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MATH 3286 Mathematics of Finance

MATH 3286 Mathematics of Finance. Alex Karassev. COURSE OUTLINE. Theory of Interest Interest: the basic theory Interest: basic applications Annuities Amortization and sinking funds Bonds Life Insurance Preparation for life contingencies Life tables and population problems

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MATH 3286 Mathematics of Finance

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  1. MATH 3286Mathematicsof Finance Alex Karassev

  2. COURSE OUTLINE • Theory of Interest • Interest: the basic theory • Interest: basic applications • Annuities • Amortization and sinking funds • Bonds • Life Insurance • Preparation for life contingencies • Life tables and population problems • Life annuities • Life insurance

  3. Chapter 1INTEREST: THE BASIC THEORY • Accumulation Function • Simple Interest • Compound Interest • Present Value and Discount • Nominal Rate of Interest • Force of Interest

  4. 1.1 ACCUMULATION FUNCTION Definitions • The amount of money initially invested is called the principal. • The amount of money principal has grown to after the time period is called theaccumulated value and is denoted byA(t) – amount function. • t ≥0 is measured in years (for the moment) • DefineAccumulation function a(t)=A(t)/A(0) • A(0)=principal • a(0)=1 • A(t)=A(0)∙a(t)

  5. Natural assumptions on a(t) • increasing • (piece-wise) continuous a(t) a(t) a(t) (0,1) (0,1) (0,1) t t t Note: a(0)=1

  6. Definition of Interest andRate of Interest • Interest = Accumulated Value – Principal:Interest = A(t) – A(0) • Effective rate of interest i (per year): • Effective rate of interest in nth year in:

  7. Example (p. 5) a(t)=t2+t+1 • Verify that a(0)=1 • Show that a(t) is increasing for all t ≥ 0 • Is a(t) continuous? • Find the effective rate of interest i for a(t) • Find in

  8. Two Types of Interest ( ≡ Two Types of Accumulation Functions) • Simple interest: • only principal earns interest • beneficial for short term (1 year) • easy to describe • Compound interest: • interest earns interest • beneficial for long term • the most important type of accumulationfunction

  9. a(t) =1+it 1+i (0,1) t 1 1.2 SIMPLE INTEREST a(t)=1+it, t ≥0 • Amount function:A(t)=A(0) ∙a(t)=A(0)(1+it) • Effective rate isi • Effective rate in nth year:

  10. a(t)=1+it Example (p. 5) Solution Jack borrows 1000 from the bank on January 1, 1996 at a rate of 15% simple interest per year. How much does he owe on January 17, 1996? A(0)=1000 i=0.15 A(t)=A(0)(1+it)=1000(1+0.15t) t=?

  11. How to calculatetin practice? • Exact simple interestnumber of days 365 • Ordinary simple interest (Banker’s Rule)number of days 360 t = t= Number of days: count the last day but not the first

  12. A(t)=1000(1+0.15t) Number of days (from Jan 1 to Jan 17) = 16 • Exact simple interest • t=16/365 • A(t)=1000(1+0.15 ∙ 16/365) = 1006.58 • Ordinary simple interest (Banker’s Rule) • t=16/360 • A(t)=1000(1+0.15 ∙ 16/360) = 1006.67

  13. 1.3 COMPOUND INTEREST Interest earns interest • After one year:a(1) = 1+i • After two years:a(2) = 1+i+i(1+i) = (1+i)(1+i)=(1+i)2 • Similarly after n years:a(n) = (1+i)n

  14. a(t)=(1+i)t 1+it 1+i (0,1) t 1 COMPOUND INTEREST Accumulation Function a(t)=(1+i)t • Amount function:A(t)=A(0) ∙a(t)=A(0) (1+i)t • Effective rate isi • Moreover effective rate in nth year is i (effective rateis constant):

  15. a(t)=(1+i)t (1+i)2 1+i 1 t 2 1 How to evaluate a(t)? • If t is not an integer, first find the value for the integral values immediately before and after • Use linear interpolation • Thus, compound interest is used for integral values of t and simple interest is used between integral values

  16. Example (p. 8) a(t)=(1+i)t Jack borrows 1000 at 15% compound interest. • How much does he owe after 2 years? • How much does he owe after 57 days, assuming compound interest between integral durations? • How much does he owe after 1 year and 57 days, under the same assumptions asin (b)? • How much does he owe after 1 year and 57 days, assuming linear interpolation between integral durations • In how many years will his principal have accumulated to 2000? A(t)=A(0)(1+i)t A(0)=1000, i=0.15 A(t)=1000(1+0.15)t

  17. PRINCIPAL ACCUMULATEDVALUE PRESENTVALUE 1.4 PRESENT VALUE AND DISCOUNT The amount of money that will accumulate to the principal over t years is called the present valuet years in the past t -t

  18. Calculation of present value • t=1, principal = 1 • Let vdenote the present value • v (1+i)=1 • v=1/(1+i)

  19. v=1/(1+i) In general: • t is arbitrary • a(t)=(1+i)t • [the present value of 1 (t years in the past)]∙ (1+i)t = 1 • the present value of 1 (t years in the past) = 1/ (1+i)t = vt

  20. a(t)=(1+i)t (0,1) t a(t)=(1+i)t gives the valueof one unit(at time 0)at any time t,past or future

  21. PRINCIPALA (0) ACCUMULATEDVALUEA(0) (1+i)t PRESENTVALUEA(0) (1+i)t If principal is not equal to 1… present value = A(0) (1+i)t t > 0 t < 0 t = 0

  22. Solution Example (p. 11) a(t)=(1+i)t The Kelly family buys a new house for 93,500 onMay 1, 1996.How much was this house worth on May 1, 1992 if real estate prices have risen at a compound rate for 8 % per year during that period? • Find the present value ofA(0) = 93,500 • 996 - 1992 = 4 yearsin the past • t = - 4, i = 0.08 • Present value = A(0) (1+i)t= 93,500 (1+0.8) -4 = 68,725.29

  23. If simple interest is assumed… • a (t) = 1 + it • Let x denote the present value of one unit t years in the past • x ∙a (t) = x (1 + it) =1 • x = 1 / (1 + it) NOTE: In the last formula, t > 0

  24. a(t) =1+it a(t) =1+it 1 1 / (1 - it) 1 1 / (1 + it) t Thus, unlikely to the case of compound interest, we cannot use the same formula for present value and accumulated value in the case of simple interest

  25. Discount Alternatively: • Look at 112 as a basic amount • Imagine that 12 were deducted from 112 at the beginning of the year • Then 12 is amount of discount • We invest 100 • After one year it accumulates to 112 • The interest 12 was added at the end of the term

  26. Rate of Discount DefinitionEffective rate of discountd accumulated value after 1 year – principal accumulated value after 1 year A(1) – A(0) A(1) d = = A(0) ∙a(1)– A(0) A(0) ∙a(1) a(1) – 1 a(1) = = Recall: accumulated value after 1 year – principal principal i = a(1) – 1 a(0) =

  27. In nth year…

  28. Identities relating d to iandv Note:d < i

  29. Present and accumulated values in terms of d: • Present value = principal * (1-d)t • Accumulated value = principal * [1/(1-d)t] If we consider positive and negative values of t then: a(t) = (1 - d)-t

  30. Examples (p. 13) • 1000 is to be accumulated by January 1, 1995 at a compound rate of discount of 9% per year. • Find the present value on January 1, 1992 • Find the value of i corresponding to d • Jane deposits 1000 in a bank account on August 1, 1996. If the rate of compound interest is 7% per year, find the value of this deposit on August 1, 1994.

  31. 1.5 NOMINAL RATE OF INTEREST Example (p. 13) A man borrows 1000 at an effective rate of interest of 2% per month. How much does he owe after 3 years? Note: t is the number ofeffective interest periodsin any particular problem

  32. More examples… (p. 14) • You want to take out a mortgage on a house and discover that a rate of interest is 12% per year. However, you find out that this rate is “convertible semi-annually”. Is 12% the effective rate of interest per year? • Credit card charges 18% per year convertible monthly. Is 18% the effective rate of interest per year? In both examples the given ratesof interest (12% and 18%) werenominal rates of interest

  33. …yet another example • You have two credit card offers: • 17% convertible semi-annually • 16% convertible monthly • Which is better?

  34. Definition • Suppose we have interest convertible m times per year • The nominal rate of interest i(m) is defined so that i(m) / m is an effective rate of interest in 1/m part of a year

  35. Note: If i is the effective rate of interest per year, it follows that In other words,i is the effective rate of interestconvertible annually which is equivalent to the effective rate of interest i(m) /m convertible mthly Equivalently:

  36. Examples (p. 15) • Find the accumulated value of 1000 after three years at a rate of interest of 24 % per year convertible monthly • If i(6)=15% find the equivalent nominal rate of interest convertible semi-annually

  37. Formula that relatesnominal rates of interest

  38. Nominal rate of discount • The nominal rate of discountd(m) is defined so that d(m) / m is an effective rate of discount in 1/m part of a year • Formula:

  39. Formula relating nominal rates of interest and discount

  40. Example • Find the nominal rate of discount convertible semiannualy which is equivalent to a nominal rate of interest of 12% convertible monthly

  41. 1.6 FORCE OF INTEREST • What happens if the number m of periods is very large? • One can consider mathematical model of interest which is convertible continuously • Then the force of interest is the nominal rate of interest, convertible continuously

  42. Definition Nominal rate of interest equivalent to i: Let m approach infinity: We define theforce of interest δequal to this limit:

  43. Formula • Force of interestδ = ln (1+i) • Thereforeeδ = 1+i • anda (t) = (1+i)t =eδt • Practical use of δ: the previous formula gives good approximation to a(t) when m is very large

  44. Example • A loan of 3000 is taken out on June 23, 1997. If the force of interest is 14%, find each of the following: • The value of the loanon June 23, 2002 • The value of i • The value of i(12)

  45. Remark The last formula shows that it is reasonable to define forceof interest for arbitrary accumulation function a(t)

  46. Definition The force of interest corresponding to a(t): • Note: • in general case,force of interest depends on t • it does not depend on t ↔a(t)= (1+i)t !

  47. Example (p. 19) • Find in δtthe case of simple interest • Solution

  48. How to find a(t)if we are given by δt ? We have: Consider differential equation in which a =a(t)is unknown function: Since a(0) = 1its solution is given by

  49. Applications • Prove that if δt = δ is a constant thena(t) = (1+i)t for some i • Prove that for any amount function A(t) we have: • Note:δtdtrepresents the effective rate of interest over the infinitesimal “period of time” dt . Hence A(t)δtdtis the amount of interest earned in this period and the integral is the total amount

  50. Remarks • Do we need to define the force of discount? • It turns out that the force of discount coincides with the force of interest!(Exercise: PROVE IT) • Moreover, we have the following inequalities: • and formulas:

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