MATH 3286

Mathematics of Finance

Instructor:

Dr. Alexandre Karassev

COURSE OUTLINE

• Theory of Interest1. Interest: the basic theory

2. Interest: basic applications

3. Annuities

4. Amortization and sinking funds

5. Bonds

• Life Insurance6. Preparation for life contingencies

7. Life tables and population problems

8. Life annuities

9. Life insurance

Chapter 1

INTEREST: THE BASIC THEORY

• Accumulation Function

• Simple Interest

• Compound Interest

• Present Value and Discount

• Nominal Rate of Interest

• Force of Interest

1.1 ACCUMULATION FUNCTION

• 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)

• Define Accumulation function a(t)=A(t)/A(0)• A(0)=principal• a(0)=1• A(t)=A(0)∙a(t)

Definitions

Natural assumptions on a(t)• increasing• (piece-wise) continuous

(0,1)t

a(t)

(0,1)

a(t)

t(0,1)

a(t)

t

Note: a(0)=1

Definition of Interest and

Rate 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:

a(t)) A(0) A(t) (since

A(0)

A(0)A(1)

a(0)

a(0)a(1)1a(1)

i

1)-a(n

1)-a(na(n)

1)-A(n

1)-A(nA(n)

ni

Example (p. 5)

• 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

a(t)=t2+t+1

Two Types of Interest

• 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 accumulation

function

( ≡ Two Types of Accumulation Functions)

1.2 SIMPLE INTERESTa(t)=1+it, t ≥0

(0,1)t

a(t) =1+it

1

1+i•Amount function:A(t)=A(0) ∙a(t)=A(0)(1+it)

•Effective rate is i

•Effective rate in nth year:

)1(1

ni

iin

Example (p. 5)

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(t)=1+it

SolutionA(0)=1000 i=0.15

A(t)=A(0)(1+it)=1000(1+0.15t)

t=?

How to calculate t in practice?

• Exact simple interest number 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

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

A(t)=1000(1+0.15t)

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

1+it

COMPOUND INTEREST Accumulation Function

a(t)=(1+i)t

(0,1)

t

a(t)=(1+i)t

1

1+i

•Amount function:A(t)=A(0) ∙a(t)=A(0) (1+i)t

•Effective rate is i

•Moreover effective rate in nth year is i (effective rate is constant):

iii

iii

n

nn

n

11)1(

)1()1(1

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

1

t

a(t)=(1+i)t

1

1+i

2

(1+i)2

Example (p. 8)

Jack borrows 1000 at 15% compound interest.

a) How much does he owe after 2 years?b) How much does he owe after 57 days,

assuming compound interest between integral durations?

c) How much does he owe after 1 year and 57 days, under the same assumptions asin (b)?

d) How much does he owe after 1 year and 57 days, assuming linear interpolation between integral durations

e) In how many years will his principal have accumulated to 2000?

a(t)=(1+i)t

A(t)=A(0)(1+i)t

A(0)=1000, i=0.15

A(t)=1000(1+0.15)t

1.4 PRESENT VALUE AND DISCOUNT

Definition

The amount of money that will accumulate to the principal over t years is called the present value t years in the past.

PRINCIPAL ACCUMULATEDVALUE

PRESENTVALUE

Calculation of present value

• t=1, principal = 1

• Let v denote the present value• v (1+i)=1

• 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

v=1/(1+i)

t

tt i

iv )1(

1

1

a(t)=(1+i)t

gives the value of one unit (at time 0)at any time t, past or future

(0,1)

t

a(t)=(1+i)t

If principal is not equal to 1…

present value = A(0) (1+i)t

PRINCIPALA (0)

ACCUMULATEDVALUE

A(0) (1+i)t

PRESENTVALUE

A(0) (1+i)t

t < 0 t = 0 t > 0

Example (p. 11)The Kelly family buys a new house for 93,500 on May 1, 1996.How much was this house worth on May 1, 1992 if real estateprices have risen at a compound rate for 8 % per year duringthat period?

Solution a(t)=(1+i)t

• Find present value of A(0) = 93,500 996 - 1992 = 4 years in 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 is positive

t > 0

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

1

t

a(t) =1+it

1 / (1 + it)1

t

a(t) =1+it

1 / (1 - it)

Discount

• We invest 100

• After one year it accumulates to 112

• The interest 12 was added at the end of the term

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

Rate of DiscountDefinition Effective rate of discount d

d = accumulated value after 1 year – principal accumulated value after 1 year

= A(1) – A(0) A(1)

= A(0) ∙a(1)– A(0) A(0) ∙a(1)

= a(1) – 1 a(1)

i = accumulated value after 1 year – principal principal = a(1) – 1

a(0)

Recall:

In nth year…

)(

)1()(

na

nanadn

Identities relating d to i and v

i

i

i

i

a

aad

11

1)1(

)1(

)0()1(

i

id

1Note: d < i

vii

ii

i

id

1

1

1

)1(

111 vd 1

d

di

1

Present and accumulated values in terms of d:

• Present value = principal * (1-d)t

• Accumulated value = principal * [1/(1-d)t]

ivd

1

11

If we consider positive and negative values of t then:

a(t) = (1 - d)-t

Examples (p. 13)

1. 1000 is to be accumulated by January 1, 1995 at a compound rate of discount of 9% per year.

a) Find the present value on January 1, 1992

b) Find the value of i corresponding to d

2. 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

Note: t is the number of effective interest periods in any particular problem

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?

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?

Note: in both examples the given ratesof interest (12% and 18%) were nominal rates of interest

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

mm

m

ii

)(

11

Equivalently:

1]1[ /1)(

mm

im

i

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.

Examples (p. 15)

1. Find the accumulated value of 1000 after three years at a rate of interest of 24 % per year convertible monthly

2. If i(6)=15% find the equivalent nominal rate of interest convertible semi-annually

Nominal rate of discount

• The nominal rate of discount d(m) is

defined so that d(m) / m is an effective rate of interest in 1/m part of a year

• Formula:

mm

m

dd

)(

11

Formula relating nominal rates of interest and discount

nn

n

dd

)(

11

1)1(1

1

11

di

id

mm

m

ii

)(

11

nnmm

n

d

m

i

)()(

11

Example

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

nnmm

n

d

m

i

)()(

11

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

]1)1[( /1)( mm imiNominal rate of interest equivalent to i:

Let m approach infinity: ]1)1[(limlim/1)(

m

m

m

mimi

We define the force of interest δ equal to this limit:

]1)1[(limlim/1)(

m

m

m

mimi

Formula

• Force of interest δ = ln (1+i)• Therefore eδ = 1+i • and a (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 loan on June 23, 2002

– The value of i– The value of i(12)

Remark

)(

)(

)1(

])1[(

)1()1ln()1(])1[(

ta

ta

i

i

iiii

t

t

ttt

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

Definition

)(

)(

ta

tat

Note: 1) in general case,

force of interest depends on t2) it does not depend on t ↔ a(t)= (1+i)t !

The force of interest corresponding to a(t):

Example (p. 19)

• Find in δt the case of simple interest

• Solution

it

i

it

it

ta

tat

11

)1(

)(

)(

How to find a(t)

if we are given by δt ?

Consider differential equation in which a = a(t) is unknown function:

)(

)(

ta

tat

We have:

at

a

Since a(0) = 1 its solution is given by

t

rdr

eta 0)(

Applications• Prove that if δt = δ is a constant then

a(t) = (1+i)t for some i• Prove that for any amount function A(t) we

have:

• Note: δt dt represents the effective rate of interest over the infinitesimal “period of time” dt . Hence A(t)δt dt is the amount of interest earned in this period and the integral is the total amount

)0()()(0

AnAdttAn

t

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:

iiiddd mmmm )()1()1()(

midid mm

111 and 1

11)()(