceeding capacity
Gai
ns f
rom
dec
line
in p
rice
µ = 0.5, In dem = 50, Cap = 150 , Pr decline = 0.3
σ = 0.20σ = 0.40σ = 0.60σ = 0.80σ = 0.90
2500 iterations - days1500 experiments
Cost benefit analysis
• Delaying capacity instalment is not only a question of making monetary savings
• What are the implications for deterioration in QoS on customers?
• The expected gains from delaying investment
• We assume the following loss as a function of time( ) ( ) ( )( )tptg κ−−= exp10
( ) ( ) ( )
≤>
=−=0;00;1
;xifxif
xttttc c θαθ
Cost benefit analysis
• Benefits from delaying capacity investment
• Disadvantage from exceeding installed capacity –resulting service deterioration
• “Optimal” instalment timedepends on the assumptionsmade about
– price decline
– losses from QoS depreciation
0 500 1000 1500 2000 2500 30000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Days
Cos
t/b
enef
it
µ = 0.3, σ = 0.25, In dem = 50, Cap = 100 , Pr decline = 0.3, α = 1, tc = 10
Hedge against price risk
• Hedge ratio, – h=(size of futures contract/size of exposure)
• Consider short hedge
• This assumes known demand but uncertain price
HSFSh hh σρσσσσ 22222 −+=
( )( )
( )FS
F
S
FS
F
S
σσρ
σ
σ
,cov
var
var2
2
=
=
=
F
SFSF
h hhh σ
σρσρσσ∂
∂σ =⇒=−= 022 22
FhShFS ∆−∆=∆Π⇒−=Π
Hedge against price and demand risk
• Demand for and price of future capacity are unknown
• Then, the correlation between demand and price matters
• Put together a portfolio
• Optimal hedge ratio
( ) ( ) ( )FDShFhDShFDS ,cov2varvar2 −+=⇒−=Π Πσ
( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )( )( )( )F
FEFSESDEDEFDSEFSDEh
var,cov,cov −−−++
=
Real options approach
• The starting point is that demand follows
• If F = F(D,C,P,t) is value of investment which depends – Demand, D(t)
– Instalment strategy, C(t)
– Price evolution, P(t)
• The conditions F = F(D,C,P,t) has to satisfy can be derived from Ito’s Lemma
tttt dWDdtDdD σµ +=
Real options approach
• Differential equation for value of investment
• With κ, market price of risk, r risk free interest rate, C(t) presently installed capacity
• Market price of risk captures the tradeoffs between risk and return for investments in capacity. The expected return on investment is
( ) ( )( ) ( ) 0,min21
2
222 =−+
∂∂−−
∂∂−
∂∂ tLPtCDrF
DF
DFD
tF κσµσ
κσ+= rR
Summary
• Stochastic models for bandwidth demand and price evolution are considered
• In spite of falling bandwidth prices there is still a considerable risk exposure
• Stochastic modelling of bandwidth demand and price evolution allow – Operator risk exposure to be quantified
– Bandwidth instalment strategies to be formulated
• After making assumptions on the cost of running out of bandwidth the optimal timing of capacity instalment is decided
• We consider real options approach where value is controlled by price evolution and instalment strategy
