computational finance 1/36 panos parpas computational finance imperial college london
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Computational Finance 1/36
Panos Parpas
Computational Finance
Imperial College
London
Computational Finance 2/36
Computational Finance Course
Contact
Panos Parpas (Huxley Building, Room 347)Panos Parpas (Huxley Building, Room 347)
Email: pp500@doc.ic.ac.uk
and tutorial helpers.
Look at the web for lecture notes and tutorials
http://www.doc.ic.ac.uk/~pp500
Course material courtesy of Nalan Gulpinar.
Computational Finance 3/36
Course will provide
to bring a level of confidence to students to the finance fieldan experience of formulating finance problems into computational problemto introduce the computational issues in financial problemsan illustration of the role of optimization in computational finance such as single period mean-variance portfolio managementan introduction to numerical techniques for valuation, pricing and hedging of financial investment instruments such as options
Computational Finance 4/36
Useful Information
The course will be mainly based on lecture notes Recommended Books
D. Duffie, Dynamic Asset Pricing Theory, Princeton University Press, 1996.
E.J., Elton, M.J. Gruber, Modern Portfolio Theory and Investment Analysis, 1995.
J. Hull, Options, Futures, and Other Derivative Securities, Prentice Hall, 2000.
D.G. Luenberger, Investment Science, 1998.
S. Pliska, Discrete Time Models in Finance, 1998.
P. Wilmott, Derivatives: The Theory and Practice of Financial Engineering, 1998.
P. Wilmott, Option Pricing: Mathematical Models and Computation, 1993.
Two course works MEng test - for MEng students Final exam - for BEng, BSci, and MSc students
Computational Finance 5/36
Contents of the Course
1. Introduction to Investment Theory
2. Bonds and Valuation
3. Stocks and Valuation
4. Single-period Markowitz Model
5. The Asset Pricing Models
6. Derivatives
7. Option Pricing Models: Binomial Lattices
Computational Finance 6/36
Panos Parpas
Introduction to
Investment Theory
381 Computational Finance
Imperial College
London
Computational Finance 7/36
Topics Covered
Basic terminology and investment problems
The basic theory of interest rates
simple interestsimple interest
compound interestcompound interest
Future Value
Present Value
Annuity and Perpetuity Valuation
Computational Finance 8/36
Terminology
Finance – commercial or government activity of managing money, debt,
credit and investment
Investment – the current commitment of resources in order to achieve later benefits
ppresent commitment of money for the purpose of receiving more money later – invest amount of money then your capital will increaseInvestor is a person or an organisation that buys shares or pays money into a bank in order to receive a profit
Investment Science – application of scientific tools to investments pprimarily mathematical tools – modelling and solving financial problem
–optimisation
–statistics
Computational Finance 9/36
Basic Investment Problems
Asset Pricing – known payoff (may be random) characteristics,
what is the price of an investment?
what price is consistent with other securities that are available?
Hedging – the process of reducing financial risks: for example an
insurance you can protect yourself against certain possible losses.
Portfolio Selection – to determine how to compose optimal
portfolio, where to invest the capital so that the profit is maximized as well as
the risk is minimized.
Computational Finance 10/36
Terminology
Cash Flows: If expenditures and receipts are denominated in cash, receipts at If expenditures and receipts are denominated in cash, receipts at any time period are termed any time period are termed cash flowcash flow..An investment is defined in terms of its resulting cash flow An investment is defined in terms of its resulting cash flow sequencesequence
–– amount of money that will flow TO and FROM an investor over time– bank interest receipts or mortgage payments – a stream is a sequence of numbers (+ or –) to occur at known time periods
A cash flow at discrete time periods t=0,1,2,…,n
Example1- Cash flow (-1, 1.20) means: investor gets £1.20 after 1 year if ash flow (-1, 1.20) means: investor gets £1.20 after 1 year if £1 is invested£1 is invested2- Cash flow (-1500,-1000,+3000)Cash flow (-1500,-1000,+3000)
300010001500 2Year1 Year0 Year
),,,( 210 naaaa
Computational Finance 11/36
Interest Rates
Interest – defined as the time value of money in financial market, it is the price for credit determined by demand and in financial market, it is the price for credit determined by demand and
supply of creditsupply of credit
summarizes the returns over the different time periods summarizes the returns over the different time periods
useful comparing investments and scales the initial amount useful comparing investments and scales the initial amount
different markets use different measures in terms of year, month, week, different markets use different measures in terms of year, month, week,
day, hour, even seconds day, hour, even seconds
Simple interest and Compound interest
Computational Finance 12/36
Simple Interest
Assume a cash flow with no risk.Invest and get back amount of after a year, at Invest and get back amount of after a year, at
Ways to describe how becomes ?Ways to describe how becomes ?
If one-period simple interest rate is then amount of money If one-period simple interest rate is then amount of money
at the end of time period isat the end of time period is
Initial amount is called principalInitial amount is called principal
A 1W 1t
A 1W
trt at time
r when 1
at )1(
2at )1(
1at )1(
21
21
212
11
nn
nn
rrrnr) A(W
ntrrrAW
trrAW
trAW
t
Computational Finance 13/36
Example: Simple interest
If an investor invest £100 in a bank account that pays 8% interest per year, then at the end of one year, he will have in the account the original amount of £100 plus the interest of 0.08.
)08.01(100108£
1at )1( 11
trAW
Computational Finance 14/36
Compound Interest
Invest amount of for n years period and one period
compound interest rate is given by
the amount of money is computed as follows;
A
ntrrr n ,,2,1at ,,, 21
r if W
)
n
nn
nn
rrrrA
ntrrrrAW
trWrrAW
trAW
21
321
21212
11
)1(
)1(1)(1)(1(
2)1()1)(1(
1)1(
Computational Finance 15/36
Simple versus Compound Interest Rates
Linear growth and Geometric growth
0
200
400
600
800
1000
1200
0 2 4 6 8 10 12 14 16 18 20
Years
Va
lue
Simple Interest
Compound Interest
Computational Finance 16/36
Example: Simple & Compound Interest
If you invest £1 in a bank account that pays 8% interest per year, what will you have in your account after 5 years?
Simple interest:
Linear growth
Compound interest:
Geometric growth
5)08.008.008.008.008.01(140.1
4)08.008.008.008.01(132.1
3)08.008.008.01(124.1
2)08.008.01(116.1
1)08.01(108.1
t
t
t
t
t
for
for
for
for
for
5)08.01)(08.01)(08.01)(08.01)(08.01(14693280768.1
4)08.01)(08.01)(08.01)(08.01(136048896.1
3)08.01)(08.01)(08.01(1259712.1
2)08.01)(08.01(11664.1
1)08.01(108.1
t
t
t
t
t
for
for
for
for
for
http://www.moneychimp.com/features/simple_interest_calculator.htm
Computational Finance 17/36
Example: Compound Interest
Assume that the initial amount to invest is A=£100 and the
interest rate is constant. What is the compound interest rate and
the simple interest rate in order to have £150 after 5 years?
%4.8
1084.1
100
1501
)1(100150
)1(
5
1
5
55
r
r
r
r
rAW
%10
5.05
100
15051
)51(100150
)51(5
r
r
r
r
rAW
Compound Interest Simple Interest
Computational Finance 18/36
Compounding Continued
At various intervals – for investment of A if an interest rate for
each of m periods is r/m, then after k periods
Continuous compounding –
k
m
rAW
1
rt
tmk
tmmtk
e
m
r
m
r
m
r
m
r
m
r
1lim1lim
111
mm
rtAeW Exponential Growth
Computational Finance 19/36
The effective & nominal interest rate
The effective of compounding on yearly growth is highlighted by stating The effective of compounding on yearly growth is highlighted by stating
an an effective interest rate
yearly interest rate that would produce the same result after 1 year yearly interest rate that would produce the same result after 1 year
without compoundingwithout compounding
The basic yearly rate is called The basic yearly rate is called nominal interest rate
Example: Annual rate of 8% compounded quarterly produces an
increase
%8
%24.8
0824.1)02.1()02.01(%24
%8 44
:rate interest nominalThe
:rate interesteffective The
Computational Finance 20/36
Example: Compound Interest
i ii iii iv vPeriods Interest Ann perc. Value Effectivein year per period rate APR after 1 year interest rate
1 6 6 1.061 = 1.06 6.000
2 3 6 1.032 = 1.0609 6.090
4 1.5 6 1.0154 = 1.06136 6.136
12 0.5 6 1.00512 = 1.06168 6.168
52 0.1154 6 1.00115452 = 1.06180 6.180
365 0.0164 6 1.000164365 = 1.06183 6.183
Computational Finance 22/36
Example: Future Value
year cash inflow interest balance 0 5000.00 0.00 5,000.001 5000.00 250.00 10,250.002 0.00 512.50 10,762.50 3 0.00 538.13 11,300.63 4 0.00 565.03 11,865.665 0.00 593.28 12,458.94
Suppose you get two payments: £5000 today and £5000 exactly one year from now. Put these payments into a savings account and earn interest at a rate of 5%. What is the balance in your savings account exactly 5 years from now.
The future value of cash flow:
94.458,12£
)05.01(5000)05.01(5000 45
FV
Computational Finance 23/36
Present Value (PV) - Discounting
Investment today leads to an increased value in future as result of interest.
reversed in time to calculate the value that should be assigned now, in the
present, to money that is to be received at a later time.
The value today of a pound tomorrow: how much you have to put into your
account today, so that in one year the balance is W at a rate of r %
)10.01(110100 PV
£110 in a year = £100 deposit in a bank at 10% interest
Discounting process of evaluating future obligations as an equivalent PV the future value must be discounted to obtain PV
Computational Finance 24/36
Present Value at time k
kkk r
WWdPV
)1(
Present value of payment of W to be received k th periods in the future
krkd)1(
1
where the discount factor is
If annual interest rate r is compounded at the end of each m equal periods per year and W will be received at the end of k th period
k
m
rkd
1
1
Computational Finance 26/36
PV for Frequent Compounding
For a cash flow stream (a0, a1,…, an) if an interest rate for each of the m
periods is r/m, then PV is
PV of Continuous Compounding
n
kk
k
nn
m
r
aPV
m
r
a
m
r
a
m
r
aaPV
0
22
11
0
1
111
rtn
tteaPV
0
Computational Finance 27/36
Example 1: Present Value
You have just bought a new computer for £3,000. The payment You have just bought a new computer for £3,000. The payment terms are 2 years same as cash. If you can earn 8% on your terms are 2 years same as cash. If you can earn 8% on your money, how much money should you set aside today in order to money, how much money should you set aside today in order to make the payment when due in two years?make the payment when due in two years?
02.572,2£2)08.1(3000 PV
Computational Finance 28/36
Example 2: Present Value
Consider the cash flow stream (-2,1,1,1). Calculate the PV and Consider the cash flow stream (-2,1,1,1). Calculate the PV and FV using interest rate of 10%.FV using interest rate of 10%.
Example 3Example 3: Show that the relationship between PV and FV of a : Show that the relationship between PV and FV of a cash flow holds. cash flow holds.
487.0331.1
648.0
)1.1(
648.01)1.1(1)1.1(1)1.1(2
487.01.1
1
1.1
1
1.1
12
3
123
32
FVPV
FV
PV
nr
FVPV
)1(
Computational Finance 29/36
Net Present Value (NPV)
time value of money has an application in investment
decisions of firms
in deciding whether or not to undertake an investment
invest in any project with a positive NPV
NPV determines exact cost or benefit of investment
decision
PVCostNPV
Computational Finance 30/36
Example 1: NPV
Buying a flat in London costs £150,000 on average. Experts predict that a year from
now it will cost £175,000. You should make decision on whether you should buy a flat
or government securities with 6% interest. You should buy a flat if PV of the expected £175,000 payoff is greater than the
investment of £150,000 –
What is the value today of £175,000 to be received a year from now? Is that PV
greater than £150,000?
Rate of return on investment in the residential property is
094,15
000,150094,165
094,16506.01
000,175
NPV
VP
%7,16000,150
000,150000,175
return of Rate InvestmentProfit
Computational Finance 31/36
Example 2: NPV
Assume that cash flows from the construction and sale of an office building is as follows. Given a 7% interest rate, create a present value worksheet and show the net present value, NPV.
000,300000,100000,150
2Year 1Year 0Year
400,18£
900,261000,300873.2
500,93000,100935.1
000,150000,1500.10
207.1
1
07.11
NPV
t
PVad ttt
Computational Finance 32/36
Annuity Valuation
Cash flow stream which is equally spaced and equal
amount a1 =, …,= an =a payments per year t=1,2,…, n
An annuity pays annually at the end of each year
£250,000 mortgage at 9% per year which is paid off with a
180 month annuity of £2,535.67
rd
d
ddaPV
n
A
1
1 where
1
)1(.
Present value of n period annuity
Computational Finance 33/36
Annuity Valuation
For a cash flow a1 =, …,= an =a
d
ddaPV
rr
aPV
r
a
r
aPV
r
r
r
a
r
aPV
r
r
a
r
a
r
aPV
r
r
a
r
a
r
aPV
n
A
nA
nA
nA
nA
nA
1
1.
1
11
111
111
11
1111
1
111
1
1
132
2
Computational Finance 34/36
Annuity Valuation
For m periods per year
i
iaPVi
mr
dm
ri
n
A
11
11
1 and 1
The present value of growing annuity: payoff grows at a rate of g per year: k th payoff is a(1+g)k
n
GA r
g
gr
aPV
1
11
Computational Finance 35/36
Example: Annuity
Suppose you borrow £250,000 mortgage and repay over 15 years. The interest rate is 9% and payments are made monthly. What is the monthly payment which is needed to pay off the mortgage?
67.535,2£0.99255581
0.992555810.9925558.000,250
1
1.PV
0.9925558
120.09
1
1
1
1
000,250£%,9
,15,12,180Given
180
A
a
a
d
dda
mr
d
PVr
Tmn
n
A
Computational Finance 36/36
Perpetuity Valuation
perpetuities are assets that generate the same cash flow forever pay a coupon at the end of each year and never matures annuity is called a perpetuity when number of payments becomes
infinite
For m periods per year;
Present value of growing perpetuity at a rate of g
r
aPVP
i
a
mra
PVP
gr
aPVGP
)( n
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