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Computational Finance */36Example: Future Valueyear cash inflow interest balance 0 5000.00 0.00 5,000.001 5000.00 250.0010,250.002 0.00 512.5010,762.50 3 0.00 538.1311,300.63 4 0.00 565.0311,865.665 0.00 593.2812,458.94Suppose 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:Computational Finance */36Present 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 %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 */36Present Value at time kPresent 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 periodComputational Finance */36PV 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 Computational Finance */36Example 1: Present ValueYou 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 */36Example 2: Present ValueConsider 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. Computational Finance */36Net 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 decisionComputational Finance */36Example 1: NPVBuying 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 isComputational Finance */36Example 2: NPVAssume 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.Computational Finance */36Annuity 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 2,535.67Present value of n period annuityComputational Finance */36Annuity Valuation For a cash flow a1 =, ,= an =a Computational Finance */36Annuity 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)kComputational Finance */36Example: 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 */36Perpetuity 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************************************