Panos Parpas

1. Computational Finance Panos Parpas Imperial College London

2. Computational Finance Course Contact 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.

3. Course will provide to bring a level of confidence to students to the finance field an experience of formulating finance problems into computational problem to introduce the computational issues in financial problems an illustration of the role of optimization in computational finance such as single period mean-variance portfolio management an introduction to numerical techniques for valuation, pricing and hedging of financial investment instruments such as options

4. 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

5. Contents of the Course Introduction to Investment Theory Bonds and Valuation Stocks and Valuation Single-period Markowitz Model The Asset Pricing Models Derivatives Option Pricing Models: Binomial Lattices

6. Introduction to Investment Theory Panos Parpas 381 Computational Finance Imperial College London

7. Topics Covered Basic terminology and investment problems The basic theory of interest rates simple interest compound interest Future Value Present Value Annuity and Perpetuity Valuation

8. 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 present commitment of money for the purpose of receiving more money later – invest amount of money then your capital will increase Investor 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 primarily mathematical tools – modelling and solving financial problem –optimisation –statistics

9. 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.

10. Terminology Cash Flows: If expenditures and receipts are denominated in cash, receipts at any time period are termed cash flow. An investment is defined in terms of its resulting cash flow sequence –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 Example 1-Cash flow (-1, 1.20) means: investor gets £1.20 after 1 year if £1 is invested 2-Cash flow (-1500,-1000,+3000)

11. Interest Rates Interest – defined as the time value of money in financial market, it is the price for credit determined by demand and supply of credit summarizes the returns over the different time periods useful comparing investments and scales the initial amount different markets use different measures in terms of year, month, week, day, hour, even seconds Simple interest and Compound interest

12. Simple Interest Assume a cash flow with no risk. Invest and get back amount of after a year, at Ways to describe how becomes ? If one-period simple interest rate is then amount of money at the end of time period is Initial amount is called principal

13. 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.

14. 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;

15. Simple versus Compound Interest Rates Linear growth and Geometric growth

16. 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 http://www.moneychimp.com/features/simple_interest_calculator.htm

17. 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? Compound Interest Simple Interest

18. 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 – Exponential Growth

19. The effective & nominal interest rate The effective of compounding on yearly growth is highlighted by stating an effective interest rate yearly interest rate that would produce the same result after 1 year without compounding The basic yearly rate is called nominal interest rate Example:Annual rate of 8% compounded quarterly produces an increase

20. Example: Compound Interest i ii iii iv v Periods Interest Ann perc. Value Effective in 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 = 1.061836.183

22. Example: Future Value 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. yearcash inflowinterestbalance 0 5000.00 0.00 5,000.00 1 5000.00 250.00 10,250.00 2 0.00 512.50 10,762.50 3 0.00 538.13 11,300.63 4 0.00 565.03 11,865.66 5 0.00 593.28 12,458.94 The future value of cash flow:

23. 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 Wat a rate ofr % • £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

24. Present Value at time k Present value of payment of W to be received k th periods in the future 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

26. PV for Frequent Compounding For a cash flow stream (a0, a1,…, an) if an interest rate for each of the mperiods is r/m, then PV is PV of Continuous Compounding

27. Example 1: Present Value 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 money, how much money should you set aside today in order to make the payment when due in two years?

28. Example 2: Present Value Consider the cash flow stream (-2,1,1,1). Calculate the PV and FV using interest rate of 10%. Example 3: Show that the relationship between PV and FV of a cash flow holds.

29. 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

30. 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

31. 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.

32. Annuity Valuation Cash flow stream which is equally spaced and equal amount a1 =, …,= an =apayments per yeart=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 Present value of n period annuity

33. Annuity Valuation For a cash flow a1 =, …,= an =a

34. Annuity Valuation For m periods per year The present value of growing annuity: payoff grows at a rate of g per year: kth payoff is a(1+g)k

35. 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?

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