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Dr Nalan Gulpinar Prof Berc Rustem

Part II Portfolio and Risk Management. Dr Nalan Gulpinar Prof Berc Rustem. Introduction to Investment Theory. 381 Computational Finance Department of Computing Imperial College 21 January 2003. Topics Covered. Basic terminology and investment problems

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Dr Nalan Gulpinar Prof Berc Rustem

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  1. Part II Portfolio and Risk Management Dr Nalan Gulpinar Prof Berc Rustem Introduction to Investment Theory 381 Computational Finance Department of Computing Imperial College 21 January 2003

  2. 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 Computational Finance

  3. 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 An 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 Computational Finance

  4. Terminology Cash Flows – If the expenditures and receipts are denominated in cash, then 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- A cash flow (-1, 1.20) means - the investor gets £1.20 after a year if £1 is invested 2- Cash flow (-1500,-1000,+3000) Computational Finance

  5. 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

  6. 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 Computational Finance

  7. 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 Computational Finance

  8. 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. £108 = 100(1+0.08) Computational Finance

  9. Compound Interest If you invest amount of for n years period and one period compound interest rate is given by then the amount of money is computed as follows; Computational Finance

  10. Simple versus Compound Interest rates Linear growth and Geometric growth Computational Finance

  11. 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 Computational Finance

  12. 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 Computational Finance

  13. 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 Computational Finance

  14. 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 callednominal interest rate Example: Annual rate of 8% compounded quarterly produces an increase Computational Finance

  15. Example: Compound Interest i ii iii iv v Periods Interest Ann perc. Value Effective in a per rate APR after interest year period(i x ii) one year 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 .5 6 1.00512 = 1.06168 6.168 52 .1154 6 1.00115452 = 1.06180 6.180 365 .0164 6 1.000164365 = 1.06183 6.183 Computational Finance

  16. Future Value (FV) • FV is what we have if we invest the cash for some period • the value tomorrow of money today • the future value of A today at r% at the end of n years is • For cash flow the future value is calculated as Computational Finance

  17. 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 is Computational Finance

  18. Present Value (PV) = Discounting Investment today leads to an increased value in the future as a result of interest. This can be 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% • £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 the PV Computational Finance

  19. Present Value at time k • The present value of payment of W to be received kth periods in the future where the discount factor is • If the 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 the kth period Computational Finance

  20. Present Value of Cash Flow Stream For given cash flow stream, Computational Finance

  21. PV for Compounding PV of Frequent Compounding – for a cash flow stream given by (a0, a1,…, an) if an interest rate for each of the m periods is r/m, then PV is PV of Continuous Compounding – Computational Finance

  22. 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? Computational Finance

  23. Example 2: Present Value Assume that the cash flows from the construction and sale of an office building is as follows. Given a 7% required rate of return, create a present value worksheet and show the net present value, NPV. Computational Finance

  24. Internal Rate of Return For a cash flow stream (a0, a1, …, an) IRR is a number r satisfying the following polynomial equation Computational Finance

  25. Annuity Valuation Cash flow stream which is equally spaced and equal amount a0= a1 =, …,= an =a 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 Computational Finance

  26. Annuity Valuation For a cash flowa0= a1 =, …,= an =a Computational Finance

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

  28. Perpetuity Valuation infinite number of payments in annuity for m periods per year; Present value of growing perpetuity at a rate of g Computational Finance

  29. 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? Computational Finance

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