LOGO
MATH 2040
Introduction to
Mathematical Finance
Dr. Ken Tsang
1

Instructor: Dr Ken Tsang & Miss Liu Youmei
Phone: 3620606(Tsang);3620630(Liu)
Email: kentsang@uic.edu.hk
ymliu@uic.edu.hk
Website: http://www.uic.edu.hk/~kentsang/math2040/math2040.htm
2

Assessment of performance (apprpx.)
Quiz
Mid-Term Exam
10%
10%
20%
Assignments
Final Exam
60%
3

• Quizzes and Assignments
10-15%
• Mid-term test (s)
20-30%
• Projects
10-20%
• Final Examination
50-60%
4

Assessment grade system:

A and A- (Not more than
10%)

A + B (Not more than
65%)

C + D (No limit ).
Letter
Grade
B-
C+
C
A
A-
B+
B
C-
D
F
Academic
Performance
Excellent
Excellent
Good
Good
Good
Satisfactory
Satisfactory
Barely
Satisfactory
Marginal Pass
Fail
Grade Point
Per Unit
4.00
3.70
3.30
3.00
2.70
2.30
2.00
1.70
1.00
0.00
5


Assignments must be handed in before the deadline.

In the mid-term test and Final Exam, the ONLY
THING you can bring is a calculator. Any other electronic device, e.g. mobile phone, is not allowed.

Result of the final examination is released by
AR only. We cannot tell you the score before AR inform you the official results.
6

General Information
• Textbook
The Theory of Interest, Third Edition,
Stephen G. Kellison, McGraw Hill
International Edition(2009).
7

• The Theory of Interest Irving Fisher
– He was an American (celebrity) economist, his reputation today is probably higher than it was in his lifetime. This book summed up Fisher’s work on capital, capital budgeting, credit markets, and the determinants of interest rates, including the rate of inflation.
• 利息理论 刘占国 主编
中国财经出版社
• 利息理论及应用 刘明亮 ， 邓庆彪 主编
中国金融出版社
8

• We start the “ Introduction to Mathematical
Finance ” by studying the “ Theory of Interest ” because of the importance of Interest in finance.
• Interest policy is often being used as a tool to regulate the economy in a modern society.
9

Central bank – the most important financial institution of a country
The Bank of England
United States Federal Reserve
The People's
Bank of China
10

Important functions of a central bank
• implementing monetary policy
• determining Interest rates
• controlling the nation's entire money supply
• setting the official interest rate – used to manage both inflation and the country's exchange rate – and ensuring that this rate takes effect via a variety of policy mechanisms
• regulating and supervising the banking industry
11

the process a government, central bank, or monetary authority of a country uses to control
– the supply of money,
– availability of money, and
– cost of money or rate of interest so as to attain a set of objectives oriented towards the growth and stability of the economy.
12




Interest rates is the main and popular tool of monetary policy.


Discount rate- interest rate a central bank charges depository institutions that borrow short-term funds from it.
Federal funds rate- interest rate banks charge each other for loans
The level of interest rate has a profound effect on economic growth and inflation.
Low interest rate generally leads to economic expansion, high interest rate generally leads to economic contraction.
13

To study the practical and theoretical concepts involved in computing interest
To acquire sufficient knowledge to handle all normal interest computations including bonds and mortgages
To be familiar with current methods of computing approximate interest rates for commercial transactions
To motivate students to appreciate the fluctuations of interest on prices of stocks and bonds.
14

• Financial decision
• Business decision
• Valuation and yield rates of bonds
• Compute proper reserves for bonds
• Loan amortizations, mortgage and
• Many others
15

Chapter 1: The Measurement of Interest
The effective rate of interest & discount
Simple & Compound Interest
Present Value
Nominal rates of interest & discount
Forces of interest & discount
16

Scenario
 Suppose John has won a lottery with two options to collect the prize (one million).
 Option one is to get 20 payments of
$60,000 on the first day of each year as from
2010.
 Option two is to get one million on the first day of 2010.
 Which option should John choose?
17

• How badly does John need the cash?
• How much debt has John?
• Does John like to spend money as he receives it?
• How much return can John make with his cash?
• All of the above and other factors related to interest rate.
18

When a landowner allow a farmer to use the land he owns, the farmer has to pay “rent” to the landowner.
Note: The “rent” can be in the form of a share of the crop harvested from the land.
When a banker lets a borrower to use a certain amount of money, the banker will charge the borrower something.
19

•
Interest may be defined as the compensation paid by a borrower of capital to a lender.
• Thus we can view interest as the rent paid by borrower to a lender for the loss of use of the capital.
• In theory, interest and capital need not be expressed in terms of the same commodity.
20

• A common financial transaction is the investment of money for interest.
• For example, a person may make a (fixed) term deposit at a bank.
• In this case, the person is the lender, the bank is the borrower, and the bank might pay interest to the person.
21

• The initial sum of money invested is called the principal .
• The total amount received after a period of time is called the accumulate value .
• The difference between the accumulate value and the principal is the amount of interest, or, simply interest .
22

• Once the principal is given, then the accumulated value at any point of time can be determined.
• We assume that no principal is added or taken away during the period of investment, so any change in the fund is strictly due to interest.
• Let t be the time elapsed from the beginning of investment. The unit in which time is measured is called the period.
• The period is normally a year, but any other time unit is acceptable.
23

• Consider the investment of one unit of principal.
• We can define an accumulation function a ( t ), which gives the accumulated value at some time t > 0 of with an original investment of
1,
• and a(0) = 1
24

• In general, the original principal will be some amount k > 0, and k is not necessarily one unit.
• We now define an amount function A ( t ), which gives the accumulated value at time t > 0 of an original investment of k .
• Clearly A ( t ) = k · a ( t ) and A (0) = k .
• Interest earned during the n -th period will be denoted by I n
. Then
I n
= A ( n ) – A ( n

1) for integral n
≥ 1.
25

• It is clear that a (0) = 1.
• Normally a ( t ) is a non-decreasing function.
• Note that it is possible for a ( t ) to decrease over t (in case of negative interest rate).
• If interest accrues continuously, the function is continuous.
26

27

• Figure 1(a) is a linear amount function
• Figure 1(b) is nonlinear, in this case an exponential function
• Figure 1(c) is a constant function, meaning that the principal is not accruing any interest
• Figure 1(d) is a step function - accruing interest at discrete time with no interest accruing between interest payment dates.
28

• Various measures of interest may be developed from the accumulation function
• In practice, two measures of interest will handle most situations which arise. They are:
– Effective rate of interest
, and
– Effective rate of discount .
29

• The effective rate of interest i is the amount of money earned at the end of one period when one unit is invested at the beginning of that period.
• In terms of the accumulation function, this is equivalent to: i = a (1)
 a (0), or a (1) = a (0) + i
30

• The effective rate of interest i is the ratio of the amount of interest earned during the period to the principal at the beginning of that period.
• In terms of the accumulation/amount function we have:
31

• The term is used for rate of interest in which interest is paid once a year , in contrast with
“ nominal rates
” of interest.
• It is often expressed as a percentage per annum
(year).
– For example, 6% p.a. means one that ¥100 will accrue an interest of ¥6 at the end of one year.
32

• The amount of principal remains constant throughout the period, i.e. principal is neither contributed nor withdrawn.
• The effective rate of interest is a measure in which interest is paid at the end of the period.
We shall encounter situations in which interest is paid at the beginning of the period.
33

For convenience, we shall use some shorter terms at times
• The term “principal” may mean “the amount of principal”.
• The term “interest” may mean “the amount of interest earned” or “the amount of interest”.
• The meaning of these words would be clear from the context.
34

• Effective rates of interest may be calculated over any measurement period in terms of the amount function.
• Let i n be the effective rate of interest during the n -th period from the date of investment. Then we have
35

• Consider the investment of one unit such that the interest in each period is constant.
• Then in general, we have a linear accumulation function a ( t ) = 1 + it for integral t
≥ 0.
• The accruing of interest in the above pattern is called simple interest.
36

• A constant rate of interest does not imply a constant effective interest rate.
• Let i be the rate of simple interest and let i n be the effective rate of interest for the th period, then for n
≥ 1 we have: n -
37

• We have defined accumulation function for simple interest only for integral values of t .
• Naturally, we want to extend the definition to nonintegral values of t > 0 as well.
• If interest is accrued only for completed periods with no credit for fractional periods, then the accumulation function become a step function – Figure 1 (d).
• Unless otherwise stated, we assume interest is accrued proportionately over fractional periods under simple interest and the accumulation function is a linear function – Figure 1 (a).
38

• If interest is accrued proportionately over fractional periods, we would like a ( t ) to have the property:
– Interest for an initial investment of 1 for t + s periods is equal to that for t periods plus that for s periods.
• In terms of accumulation function that becomes: a ( t + s )

1 = [ a ( t )

1] + [ a ( s )

1]
• So we have the formula: a ( t + s ) = a ( t ) + a ( s )

1 for t
≥ 0 and s
≥ 0.
39

• Assuming a ( t ) is differentiable, from basic definition of derivative, we have: a constant.
40

• Replacing t by r and integrating both sides between the limits 0 and t , we have
• If we let t = 1, we have 1 + i = a (1) = 1 + a' ( 0 ).
• It follows that a ( t ) = 1 + it for all t
≥ 0.
41

• Find the accumulated value of $200 invested for four years, if the rate of simple interest is 8% per annum.
• The answer is : 200[1 + (0.08) 4] = 264.
• The amount of interest earned is 264 
200 = 64, which could also have been obtained as 200(0.08)(4), or, in general as A (0) · i
· t
• The above becomes the familiar formula
I = Prt , which states that interest is equal to the product of the principal, the rate of interest and the period.
42

A Canadian T-Bill with face value $100 is a security which is exchangeable for $100 CAD on the maturity date.
Suppose that a T-Bill with face value $100 is issued on
2005.09.08 and matures on 2005.12.15 (there are 98 days between the dates). Given that the price of the T-Bill is $99.27076, find the effective annual rate of interest .
Assume simple interest.
Answer:
43

• In simple interest, the interest earned over any period is not re-invested.
• For example, if ¥100 is invested for two years at
10% simple interest, the investor will earn ¥10 during both year one and year two.
• In reality, investor would like to receive and reinvest ¥10, the interest for the first year.
• Then the interest for the second year would be ¥11, and the amount after two years would be ¥121.
44

• The theory of compound interest assumes that the interest earned is automatically re-invested.
• The word “compound” refers to “interest on interest”.
• At every point of time, the total of principal and interest earned to date is treated as the new principal .
45

Accumulation function - compound interest
• Consider the investment of 1 which accumulates to
(1 + i ) at the end of period one.
• The sum (1 + i ) becomes principal for period two.
• The balance at the end of period two now becomes:
(1 + i ) + i (1 + i ) = (1 + i ) 2 .
• Similarly, (1 + i ) 2 becomes principal for period three and the balance at the end of period three is (1
+ i ) 3 .
• Continuing this process indefinitely, we have a ( t ) = (1 + i ) t for integral t

0.
46

• Let i be the rate of compound interest and let i n the effective rate of interest for the n -th period.
• Then using the formula: be we can show that i n
= i , which is independent of n .
• Although defined differently, a rate of compound interest and an effective rate of interest are identical.
47

• The accumulation formula for compound interest has been obtained for integral values of t , namely a ( t ) = (1 + i ) t .
• For non-integral values of t , we start with the following property which we want compound interest to process:
• Investing 1 for t periods and then re-investing the proceeds for another s periods will be the same as investing 1 for s
+ t periods.
• That results in the formula a ( t + s ) = a ( t ) a ( s ) for integral t > 0 and s > 0.
48

• Assuming that a ( t ) is differentiable, from the definition of derivatives, we have:
49

• It follows that and
• Since log e a (0) = 0, so if we let t = 1 and remember that a (1) = 1 + i , we have
50

• Unless otherwise stated, compound interest will be accrued over fractional periods according to the formula a ( t ) = (1 + i ) t .
• This function is exponential as in Figure 1(b).
• For compound interest, any interest accrued may be considered paid and re-invested. So we can forget about actually paying the interest, because that is like withdrawing from the principal.
51

• Simple and compound interest produces the same result over the first period of investment.
• For longer than one period, compound interest produces more interest.
• For a fraction of the first period, interest from compound interest is less than that of simple interest.
• For simple interest , absolute amount of growth is constant, for compound interest , the relative rate of growth is constant.
52

• Let s be fixed. Under simple interest a ( t + s ) – a ( t ) is independent of t , whereas under compound interest, a ( t + s ) – a ( t ) a ( t ) is independent of t .
• Compound interest is used almost exclusively for financial transactions, unless the term is rather short.
• It is assumed that interest earned under compound interest is re-invested at the same rate. This may not be always the case.
53

• Find the accumulated value of ¥200 invested for four years, if the rate of compound interest is 8% annum.
• The answer is
200(1 + 0.08) 4 = 272.10
• The answer is in contrast with the answer of 264 in example 1 using simple interest. The extra ¥8.10 is the result of compound interest
54

comparing compound & simple interest
I deposited $1,000 in an account with an effective annual interest rate of 7.3%, how much would I have in my account after 6 months, using
(a) simple and (b) compound interest?
Using simple interest, I have:
Using compound interest, I have:
For a fraction of the first period, interest from compound interest is less than that of simple interest.
55

• The term (1 + i ) is called an accumulation factor because a principal of 1 will accumulate to (1 + i ) after one period.
• How much do we need to invest now so that the investment will accumulate to 1 after one period?
• The answer, which we call discount factor , is (1 + i )

1 .
• The discount factor (1 + i )

1 is denoted by v , and it
“discounts” the value of an investment at the end of a period to its value at the beginning of the period.
56

• In general, the investment required to produce an amount of 1 at the end of one period is a

1 ( t ).
• We will call a

1 (t) the discount function.
• We obtained the following results for t
≥ 0:
57

• In a sense, accumulating and discounting are two opposite processes.
• The term (1 + i ) t is called the accumulated value of
1 at the end of t periods .
• The term (1 + i )
 t , or equivalently v t , is called the present value of 1 to be paid at the end of t periods .
• It is clear that v t extends the definition of the accumulation function to negative values of t .
58

• Find the amount which must be invested to accumulate to ¥1000 at the end of three years at a rate of 9% per annum in (a) simple interest and (b) compound interest.
• The answer to (a) is:
• The answer to (b) is:
59

t
60

On Jan. 1, you won a “$400,000 sweepstakes”. The prize is to be paid out in 4 yearly installments of $100,000 each with the first paid immediately.
Assuming that you can invest funds at 5% interest compounded yearly, what is the present value of the prize?
General principle: the value today of a promised series of future payments is the sum of their present values, computed at the prevailing interest rate for comparable investments.
61

Solution of Example9: Future & present values
62

Future & present values
At what time would a single payment of $400,000 be equivalent the series of payments in Example 9, i.e. at what time t does the present value of $400,000 equal the present values of the payments in Example 9?
Solution: From the solution to Example 9, the present value of all of the payments in question is 372324.80. Hence, we seek t such that: which leads to
63

The time value of money is the value of money figuring in a given amount of interest earned over a given amount of time.
For example, 100 dollars of today's money invested for one year and earning 5 percent interest will be worth 105 dollars after one year. Therefore, 100 dollars paid now or 105 dollars paid exactly one year from now both have the same value to the recipient who assumes 5 percent interest; using time value of money terminology, 100 dollars invested for one year at 5 percent interest has a future value of 105 dollars.
The method allows the valuation of a likely stream of income in the future, in such a way that the incomes are discounted and then added together, thus providing a lump-sum "present value" of the entire income stream.
64

• Suppose we go to the bank and borrow ¥1000 at 7% per annum for one year. At the end of the year we have to repay the bank ¥1000 plus ¥70 for a total of ¥1070.
• Again if we borrow ¥1000, but the 7% interest has to be paid at the time the money is borrowed, then we only get
¥930 from the bank, and we have to repay the bank ¥1000.
• This latter method of interest charged by the bank is called discounting. We say that the bank is charging the loan an effective rate of discount of 7% per annum.
65

实际贴现率
• It is clear that an effective rate of interest at 7% is not the same as an effective rate of discount of 7%.
• Although the interest are the same in both cases, but the amount of loan is smaller when the effective rate is for discounting.
• The effective rate of discount d is the ratio of interest earned during the period to the amount at the end of the period.
66

• The first three observations
(p.32-33) on effective rate of interest also applies to effective rate of discount.
• The phrases “amount of interest” and “amount of discount” can be used interchangeably in situations involving discount.
• The key distinction between effective rate of interest and effective rate of discount is:
– Interest – paid at the end of the period on beginning balance.
– Discount – paid at the beginning of the period on end balance.
67

• Effective rate of discount can be calculated over any particular period.
• Let d n be the effective rate of discount for the th period. A formula analogous to that for i n is: n -
• Note again that
I n can be called either the “amount of interest” or “the amount of discount”.
68

• In general, d n periods.
may vary over different
• But if we have compound interest, in which case the effective rate of interest is constant, then the effective rate of discount is also constant.
• We refer to this situation as compound discount .
69

• Two rates of interest and discount are equivalent if given a certain principal invested for the same length of time produces the same accumulated value.
• Suppose John borrows 1 at an effective rate of discount d . The real borrowed amount is then 1
 d , and the interest (discount) paid is d . If i is the effective rate of interest, then we have
70

What is the equivalent effective rate of interest for an account that earns interest at an effective discount rate of 3.7%?
Solution:
= 1 + i or i = d / (1 - d) = 0.037 / ( 1- 0.037) = 0.0384
So the equivalent effective interest rate is 3.84%.
71

Application of “Discount”:
US Treasury bills
Treasury bills (or T-Bills), issued by the United States Government, mature in one year or less. Like zero-coupon bonds, they do not pay interest prior to maturity; instead they are sold at a discount of the par value to create a positive yield to maturity. Treasury bills are sold in auctions held weekly.
What is the effective interest rate of a one year T-Bill sold at a discount of 2.5% ?
Solution: 0.025 / ( 1 – 0.025 ) = 0.02564
72

• From the previous slide, we have
73

1
• There are important relationships between d , a rate of discount, and v , a discount factor.
• The first one is d = iv .
• A verbal interpretation of this formula is:
– The interest for an investment of 1 due at the end of one year is d .
– The present value of 1 due at the end of one year is v , and the amount of interest on that for one year is iv .
– Hence d = iv .
74

2
• Another important relationship between d and v is: d = 1
 v .
• Written in the form v = 1
 d , we see that both sides represent the present value of 1 due at the end of one year from now.
75

3
• The third important relationship between d and v is: i
 d = id .
• A person can either borrow 1 and repay 1
+ i at the end of the year or borrow 1
 d and repay 1 at the year of the year. The interest saved in the second deal is i
 d .
• In the second deal, the principal borrowed is less by amount d , hence interest saved should be id .
• Hence the formula i
 d = id .
76

Graphical relationship between d and v a(1) =
Accumulation function a(0) = 1 effective rate of discount discount factor
Time i = effective rate of interest, equivalent to discount rate d
77

• It is possible to define simple discount similar to that of simple interest.
• Suppose the amount of discount for each period is the same.
• Then the original principal which will produce an accumulated value of 1 at the end of t periods is a -1 ( t ) = 1
 dt for 0 ≤ t < 1/ d .
• We need to have t < 1/ d to keep a -1 ( t ) > 0.
78

• Simple discount has properties analogous to, but opposite, to simple interest.
• A constant amount of interest in each period implies a decrease of effective interest rate, but a constant amount of discount leads to an increasing effective rate of discount.
• For one period, simple and compound discount produces the same result.
• Over more periods, simple discount produces a smaller present value than compound discount. The opposite is true for shorter periods.
79

• Find the amount which must be invested to accumulate to ¥1000 at the end of three years at a rate of 9% per annum (a) simple discount and (b) compound discount.
• The answer to (a) is 1000{1 
(3)(0.09)} = 730
• The answer to (b) is 1000(1 – 0.09) 3 = 753.57
• Note that the amount discounted is smaller in the case of compound discount.
80

• Suppose three banks offer loans at the following rates:
– Bank A charges an effective rate of 9%,
– Bank B charges 8.75% compounded quarterly, and
– Bank C charges 8.5% payable in advance and convertible monthly.
• Bank A is charging an effective rate. However, Bank
B is charging a nominal interest rate , and Bank C is charging a nominal discount rate.
81

Symbols for nominal interest rates
• A rate of interest or discount is called
if it is charged, payable, compounded or convertible for more than once a year.
(see p.32 for effective rate)
• A nominal rate of interest payable m times a year, where m is a positive integer, is denoted by i ( m ) .
82

• Suppose i (4) = 8%. That means interest is compounded 4 times a year, i.e. every three months.
• The rate for each three month will then be 2%, being one-fourth of 8%.
• So a principal of 1 accumulates to
(1 + 0.02) 4 = 1.0 824 after one year.
83

• In general, a nominal rate of interest i ( m ) per annum is identical to an effective interest rate of i ( m ) / m for every 1/ m year.
• Thus by the definition of equivalency, we have
• This gives
84

Bank A offers a nominal rate of 5.2% interest, compounded twice a year. Bank B offers 5.1% interest, compounded daily. Which is the better deal?
85

• A nominal rate of discount payable m times a year, where m is a positive integer, is denoted by d (m ) .
• With this rate, we have a discount rate of d (m )/ m every 1/ m of a year.
• Consider an investment of 1 to be paid at the end of the year. To find the relationship between d (m ) and d , we work backwards from the end of the year to the beginning of the year.
86

(m)
• During the m -th 1/ m year, the ending balance is one, and the amount of discount for that period is d (m )/ m .
• So beginning balance for this period is
• Continuing this process to the beginning of the year, we have
87

(m)
• Rearranging we have and
88

Relationships between d (m) and i (m)
89

(m)
(m)
[ 1
 d
( m )
] m m

1
 d
 v

( 1
 i )

1 
[ 1
 i
( n )
]
 n n
90

• Find the accumulated value of ¥500 invested for five years at 8% per annum convertible quarterly.
• The answer is
500 
  0 .
08
4


4

5

500 ( 1 .
02 )
20
• It should be noted that this situation is equivalent to one in which ¥500 is invested at a rate of interest of 2% for 20 years.
91

• Find the present value of ¥1000 to be paid at the end of six years at 6% per annum payable in advance and convertible semiannually.
• The answer is
1000 
  0 .
06
2


2

6

1000 ( 0 .
97 )
12
.
• It should be noted that this situation is equivalent to one in which the present value of ¥1000 is to be paid at the end of 12 years is calculated at a rate of discount of 3%.
92

• Find the nominal rate of interest converted quarterly which is equivalent to a nominal rate of discount of 6% per annum convertible monthly.
• The answer is


 1
 i
( 4 )
4 

 4


  0 .
06
12



12
.
1
 i
( 4 )
4

( 0 .
995 )

3
.
i
( 4 ) 
4 [( 0 .
995 )

3 
1 ].
93

1
• It is important to measure the intensity with which interest is charging at each moment of time.
• Nominal rate of interest i ( m ) accrues interest every
1/ m years.
• If m is very large, the nominal rate

= i (

) accrues every moment, and we have
94

2
95

• Annual effective rate of interest and discount are applied over a one-year period
• Annual nominal rate of interest and discount are applied over a sub-period (of length 1/ m ) and converted m times a year
• Annual force of interest and discount are applied over the smallest sub-period imaginable (at a moment), i.e. m
  times a year .
96

• Recall that the interest rate over a sub-period is the ratio of the interest earned during that period to the accumulated value at the beginning of the (sub)-period
• We have i
( m ) m

A ( n

1 m
)

A ( n )
A ( n ) rate for sub-period or i
( m )  m

 A ( n

1 m
)

A ( n )


A ( n ) annual rate
97

Annual force of interest at time n,
 n i
• The interest rate over a sub-period is the ratio of the interest earned during that period to the accumulated value at the beginning of the period, i.e.
• If m = 12, = monthly rate = annual rate/12
• If m = 365, = daily rate = annual rate/365
• If m = 8760, = hourly rate = annual rate/8760
98

Annual force of interest at time n,
 n i
• Force of interest at time n is therefore
 n i  m lim
  m

 A ( n

1 m
)

A ( n )


A ( n )
 d dn

A (
A ( n ) n )

 d dn a (
  n ) n )
 n i  d dn log[ A ( n )]
 d dn log[ a ( n )]
99

• Recall that the Force of interest is defined as
 n i  d dn log[ a ( n )]
  n i  dn
 d (log[ a ( n )])
• Integrate both sides from time 0 to t results in
100

• It follows that
• Taking the exponential function on both sides results in
• The accumulation function is the exponential function where the annual force of interest is converted into an infinitesimal small rate [
 n i ·dn ]; this small rate is then applied over every existing moment from time 0 to time t .
101

• Recall that the force of interest is also defined as
• Integrating both sides from time 0 to time t , we get
• Interest earned over the period [0, t ] can be found by applying the momentary interest rate,
 n i dn , to the balance at that moment, A ( n ), and summing it up for every moment in [0, t ].
102

An account grows with a force of interest of 0.0334 per year.
What is the interest rate?
Solution:
103

Annual force of discount at time n
• The discount rate over a sub-period is the ratio of the interest earned during that period to the accumulated value at the end of the period, i.e.
• If m = 12, = monthly rate = annual rate/12
• If m = 365, = daily rate = annual rate/365
• If m = 8760, = hourly rate = annual rate/8760
104

Annual force of discount at time n,
 n d
• Force of discount at time n is therefore
 d n
 lim m
  m

 A ( n
A (
 n
1 m

)

1 m
)
A ( n )


 n d  d dn log[
A (
A n )
( n )]
• From now on, we will use  n
 d dn log[ a ( n a
) instead of
( n )]

 n i and
 i n
 n d
105

Alternative Definition
Similarly to the interest force , the force of discount can be defined as:
 t d  d a

1
( t ) dt a

1
( t )
• In fact, force of interest and force of discount are equivalent
 t d   d a dt a

1

1
( t
( t
)
)
 a

2
( a t d
)

1 dt
( t ) a ( t )
 a

2
( t ) a ( t a

1
( t )
)
 t i
  t i
106

Force of interest when interest rate is constant
• In general,  n can vary at each instantaneous moment.
• Suppose that  n
=
  i n
= i for all n , then
107

Force of interest under simple interest
• A constant rate of simple interest implies a decreasing force of interest:
108

Force of interest under simple discount
• A constant rate of simple discount implies an increasing force of interest:
109

• Find the accumulated value of $1000 invested for ten years if the force of interest is 5% p.a.
• The answer is:
1000 e (0.05)

(10) = 1000 e 0.5
110

• The first type - a continuously varying force of interest.
• Recall the basic formula:
• If  n easily.
is readily integrable, then a ( t ) can be derived
• If  n is not readily integrable, approximate methods of integration are necessary.
111

• The second type – changes in the effective rate of interest over a period of time.
• Let i n period.
be the effective interest rate for the nth
• Then for t
≥ 1, we have
• If i
1
= i
2
=

= i t
= i , then we have a ( t ) = (1 + i ) t .
112

• Present value with varying effective rates of interest can be handled similarly.
• Let i n period.
be the effective interest rate for the nth
• Then for t
≥ 1, we have
• If i
1 i )
 t .
= i
2
=

= i t
= i , then we have a

1 ( t ) = (1 +
113

• Find the accumulated value of $1000 invested for n years if
 t
= 1/(1 + t ).
• Using the formula on force of interest, the answer is: e

0 n
 t dt  e

0 n 1
1
 t dt
 e log e
( 1
 t )
 n
0

1
 n .
114

• Find the accumulated value of $1000 at the end of 15 years if the effective rate of interest is
5% for the first 5 years, 4.5% for the second
5 years, and 4% for the third 5 years.
The answer is 1000(1.05) 5 (1.045) 5 (1.04) 5 = 1000*X
• What is the annual effective (compound) rate of return?
The annual effective rate I is determined by
(1 + I )^15 = X
115

116

What is the effective annual (compound) rate of return on an investment that grows at a discount rate of 6%, compounded monthly for the first two years and at a force of interest of 5% for the next 3 years?
The annual effective rate i is determined by
(1+i)^5 = (1-0.06/12)^(-24) * exp(0.05*3)
117

• How fast can the money paid to John can grow
• Cash position of John
• Investment opportunities available to John
• Character of John
• Combination all of the above factors, together with others, leads to interest rate
118

effective rate of discount discount factor i = effective rate of interest
119