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2. T OPICS : 1. What is financial mathematics?; 2. What is annuity?; 3. Time line; 4. Kinds of annuity; 2

3. 1. W HAT IS FINANCIAL MATHEMATICS ? Mathematical finance is a field of applied mathematics, concerned with financial markets. The subject has a close relationship with the discipline of financial economics, which is concerned with the underlying theory. Generally, mathematical finance derives and extends the mathematical or numerical models suggested by financial economics. 3

4. 2. W HAT IS ANNUITY ? The term annuity is used in financial mathematics to refer to any terminating sequence of regular fixed payments over a specified period of time. Loans are usually paid off by an annuity. If payments are not at regular (irregular) periods, we are not working with an annuity. We get two types of annuities: i. The ordinary annuity :This is an annuity whose payments are made at the end of each period. (At the end of each week, month, half year, year, etc.) Paying back a car loan, a home loan, etc…; ii. The annuity due: This is an annuity whose payments are made at the beginning of each period. Deposits in savings, rent payments, and insurance premiums are examples of annuities due. 4

5. 3. O N THE TIMELINE … If we look at the timeline, we clearly see that if we are looking at investing money into an account, then we will be working with the future value of these payments. This is so because we are saving up money for some use one day in the future. If we want to consider the present value of a series of payments, then we will be looking at a scenario where a loan is being paid off. This is so because we get the money today, and pay that money with interest back to the financing company some time in the future. 5

6. G RAPHIC … 6 T T1 T2 T3T5 T6T7 T4 Future Value works forwards Present Value works backwards Regular periodic payments x XX XXXX XX

7. 4. K INDS OF ANNUITY : 7 We have four kinds of annuity; these are: A.future values of ordinary annuities; B.Future value of due annuity; C.Present value of annuity; D.Present value of due annuity; E.Perpetuities annuity;

8. A. F UTURE VALUE OF ORDINARY ANNUITY 8

9. The Future Value of Ordinary Annuity (FVoa) is the value that a stream of expected or promised future payments will grow after a given number of periods at a specific compounded interest. The Future Value of an Ordinary Annuity could be solved by calculating the future value of each individual payment in the series using the future value formula and then summing the results. A more direct formula is: 9 Fvoa= = PMT [((1 + i) n - 1) / i] Where: FVoa = Future Value of an Ordinary Annuity PMT = Amount of each payment i = Interest Rate Per Period n = Number of Periods

10. E XAMPLE … 10 What amount will be accumulated if we deposit $5,000 at the end of each year for the next 5 years? Assume an interest of 6% compounded annually. Fvoa =b is future value of annuity. Pmt = 5,000.00 i = 0.06 n = 5 years FVoa = 5,000*[(1.06)^5-1/0.06] = 5,000 *[ (1.3382255776 - 1) /0.06 ] = 5,000* (5.637092) = 28,185. 46

11. 11 Pmt Future Value works forwards 01 2345 67 FVoa Pmt Let's watch the graphic: ……

12. B. F UTURE VALUE OF DUE ANNUITY The Future Value of due Annuity is identical to an ordinary annuity except that each payment occurs at the beginning of a period rather than at the end. Since each payment occurs one period earlier, we can calculate the present value of an ordinary annuity and then multiply the result by (1 + i). 12 FVad = FVoa *(1+i) Where:  FVad = Future Value of an Annuity Due  FVoa = Future Value of an Ordinary Annuity  i = Interest Rate Per Period

13. E XAMPLE … 13 What amount will be accumulated if we deposit $5,000 at the beginning of each year for the next 5 years? Assume an interest of 6% compounded annually. PV = 5,000 i = 0.06 n = 5 FVoa = 28,185.46* (1.06) = 29,876.59

14. 14 Pmt 1 2345 6 FVoa Pmt Let's watch the graphic: …… 0

15. C. P RESENT VALUE OF AN ORDINARY ANNUITY 15

16. 16 The Present Value of an Ordinary Annuity (PVoa) is the value of a stream of expected or promised future payments that have been discounted to a single equivalent value today. It is extremely useful comparing two separate cash flows that differ in some way. PV-oa can also be thought of as the amount you must invest today at a specific interest rate so that when you withdraw an equal amount each period, the original principal and all accumulated interest will be completely exhausted at the end of the annuity. The Present Value of an Ordinary Annuity could be solved by calculating the present value of each payment in the series using the present value formula and then summing the results. A more direct formula is: PVoa = PMT [(1 – (1 + i)- n ) / i] Where: PVoa = Present Value of an Ordinary Annuity PMT = Amount of each payment i = Discount Rate Per Period n = Number of Periods

17. E XAMPLE … 17 Mario buying a car, paying € 5000 per year for 7 years, starting next year. Applying the assessment rate del15% per year, what is the price of the car? PVoa = PMT [(1 – (1 + i)- n ) / i] Pvoa= present value of ordinary annuity PMT= 5000€ i= 0.15 N= 7 years Pvoa= 5000*[(1-(1+0.15)^-7)/0.15] = 20802,10€

18. 18 Pmt 1a 2a3a4a5a 6a Pmt 07a PVoaPmt we now see the graph, how to do.....

19. D. P RESENT VALUE OF DUE ANNUITY 19

20. 20 The Present Value of due annuity is identical to an ordinary annuity except that each payment occurs at the beginning of a period rather than at the end. Since each payment occurs one period earlier, we can calculate the present value of an ordinary annuity and then multiply the result by (1 + i). PVoad= PVoa*(1+i) Where: Pvoad= present value of an ordinary annuity due; Pvoa= present value of an ordinary annuity.

21. E XAMPLE … 21 Mario buying a car, paying € 5000 per year for 7 years, starting next year. Applying the assessment rate del15% per year, what is the price of the car? PVoad = Pvoa*(1 + i) Pvoad= present value of ordinary annuity due PMT= 5000€ i= 0.15 N= 7 years Pvoad= (5000*[(1-(1+0.15)^-7)/0.15] )*(1.15)= 23922.42€

22. 22 Pmt 1a 2a3a4a5a 6a Pmt 07a PVoaPmt we now see the graph, how to do.....

23. E. P ERPETUITY ANNUITY A perpetuity is an annuity in which the periodic payments begin on a fixed date and continue indefinitely. It is sometimes referred to as a perpetual annuity. Fixed coupon payments on permanently invested (irredeemable) sums of money are prime examples of perpetuities. Scholarships paid perpetually from an endowment fit the definition of perpetuity. The value of the perpetuity is finite because receipts that are anticipated far in the future have extremely low present value (present value of the future cash flows). Unlike a typical bond, because the principle is never repaid, there is no present value for the principal. Assuming that payments begin at the end of the current period, the price of a perpetuity is simply the coupon amount over the appropriate discount rate or yield, that is: PV= A/i Where: PV = Present Value of the Perpetuity; A = Amount of the periodic payment; i = Discount rate or interest rate. 23

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25. 25 1. Present value of perpetuity annuity An example: calculate the present value of the following perpetuity: € 1254 per year, postponed to the 11% rate. PV= A/i PV= present value of perpetuity annuity; A= 1254€ i= 0.11 PV= 1254/0.11= 12400€

26. 26 1a 2a3a4a5a 6a Pmt 0 PVPmt we now see the graph, how to do..... + infinite

27. 27 2. Present value of due perpetuity annuity An example: calculate the present value of the following perpetuity: € 1254 per year, postponed to the 11% rate. PVD= (A/I)*(1+i) PV= present value of perpetuity annuity; A= 1254€ i= 0.11 PV= 1254/0.11= 12400€ PVD= (1254/0.11)*(1.11)= 12654€

28. 28 1a 2a3a4a5a Pmt 0 PVD Pmt we now see the graph, how to do..... + infinite

29. G LOSSARY Annuity = Rendite; Future values of ordinary annuities = Montante di una rendita anticipata; Future value of due annuity= montante di una rendita posticipata; Present value of ordinary annuity = Valore attuale di una renditya anticipata; Present value of due annuity= valore attuale di una rendita posticipata; Present value of Perpetuity annuity = rendita perpetua anticipata Present value of due perpetuity annuity= rendita perpetua posticipata Discount rate= interessi. N= numero di rate. 29

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