Stock Price Simulation in R

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Actuarial andfinancial applications

of simulation

EA Valdez

Modeling stock pricesThe lognormal distribution

Illustrative example

R code to generate the

stock price process

Pictorial illustration

Lognormal property

Illustration of generating

distribution of stock price

Distribution graph

Generating distribution of aportfolio of assets

Modeling aggregateclaims

Distribution assumptions

Case illustration

Simulation

Modeling in lifeinsurance

The Gompertz lifetime

distribution

Simulating from Gompertz

Simulating the loss

Parameter assumptions

Simulation results

page 1

Actuarial and financial applications

of simulationMath 276 Actuarial Models

Spring 2008 semester

EA ValdezUniversity of Connecticut - Storrs

Lecture Weeks 6 and 7


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stock price process


Lognormal property



Distribution graph




Case illustration

Simulation



distribution


Simulating the loss


Simulation results

page 2

Modeling stock prices

In finance, we are always interested in the return onstocks.

The Normal distribution is a typical distribution model forreturn on the stock; indeed equivalent to modeling thevalue of the stock as a lognormal distribution.

Assume that the return on the stock is normally distributed

with annual mean and annual standard deviation .

Denote by St the value of the asset at time t and St+tdenoting the the value t periods later. Thus, thepercentage change (or return) of the value of the stockbetween times t and t+t is approximated by

log St log St+t = logSt

St+t= log (1 + rt) rt,

where rt = (St St+t)/St.


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stock price process


Lognormal property



Distribution graph




Case illustration

Simulation



distribution


Simulating the loss


Simulation results

page 3

The lognormal distribution (and geometric diffusions)

Another way to write the stock price at time t+t is

St+t = St expt+ Z

t

,

where Z is standard normal N(0,1).

If you know diffusion processes, this is the discreteanalogue of the geometric diffusion:

dS

S= dt+ dB,

where dB is a Brownian motion (or Weiner) process withdB= Z

dt.


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stock price process


Lognormal property



Distribution graph




Case illustration

Simulation



distribution


Simulating the loss


Simulation results

page 4


Consider a stock paying no dividends with a volatility

= 0.05 per annum and with an expected return of = 0.10 per annum with continuous compounding.

The stock price process can be written asdS

S= 0.10dt+ 0.20dB or (in the discrete sense) with

small interval of timeS

S= 0.15t+ 0.20Z

t.

The figures in the following page demonstrate this price

process (by simulation) for different time intervals: year(t= 1), month (t= 1/12), week (t= 1/52), and day(t= 1/365).

Here we assume the initial stock price is 100.


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stock price process


Lognormal property



Distribution graph




Case illustration

Simulation


The Gompertz lifetimedistribution


Simulating the loss


Simulation results

page 5

R code to generate the stock price process

The following is a routine in R to generate the stock priceprocess. Function is called simstock.R.

# function to generate (discrete) stock price process

simstock


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stock price process


Lognormal property



Distribution graph




Case illustration

Simulation




Simulating the loss


Simulation results

page 6

Pictorial illustration of the stock price process

0 5 10 15 20

100

200

300

400

500

600

700

year

stock

price

0 50 100 150 200

100

200

300

400

month

stock

price

0 200 400 600 800 1000

200

400

600

800

1000

week

stock

price

0 2000 4000 6000

100

200

300

400

day

stock

price


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stock price process


Lognormal property



Distribution graph




Case illustration

Simulation




Simulating the loss


Simulation results

page 7

Lognormal property of stock prices

In the geometric Brownian motion, the change in the log Sbetween time 0 and T has a Normal distribution with

log ST log S0 N( 2/2)T, 2T

,

where S0 is the initial stock price while ST is the stockprice T periods later.

This is equivalent to ST having a lognormal distribution:

log ST N

log S0 + ( 2/2)T, 2T.

It is straightforward to show that the mean is given by

E(ST) = S0eT

,

and the variance is

Var(ST) = S20 e

2T

e2T 1

.


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stock price process


Lognormal property



Distribution graph




Case illustration

Simulation




Simulating the loss


Simulation results

page 8

R code to illustrate simulating the stock price at time T

The following is a routine in R to generate the stock price attime T. Function is called simstprice.R.

# function to generate stock price/value at time T

simstprice


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stock price process


Lognormal property



Distribution graph




Case illustration

Simulation




Simulating the loss


Simulation results

page 9

Graphical representation of the distribution

> hist(out1,br=25,xlab="initial price = $40",ylab="stock price after half year",main="1,000

simulations from logNormal with mu=30%, sigma=10%",freq=FALSE)

> lines(density(out1),xlim=c(20,80),col="blue")

1,000 simulations from logNormal with mu=30%, sigma=10%

initial price = $40

stock

price

afterhalfye

ar

20 30 40 50 60 70 80

0.

00

0

.01

0.

02

0.

03

0

.04

0.

05


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stock price process


Lognormal property



Distribution graph




Case illustration

Simulation




Simulating the loss


Simulation results

page 10

R code to illustrate simulating a portfolio of assetsThe following is a routine in R to generate the value of aportfolio of stocks at time T. Function is calledsimportfolio.R.

# function to generate a portfolio of securities at time T

# the variables required: n.gen (number to simulate), S0.vector (vector of initial# values of the stocks), mu.vector/sigma.vector (self-explanatory), n.holdings

# (the number of holdings for each corresponding stock), T (valuation date)

simportfolio


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The lognormal distribution



stock price process


Lognormal property



Distribution graph




Case illustration

Simulation




Simulating the loss


Simulation results

page 11

Graphical representation of the resulting portfoliodistribution

> hist(out1,br=25,xlab="",ylab="portfolio value",main="Distribution of Portfolio Value after 5

years",freq=FALSE,xlim=c(50000,250000))

> lines(density(out1),xlim=c(50000,250000),col="blue")

Distribution of Portfolio Value after 5 years

portfolio

value

50000 100000 150000 200000 250000

0.0

e+00

5.

0e

06

1.

0e05

1.5

e05

2.0e05

2.

5e05

A i l d


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stock price process


Lognormal property



Distribution graph




Case illustration

Simulation




Simulating the loss


Simulation results

page 12

Some statistics on the resulting portfolio distribution

> source("C:\\...\\Math276-Spring2008\\Rcodes-2008\\Week67\\Data.SummStats.R")

> Data.SummStats(out1)

Value

Number 1.000e+03

Mean 1.181e+05

5th Q 9.157e+04

25th Q 1.042e+05

Median 1.167e+05

75th Q 1.279e+05

95th Q 1.530e+05

Variance 3.533e+08

StdDev 1.880e+04

Minimum 7.529e+04

Maximum 2.382e+05

Skewness 9.300e-01

Kurtosis 2.080e+00

>

A t i l d


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stock price process


Lognormal property



Distribution graph




Case illustration

Simulation




Simulating the loss


Simulation results

page 13

The distribution of the total claim amount

In general (property/casualty) insurance, we often studythe distribution of the aggregate claim amount defined by

S= X1 + X2 + + XN,where Xi is the amount of the i-th claim and N refers to thenumber of claims.

Here we are referring to the total claims only for a fixed

period, e.g. one year, though this can be a stochasticprocess (over time).

The standard assumptions in the model are:

1 the claim amounts Xi are i.i.d. (independent and identicallydistributed) random variables; and

2 the claim amounts X1,X2, . . . and the claim count N are allindependent.

The aggregate sum S has what we call a compounddistribution, and in many instances, it is not possible to

derive explicit form of its distribution.

Actuarial andCl i i d l i di ib i i


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stock price process


Lognormal property



Distribution graph




Case illustration

Simulation




Simulating the loss


Simulation results

page 14

Claim size and claim count distribution assumptions

For purposes of illustration, we shall assume the following:

The claim size Xi has the Pareto(, ) distribution with CDF

F(x) = 1

x +

, for x > 0.

The claim count N is assumed to have a Poisson()distribution.

Note that to simulate from the Pareto, it can be shown that,using the inverse transform method, the followinggenerates a Pareto(, ) random variable:

X= (1 U)1/ 1

.


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Actuarial andCase illustration


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financial applicationsof simulation

EA Valdez





stock price process


Lognormal property



Distribution graph

Generating distribution of a

portfolio of assets



Case illustration

Simulation




Simulating the loss


Simulation results

page 15

Case illustration

Consider a portfolio of 10,000 automobile insurance

policies with the following assumptions:The period is exactly one year where each policy pays anannual premium of = $500.

Expenses include an overhead (or fixed) expense of$500,000 and a per policy expense of $2.50.

The aggregate claims distribution assume that claim sizehas a Pareto with = 625 and = 1.5 and claim count hasa Poisson with = 900.

The Profit/Loss (P/L) for this insurance portfolio is clearlyPremiums - (Claims + Expenses) where:

Premiums: 10, 000 = 10, 000(500) = 5, 000, 000

Claims: S = X1 + X2 + + XN

Expenses: 500, 000 + 2.5 10, 000 = 525, 000

Actuarial andR code to illustrate simulating the aggregate claims


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stock price process


Lognormal property



Distribution graph


portfolio of assets



Case illustration

Simulation




Simulating the loss


Simulation results

page 16

R code to illustrate simulating the aggregate claimsThe following is a routine in R to generate the aggregate claimamount for a portfolio of auto insurance policies. Function iscalled simillustrate.R.

# function to simulate the aggregate claims using the illustration

simillustrate


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stock price process


Lognormal property



Distribution graph


portfolio of assets



Case illustration

Simulation




Simulating the loss


Simulation results

page 17

Histograms of the aggregate claims distribution

> hist(out1/1000000,br=50,xlab="in millions",ylab="frequency",main="Distribution of Aggregate

Claims",freq=FALSE)

> min(log(out1))

[1] 13.41646

> max(log(out1))

[1] 17.02933

> hist(log(out1),br=50,xlab="in logarithm",ylab="frequency",main="Distribution of AggregateClaims",freq=FALSE,xlim=c(13,17))

>

Distribution of Aggregate Claims

in millions

frequency

0 5 10 15 20 25

0.

0

0.

2

0

.4

0.

6

0.

8

1.

0

Distribution of Aggregate Claims

in logarithm

frequency

13 14 15 16 17

0.

0

0.

5

1.

0

1.

5

2.

0

2.5

Actuarial andfinancial applicationsProfit/loss analysis


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stock price process


Lognormal property



Distribution graph


portfolio of assets



Case illustration

Simulation




Simulating the loss


Simulation results

page 18

Profit/loss analysis> pl Data.SummStats(pl)

Value

Number 1.000e+04

Mean 3.345e+06

5th Q 2.943e+06

25th Q 3.302e+06

Median 3.427e+0675th Q 3.518e+06

95th Q 3.620e+06

Variance 3.618e+11

StdDev 6.015e+05

Minimum -2.040e+07

Maximum 3.804e+06

Skewness -2.076e+01

Kurtosis 6.096e+02

Profit/Loss Distribution

in millions

frequency

20 15 10 5 0 5

0.

0

0.

2

0.

4

0.

6

0.

8

1.

0

1.

2

Actuarial andfinancial applicationsIntroduction - life insurance models


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stock price process


Lognormal property



Distribution graph


portfolio of assets



Case illustration

Simulation




Simulating the loss


Simulation results

page 19

Introduction life insurance models

The actuarial equivalence principle has one maindrawback: it simply looks at the mean/average of theloss-at-issue distribution.

Alternative is to also examine the variability of this loss, butthis does not give the complete picture of the lossdistribution.

A much better alternative is to examine the loss

distribution itself.

In many cases, it is impossible to derive explicit form of theloss distribution.

Simulating the loss distribution is one method to do it -

main drawback is it may require computer-intensivecalculations.

We demonstrate this only for a whole life insurance policyissued to a single person - in practice, you would be doingthis for a portfolio of insurance contracts.


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Actuarial andfinancial applicationsThe Gompertz lifetime distribution


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stock price process


Lognormal property



Distribution graph


portfolio of assets



Case illustration

Simulation




Simulating the loss


Simulation results

page 20

p

Assume that mortality follows the Gompertz with force ofmortality

x = Bcx,

where B and c are constants satisfying B> 0 and c> 1.

It is easy to show that for an issue age x, its future lifetimeTx follows the survival pattern

STx(t) = P(Tx > t) = exp

Bcxlog(c)

ct 1

,

for t 0.


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Actuarial andfinancial applicationsSimulating from Gompertz


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stock price process


Lognormal property



Distribution graph


portfolio of assets



Case illustration

Simulation



distribution


Simulating the loss


Simulation results

page 21

We can use the inverse transform method to simulate from

Gompertz.Begin with a random number U, generate a Gompertzlifetime, say T, from the following equation:

expBcx

log(c)cT

1 = U,or equivalently, we have

T =1

log(c)log

1 log(c) log(U)

Bcx

.

Running this procedure m(number of simulations) times,we can then have a simulated distribution of the Gompertzlifetime.

Actuarial andfinancial applicationsSimulating the loss-at-issue


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stock price process


Lognormal property



Distribution graph


portfolio of assets



Case illustration

Simulation



distribution


Simulating the loss


Simulation results

page 22

With a simulated value of T, we can then simulate a valueof the present value of the loss-at-issue.

For example, in a (fully continuous) whole life insurancecontract, we have

L0 = bTvT

aT

,

where v= 1/(1 + i) = e is the discount factor, is theannual premium assumed to be payable continuouslythroughout the year, and bT is the amount of insurancepayable at death.

Again, run this procedure for mnumber of times to get asimulated distribution of the loss-at-issue.


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Simulating the loss after k years


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stock price process


Lognormal property



Distribution graph


portfolio of assets



Case illustration

Simulation



distribution


Simulating the loss


Simulation results

page 23

When computing reserves, we need to evaluate the loss atthat point.

Suppose we are interested in the loss after k years, then itcan be shown that the simulated lifetime for the personwho is then aged x+ k is

T =1

log(c)log 1

log(c) log(U)Bcx+k

,where U is U(0,1) generated value.

For the same (fully continuous) whole life insurancecontract, we would have the loss after k years evaluated

asLk = bTv

T aT,

where T is the future lifetime of the person x who is nowaged x+ k.


of simulation



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stock price process


Lognormal property



Distribution graph


portfolio of assets



Case illustration

Simulation



distribution


Simulating the loss


Simulation results

page 24

To illustrate, we assume the following Gompertz parametervalues:

B= 0.0000429 and c= 1.1070839.

In addition, benefit amount is $100, premium is $0.0095

per $1 of insurance, and i= 5%.Number of simulations: 50,000.

Apart from calculating the losses at issue, we alsocalculate reserves (or losses) at the end of 10 years.

The R routine is called Gompertz.SimulationT.R - toolong to print in these slides; but is available on the website.


of simulation

Some summary statistics of the simulation results


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stock price process


Lognormal property



Distribution graph


portfolio of assets



Case illustration

Simulation



distribution


Simulating the loss


Simulation results

page 25

> source("C:\\...\\Math276-Spring2008\\Rcodes-2008\\Week67\\Gompertz.SimulationT.R")

Value

Number 50000.00Mean 41.19

5th Q 18.90

25th Q 34.41

Median 42.97

75th Q 49.71

95th Q 57.16

Variance 136.32

StdDev 11.68

Minimum 0.00

Maximum 71.08

Skewness -0.74Kurtosis 0.40

Value

Number 50000.00

Mean -0.18

5th Q -12.12

25th Q -8.90

Median -4.79

75th Q 2.82

95th Q 28.04

Variance 212.20

StdDev 14.57

Minimum -15.75

Maximum 99.99

Skewness 2.79

Kurtosis 10.19


of simulation

- continued


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stock price process


Lognormal property



Distribution graph


portfolio of assets



Case illustration

Simulation



distribution


Simulating the loss


Simulation results

page 26

Value

Number 50000.00

Mean 31.745th Q 10.85

25th Q 24.90

Median 33.18

75th Q 39.83

95th Q 47.26

Variance 119.26

StdDev 10.92

Minimum 0.01

Maximum 59.04

Skewness -0.54

Kurtosis -0.14

Value

Number 50000.00

Mean 10.16

5th Q -7.56

25th Q -2.36

Median 4.20

75th Q 15.98

95th Q 50.89

Variance 353.57

StdDev 18.80Minimum -12.77

Maximum 99.95

Skewness 1.93

Kurtosis 4.21


of simulation

Graphical displays of the simulation results


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stock price process


Lognormal property



Distribution graph


portfolio of assets



Case illustration

Simulation



distribution


Simulating the loss


Simulation results

page 27

Distribution of T(30)

t.30

frequency

0 10 30 50 70

0.

00

0.

02

Distribution of Lossatissue

loss.30

frequency

20 0 20 40 60 80

0.

00

0.

02

0.

04

0.

06

Distribution of T(40)

t.40

freq

uency

0 10 20 30 40 50 60

0.

00

0

.02

0.

04

Distribution of Loss at 10 yrs

loss.40

freq

uency

0 20 40 60 80 100

0.

00

0.

02

0.

04


of simulation

Graphical displays of simulating repeatedly


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stock price process


Lognormal property



Distribution graph


portfolio of assets



Case illustration

Simulation



distribution


Simulating the loss


Simulation results

page 28

0 20 40 60

0.0

0

0.0

2

T(30)

N = 50000 Bandwidth = 1.182

Density

20 0 20 40 60 80

0.0

0

0.0

2

0.0

4

0.0

6

loss at issue

N = 50000 Bandwidth = 0.9113

Density

0 10 20 30 40 50 60

0.0

0

0.0

2

T(40)

N = 50000 Bandwidth = 1.124

De

nsity

20 0 20 40 60 80

0.0

0

0.0

2

0.0

4

loss after 10 yrs

N = 50000 Bandwidth = 1.397

De

nsity

Stock Price Simulation in R

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