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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 3286Mathematicsof 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 • Life annuities • Life insurance
Chapter 1INTEREST: THE BASIC THEORY • Accumulation Function • Simple Interest • Compound Interest • Present Value and Discount • Nominal Rate of Interest • Force of Interest
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)
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
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:
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
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
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:
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=?
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
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
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
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):
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
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
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
Calculation of present value • t=1, principal = 1 • Let vdenote the present value • v (1+i)=1 • v=1/(1+i)
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
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
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
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
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
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
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
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) =
Identities relating d to iandv Note:d < i
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
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.
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
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
…yet another example • You have two credit card offers: • 17% convertible semi-annually • 16% convertible monthly • Which is better?
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
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:
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
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:
Example • Find the nominal rate of discount convertible semiannualy which is equivalent to a nominal rate of interest of 12% convertible monthly
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
Definition Nominal rate of interest equivalent to i: Let m approach infinity: We define theforce of interest δequal to this limit:
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
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)
Remark The last formula shows that it is reasonable to define forceof interest for arbitrary accumulation function a(t)
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 !
Example (p. 19) • Find in δtthe case of simple interest • Solution
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
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
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: