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Managing the risk associated with bandwidth demand uncertainty - - PowerPoint PPT Presentation

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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

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Managing the risk associated with bandwidth demand uncertainty

Sverrir Olafsson Mobility Research Centre

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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
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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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Sverrir.Olafsson@bt.com

Bandwidth price risk

  • Unknown price of bandwidth

– Prices are on their way down – The rate of decline is unknown ⇒ risk

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Sverrir.Olafsson@bt.com

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 known Uncertainties

  • Price of bandwidth

Analogy to commodity market were magnitude is known but the price

  • f commodity is unknown
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Sverrir.Olafsson@bt.com

Aim of risk management

  • Identify risk sources
  • Quantify the risk caused
  • Control the risk

– Hedge

  • nly some risks can be hedged (commodities markets,….)

– More efficient decision making

  • perational caution
  • insurance
  • real options
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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

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Sverrir.Olafsson@bt.com

Modelling bandwidth demand

  • Evidence for “exponential growth”
  • But, rate of growth is uncertain
  • Model as geometric Brownian motion
  • Therefore

t t t t

dW D dt D dD σ µ + =       +       − =

t t

W t D D σ σ µ

2

2 1 exp

[ ] ( ) [ ]=

=

t t

D t D D E var ; exp µ

( )

1 , N dt dW

t t t

∈ = η η

100 200 300 400 500 600 700 800 900 1000 2 0 4 0 6 0 8 0 100 120 100 200 300 400 500 600 700 800 900 1000 20 40 60 80 100 120 140 Time [days] Value 100 200 300 400 500 600 700 800 900 1000 2 0 4 0 6 0 8 0 100 120 Standard deviation x0 = 50, µ = 0.3, σ = 0.5 Process Mean Standard deviation

gbmo(50,0.3,0.50,1/360,1000,1)

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Sverrir.Olafsson@bt.com

Possible demand scenarios

  • Monte Carlo simulations – iterate a large number of

scenarios each compatible with the assumptions

100 200 300 400 500 600 700 800 900 1000 50 100 150 200 250 Time (Days) Required capacity Capacity evolution, µ = 0.3, σ = 0.25, D0 = 50

justgbm.m

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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

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Sverrir.Olafsson@bt.com

Probability of exceeding capacity

  • Probability of exceeding installed capacity
  • Probability density function
  • Cumulative probability function

500 1000 1500 2000 2500 0.005 0.01 0.015 0.02 0.025 0.03 0.035 Days Probability density Parameter estimates, a = 6.2112, b = 141.9538 Empirical data Gamma fit 500 1000 1500 2000 2500 3000 0.2 0.4 0.6 0.8 1 Days Cumulative probability

µ = 0.35, σ = 0.25, Initial demand = 50, Capacity = 150

Log-normal Empirical Gamma

gmbreach.m gmbreach.m

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Sverrir.Olafsson@bt.com

Impact of uncertainty

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Days Probability density

µ = 0.5, σ = 0.2, Initial demand = 50, Capacity = 150

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Days Cumulative probability

µ = 0.5, σ = 0.2, Initial demand = 50, Capacity = 150

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Days Probability density

µ = 0.5, σ = 0.4, Initial demand = 50, Capacity = 150

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Days Cumulative probability

µ = 0.5, σ = 0.4, Initial demand = 50, Capacity = 150

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Days Probability density

µ = 0.5, σ = 0.8, Initial demand = 50, Capacity = 150

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Days Cumulative probability

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 same mean capacity life

        = log 1 D C t µ

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Sverrir.Olafsson@bt.com

Justification for log-normal modelling

  • The empirical cumulative

probability is well approximated by the log-normal distribution

1000 2000 3000 4000 5000 6000 0.2 0.4 0.6 0.8 1 Days Cumulative probability

µ = 0.25, σ = 0.4, Initial demand = 50, Capacity = 150

Log-normal Empirical Gamma

( ) ( ) (

)

              − − + = ≤ t t D C erf t C D CP

t

2 2 / / log 1 2 1 ,

2

σ σ µ

  • Deviations are explained by demand

exceeding installed capacity and then go down below installed capacity again

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Sverrir.Olafsson@bt.com

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

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Sverrir.Olafsson@bt.com

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

{ }

;... , ,..., ,

1 1 k k

C t C t ∆ ∆ = Ω

( ) ( ) ( )

1 Pr → ≥ t D t C

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Sverrir.Olafsson@bt.com

Controlled instalments

  • Probability to exceed installed capacity
  • The instalment process will depend on

– Expected growth – Expected volatility – Required QoS

( ) ( ) (

)

              − − + = ≤ t t D C erf t C D CP

t

2 2 / / log 1 2 1 ,

2

σ σ µ ] [ ] , [

1 1

instalment controlled dC C C C ed uncontroll GBM dD D D D

T T T T t t t t

+ = → + = →

+ +

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Sverrir.Olafsson@bt.com

Instalment strategy

  • Probability that demand does not exceed installed

capacity for different instalment strategies

Program:tempcapincrease.m

1000 2000 3000 4000 5000 0.5 0.6 0.7 0.8 0.9 1 Initial demand = 50, Initial capacity = 100, µ = 0.2, σ = 0.50 Days Probability that demand is below the installed capacity

µinc=0.05 µinc=0.10 µinc=0.15 µinc=0.20 µinc=0.25 µinc=0.30

200 400 600 800 1000 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Initial demand = 50, Initial capacity = 100, µ = 0.2, σ = 0.50 Days Probability that demand is below the installed capacity

µinc=0.05 µinc=0.10 µinc=0.15 µinc=0.20 µinc=0.25 µinc=0.30

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Sverrir.Olafsson@bt.com

Time to capacity exhaustion

  • Assume additional capacity is installed at the average

rate µi

  • Time to exhaustion

( ) ( )

t C t t W t D D

in t

µ σ σ µ exp 2 exp

2

=         +       − =

( )

( )

( )

2 log 2 log

2 2

σ µ µ σ σ µ µ − −         = ⇒ + − −         =

= i W E i

D C t E t W D C t

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Sverrir.Olafsson@bt.com

Different instalment strategies

  • Installing capacity at different

– Rates – Time intervals

10 20 30 40 50 50 100 150 200 90 92 94 96 98 100 Excess instalment [%] Initial demand =50, Initial capacity = 75, µ = 0.35, σ = 0.25 Days between instalments Capacity coverage [%] 10 20 30 40 50 200 400 600 75 80 85 90 95 100 Excess instalment [%] Initial demand =50, Initial capacity = 75, µ = 0.35, σ = 0.25 Days between instalments Capacity coverage [%] 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);

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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

( ) ( )

1

1

+

+ = +

t

t aS t S η

( )

{ } ( )

{ }

2 2 1

var , , ,

η η

σ η η η σ η = = = ∈

+ t t t t t

E E N

( ) ( ) ∑

= + −

+ = +

k i i t i k k

a t S a k t S

1

η

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Sverrir.Olafsson@bt.com

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

( ) { } ( )

t S a k t S E

k t

= +

( ) ( )

( )

σ σ σ

η η S n n k k

k S t k a a a

2 2 2 1 2 2 2

1 1 = + = = − −      

= −

∑

var

( )

{ }

( )

{ }

( )

{ }

E S t k E S t k E S t + > + + > > + ∞ 1 ...

( ) ( ) ( )

σ σ σ

S S S

k k

2 2 2

1 < + < < ∞ ...

50 100 150 200 250 300 350 400 2 4 6 50 100 150 200 250 300 350 400 40 50 60 70 80 90 100 Time Value 50 100 150 200 250 300 350 400 1 2 3 4 5 6 Standard deviation s0 = 100, a = 0.99858,

σ = 0.35, Expected annual price change [%] = -0.4

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Sverrir.Olafsson@bt.com

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

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Sverrir.Olafsson@bt.com

When to install additional capacity?

  • Delaying capacity instalment

– Provides monetary benefits – Incurs risk

500 1000 1500 2000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Days Cumulative probability

µ = 0.3, σ = 0.25, Initial demand = 50, Capacity = 100

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Probability of exceeding capacity Gains from decline in price

µ = 0.3, σ = 0.25, In dem = 50, Cap = 100 , Pr decline = 0.3

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Sverrir.Olafsson@bt.com

When to install additional capacity?

  • The impact of volatility on expected benefits

gmbreach.m

0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Probability of exceeding capacity Gains from decline in price

µ = 0.5, In dem = 50, Cap = 150 , Pr decline = 0.3 σ = 0.20 σ = 0.40 σ = 0.60 σ = 0.80 σ = 0.90

10000 iterations - days 1500 experiments 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Probability of exceeding capacity Gains from decline in price

µ = 0.5, In dem = 50, Cap = 150 , Pr decline = 0.3 σ = 0.20 σ = 0.40 σ = 0.60 σ = 0.80 σ = 0.90

2500 iterations - days 1500 experiments

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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
  • We assume the following loss as a function of time

( ) ( ) ( ) ( )

t p t g κ − − = exp 1

( ) ( ) ( )

   ≤ > = − = ; ; 1 ; x if x if x t t t t c

c

θ αθ

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Sverrir.Olafsson@bt.com

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

– price decline – losses from QoS depreciation

500 1000 1500 2000 2500 3000 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Days Cost/benefit

µ = 0.3, σ = 0.25, In dem = 50, Cap = 100 , Pr decline = 0.3, α = 1, tc = 10

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Sverrir.Olafsson@bt.com

Hedge against price risk

  • Hedge ratio,

– h=(size of futures contract/size of exposure)

  • Consider short hedge
  • This assumes known demand but uncertain price

H S F S h

h h σ ρσ σ σ σ 2

2 2 2 2

− + =

( ) ( ) ( )

F S F S

F S F S σ σ ρ σ σ , cov var var

2 2

= = =

F S F S F h

h h h σ σ ρ σ ρσ σ ∂ ∂σ = ⇒ = − = 2 2

2 2

F h S hF S ∆ − ∆ = ∆Π ⇒ − = Π

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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
  • Optimal hedge ratio

( ) ( ) ( )

F DS h F h DS hF DS , cov 2 var var

2

− + = ⇒ − = Π

Π

σ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

F F E F S E S D E D E F D S E F S D E h var , cov , cov − − − + + =

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Sverrir.Olafsson@bt.com

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

t t t t

dW D dt D dD σ µ + =

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Sverrir.Olafsson@bt.com

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

( ) ( ) ( ) ( )

, min 2 1

2 2 2 2

= − + ∂ ∂ − − ∂ ∂ − ∂ ∂ t LP t C D rF D F D F D t F κσ µ σ κσ + = r R

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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

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Sverrir.Olafsson@bt.com

References