Managing the risk
associated with bandwidth
demand uncertainty
Sverrir Olafsson
Mobility Research Centre
Sverrir.Olafsson@bt.com
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
Sverrir.Olafsson@bt.com
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
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Bandwidth price risk
• Unknown price of bandwidth
– Prices are on their way down
– The rate of decline is unknown ⇒ risk
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What uncertainties?
• Possible scenarios
Demand known
Uncertainties
• Price of bandwidth
Analogy to commodity market were
magnitude is known but the price
of commodity is unknown
Uncertainties
• Required bandwidth
• Price of bandwidth
Analogy to commodity market were
both the magnitude and price of
commodity are unknown
• Require different risk management
• Most risk management procedures assume
– Known demand (currency, commodity,…)
– Uncertain price
Sverrir.Olafsson@bt.com
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
Sverrir.Olafsson@bt.com
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
Sverrir.Olafsson@bt.com
Modelling bandwidth demand
• Evidence for “exponential growth”
• But, rate of growth is uncertain
• Model as geometric Brownian motion
dDt = µDt dt + σDt dWt
• Therefore
dWt = ηt dt
ηt ∈ N (0,1)
120
Process
Mean
120 Standard deviation
100
100
80
80
60
60
40
40
20
20
0
100
200
300
400 5 0 0 600
Time [days]
700
gbmo (50,0.3,0.50,1/360,1000,1)
800
0
900 1000
Standard deviation
Value
x 0 = 50, µ = 0.3, σ = 0.5
140
1 2


Dt = D0 exp  µ − σ t + σWt 
2 


E [Dt ] = D0 exp (µt ) ; var [Dt ] =
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Possible demand scenarios
• Monte Carlo simulations – iterate a large number of
scenarios each compatible with the assumptions
Capacity evolution, µ = 0.3, σ = 0.25, D 0 = 50
250
Required capacity
200
150
100
50
justgbm.m
0
0
100
200
300
400 500 600
Time (Days)
700
800
900 1000
Sverrir.Olafsson@bt.com
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
Sverrir.Olafsson@bt.com
Probability of exceeding capacity
• Probability of exceeding installed capacity
• Probability density function
• Cumulative probability function
Parameter estimates, a = 6.2112, b = 141.9538
µ = 0.35, σ = 0.25, Initial demand = 50, Capacity = 150
1
Empirical data
Gamma fit
0.035
0.8
Cumulative probability
Probability density
0.03
Log-normal
Empirical
Gamma
0.025
0.02
0.015
0.01
0.6
0.4
0.2
0.005
0
0
gmbreach.m
500
1000
1500
Days
2000
2500
0
0
500
1000
1500
Days
2000
2500
3000
gmbreach.m
All scenarios have the same
mean capacity life
Sverrir.Olafsson@bt.com
Impact of uncertainty
µ = 0.5 , σ = 0.2
t=
µ = 0.5 , σ = 0.4
µ = 0.5, σ = 0.2, Initial demand = 50, Capacity = 150
µ = 0.5 , σ = 0.8
µ = 0.5, σ = 0.8, Initial demand = 50, Capacity = 150
µ = 0.5, σ = 0.4, Initial demand = 50, Capacity = 150
0.09
0.05
0.04
0.045
0.03
0.025
0.02
0.015
0.07
0.04
0.035
0.03
0.025
0.02
0.06
0.05
0.04
0.03
0.015
0.02
0.01
0.01
0.005
0.005
0
0.08
Probability density
Probability density
0.035
0
0
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Days
0
µ = 0.5, σ = 0.2, Initial demand = 50, Capacity = 150
0.9
0.9
0.8
0.8
0.6
0.5
0.4
0.3
0
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Days
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.7
0.6
0.5
0.4
0.3
0.2
0.2
0.1
0.1
0.1
0
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Days
0
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Days
µ = 0.5, σ = 0.8, Initial demand = 50, Capacity = 150
0.2
0
0
1
Cumulative probability
1
0.7
0.01
µ = 0.5, σ = 0.4, Initial demand = 50, Capacity = 150
1
Cumulative probability
Cumulative probability
Probability density
0.05
0.045
C 
1
log 0 
µ
 D0 
0
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Days
0
0
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Days
Sverrir.Olafsson@bt.com
Justification for log-normal modelling
•
The empirical cumulative
probability is well
1
 log(C / D0 ) − µ − σ 2 / 2 t  
approximated by the
CP( Dt ≤ C , t ) = 1 + erf 
 
log-normal distribution
2
σ 2t
(

)

µ = 0.25, σ = 0.4, Initial demand = 50, Capacity = 150
1
•
Cumulative probability
0.8
0.6
0.4
0.2
0
Log-normal
Empirical
Gamma
0
1000
2000
3000
Days
4000
5000
6000
Deviations are explained by demand
exceeding installed capacity and then go
down below installed capacity again

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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
Sverrir.Olafsson@bt.com
Bandwidth demand risk
• Bandwidth evolution is a stochastic process D(t)
• Match installed capacity optimally to demand
• Upgrade sequence
Ω = {t1 , ∆C1 ,..., t k , ∆Ck ;...}
• The process C(t) should stochastically dominate D(t)
Pr (C (t ) ≥ D(t )) → 1
• Approach
– Dynamic programming, simulation, real options
Sverrir.Olafsson@bt.com
Controlled instalments
• Probability to exceed installed capacity
 log (C0 / D0 ) − (µ − σ 2 / 2 )t  
1
CP ( Dt ≤ C0 , t ) = 1 + erf 
 
2
σ 2t


Dt → Dt +1 = Dt + dDt
[GBM , uncontroll ed ]
CT → CT +1 = CT + dCT
[controlled instalment ]
• The instalment process will depend on
– Expected growth
– Expected volatility
– Required QoS
Sverrir.Olafsson@bt.com
Instalment strategy
Initial demand = 50, Initial capacity = 100, µ = 0.2, σ = 0.50
1
0.95
0.9
0.85
µ
inc
0.8
=0.05
µinc =0.10
µ
inc
0.75
=0.15
µ
inc =0.20
µ
0.7
inc
=0.25
µ
inc =0.30
0.65
0
200
400
600
Days
800
1000
Probability that demand is below the installed capacity
Probability that demand is below the installed capacity
• Probability that demand does not exceed installed
capacity for different instalment strategies
Initial demand = 50, Initial capacity = 100, µ = 0.2, σ = 0.50
1
0.9
0.8
µ
µ
0.7
inc
=0.05
inc =0.10
µ inc =0.15
µ
0.6
µ
inc
inc
=0.20
=0.25
µ inc =0.30
0.5
0
1000
2000
3000
Program:tempcapincrease.m
Days
4000
5000
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Time to capacity exhaustion
• Assume additional capacity is installed at the average
rate µi


σ2
Dt = D0 exp   µ − t + σW (t )t  = C0 exp ( µin t )
2


• Time to exhaustion
 C0 
log 
D0 

t=
σ2
µ − µi −
+ σW (t )
2
⇒
E (W )= 0
 C0 
log 
D0 

E (t ) =
σ2
µ − µi −
2
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Different instalment strategies
• Installing capacity at different
– Rates
– Time intervals
Initial demand =50, Initial capacity = 75, µ = 0.35, σ = 0.25
Initial demand =50, Initial capacity = 75, µ = 0.35, σ = 0.25
100
Capacity coverage [%]
Capacity coverage [%]
100
98
96
94
92
90
200
150
100
20
50
Days between instalments
0
0
30
40
50
95
90
85
80
75
600
200
10
Excess instalment [%]
capacityplot([0:0.05:0.45],[1:50:200],50,75,0.35,0.25,1/360,1000,1000,1.5);
50
400
Days between instalments
20
0
0
30
40
10
Excess instalment [%]
capacityplot([0:0.05:0.45],[1:50:500],50,75,0.35,0.25,1/360,1000,1000,1.5)
Sverrir.Olafsson@bt.com
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
S (t + 1) = aS (t ) + η t +1
ηt ∈ N (0, σ η ), Et {η t +1} = 0, var (ηt ) = E {ηt2 } = σ η2
k
S (t + k ) = a S (t ) + ∑ a k −iηt +i
k
i =1
Sverrir.Olafsson@bt.com
Simple model for price of bandwidth
Value
• Price expectations and variance
Et {S (t + k )} = a S (t )
k
σ ( k ) = var ( S (t + k )) = σ
2
S
90
5
80
4
70
3
60
2
50
1
40
0
k −1
2
η
∑a
n =0
2n
1 − a2 k 
=σ 

 1 − a2 
50
100
150
200
Time
2
η
• Therefore - even if the future expected spot price
decreases its variance increases as long as a < 1
E {S ( t + k )} > E {S (t + k + 1)} >... > E {S (t + ∞ )}
σ S2 ( k ) < σ S2 ( k + 1) <... < σ S2 ( ∞)
250
300
350
0
400
Standard deviation
s0 = 100, a = 0.99858, σ = 0.35, Expected annual price change [%] = -0.4
100
6
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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
Sverrir.Olafsson@bt.com
When to install additional capacity?
• Delaying capacity instalment
– Provides monetary benefits
– Incurs risk
µ = 0.3, σ = 0.25, Initial demand = 50, Capacity = 100
µ = 0.3, σ = 0.25, In dem = 50, Cap = 100 , Pr decline = 0.3
1
0.9
0.9
0.8
Gains from decline in price
Cumulative probability
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.6
0.5
0.4
0.3
0.2
0.1
0.1
0
0.7
0
500
1000
Days
1500
2000
0
0
0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8
Probability of exceeding capacity
0.9
1
Sverrir.Olafsson@bt.com
When to install additional capacity?
• The impact of volatility on expected benefits
µ = 0.5, In dem = 50, Cap = 150 , Pr decline = 0.3
µ = 0.5, In dem = 50, Cap = 150 , Pr decline = 0.3
0.9 10000 iterations - days
1500 experiments
0.8
0.8 2500 iterations - days
1500 experiments
Gains from decline in price
Gains from decline in price
0.7
0.6
0.5
0.4
0.3
σ = 0.20
0.2
σ = 0.40
σ = 0.60
0.1
σ = 0.80
σ = 0.90
0
0
0.2
0.4
0.6
0.8
Probability of exceeding capacity
1
0.7
0.6
0.5
0.4
0.3
σ = 0.20
0.2
σ = 0.40
σ = 0.60
0.1
σ = 0.80
0
σ = 0.90
0
0.2
0.4
0.6
0.8
Probability of exceeding capacity
1
gmbreach.m
Sverrir.Olafsson@bt.com
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
g (t ) = p(0)(1 − exp (− κt ))
• We assume the following loss as a function of time
1 ; if x > 0
c(t ) = αθ (t − tc )t ; θ ( x ) = 
0 ; if x ≤ 0
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Cost benefit analysis
• Benefits from delaying capacity investment
• Disadvantage from exceeding installed capacity –
resulting service deterioration
• “Optimal” instalment time
depends on the assumptions
made about
µ = 0.3, σ = 0.25, In dem = 50, Cap = 100 , Pr decline = 0.3, α = 1, t = 10
c
0.5
0.45
0.4
– price decline
Cost/benefit
– losses from QoS depreciation
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
500
1000
1500
Days
2000
2500
3000
Sverrir.Olafsson@bt.com
Hedge against price risk
• Hedge ratio,
– h=(size of futures contract/size of exposure)
• Consider short hedge
Π = S − hF ⇒ ∆Π = ∆S − h∆F
σ = σ + h σ − 2 hρσ Sσ H
2
h
2
S
2
2
F
var (S ) = σ
var ( F ) = σ
ρ =
2
S
2
F
cov ( S , F )
σ Sσ F
∂σ h2
σ
= 2hσ F2 − 2 ρσ Sσ F = 0 ⇒ h = ρ S
∂h
σF
• This assumes known demand but uncertain price
Sverrir.Olafsson@bt.com
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
Π = DS − hF ⇒ σ Π2 = var ( DS ) + h var ( F ) − 2 h cov( DS , F )
• Optimal hedge ratio
E (D ) cov (S , F ) + E (S ) cov( D, F ) + E (( D − E ( D ))(S − E (S ))( F − E ( F )))
h=
var (F )
Sverrir.Olafsson@bt.com
Real options approach
• The starting point is that demand follows
dDt = µDt dt + σDt dWt
• 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
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Real options approach
• Differential equation for value of investment
∂F 1 2 2 ∂ 2 F
∂F
− σ D
−
(
µ
−
κσ
)
+ rF − min ( D, C (t ))LP(t ) = 0
2
∂t 2
∂D
∂D
• 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
R = r + κσ
Sverrir.Olafsson@bt.com
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
Sverrir.Olafsson@bt.com
References