logo 1 math 2040 introduction to mathematical finance dr. ken tsang
TRANSCRIPT
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LOGO
MATH 2040 Introduction to
Mathematical Finance
Dr. Ken Tsang
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Instructor: Dr Ken Tsang & Miss Liu Youmei
Phone: 3620606(Tsang);3620630(Liu)
Email: [email protected] [email protected]: http://www.uic.edu.hk/~kentsang/math2040/math2040.htm
Instructor Info
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Assessment of performance (apprpx.)Assessment of performance (apprpx.)
60%
10%10%
20%
QuizQuiz
Mid-Term ExamMid-Term Exam
AssignmentsAssignments
Final ExamFinal Exam
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Assessment (alternative)
• Quizzes and Assignments 10-15%
• Mid-term test (s) 20-30%
• Projects 10-20%
• Final Examination 50-60%
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Score System
Assessment grade system: Assessment grade system: A and A- (Not more than A and A- (Not more than
10%) 10%) A + B (Not more than A + B (Not more than
65%)65%) C + D (No limit ).C + D (No limit ).
LetterGrade
AcademicPerformance
Grade PointPer Unit
A Excellent 4.00
A- Excellent 3.70
B+ Good 3.30
B Good 3.00
B- Good 2.70
C+ Satisfactory 2.30
C Satisfactory 2.00
C-Barely
Satisfactory1.70
D Marginal Pass 1.00
F Fail 0.00
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What you should note on this Course:
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.
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General Information
• Textbook
The Theory of Interest, Third Edition, Stephen G. Kellison, McGraw Hill International Edition(2009).
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References
• 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
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Why the Theory of Interest?
• 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.
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Central bank – the most important financial institution of a country
The People's
Bank of China
United States Federal ReserveThe Bank of England
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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
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Monetary policy is …
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.
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The importance of Interest Rate
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.
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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
Objective of this Course
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.
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Application of this subject
• Financial decision
• Business decision
• Valuation and yield rates of bonds
• Compute proper reserves for bonds
• Loan amortizations, mortgage and
• Many others
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Forces of interest & discount
Chapter 1: The Measurement of Interest
The effective rate of interest & discount
Nominal rates of interest & discount
Present Value
Simple & Compound Interest
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Scenario Scenario
Typical problem
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?
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• 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.
Reasons/Motivations for John’s choice
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What is Interest?
When a landowner allow a farmer to use the land he owns, the farmer has to pay “rent” to the landowner.
When a banker lets a borrower to use a certain amount of money, the banker will charge the borrower something.
Note: The “rent” can be in the form of a share of the crop harvested from the land.
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Definition of Interest
• 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.
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Investment – bank deposit
• 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.
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Accumulated value
• 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.
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Period of investment
• 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.
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Accumulation function
• 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
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Amount function
• 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 In. Then
In = A(n) – A(n 1) for integral n ≥ 1.
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Properties of accumulation function
• 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.
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Examples of amount functions
• 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.
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Measures of interest
• 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.
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Effective rate of interest
• 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
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Alternative definition
• 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:
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Observations on the term “effective rate”
• 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.
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• 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.
Observations on “effective rate”
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Some terminologies
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.
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Calculation of effective rates
• Effective rates of interest may be calculated over any measurement period in terms of the amount function.
• Let in be the effective rate of interest during the n-th period from the date of investment. Then we have
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Simple interest
• 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.
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Effective rate for simple interest
• A constant rate of interest does not imply a constant effective interest rate.
• Let i be the rate of simple interest and let in be the effective rate of interest for the n-th period, then for n ≥ 1 we have:
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Simple interest – in between periods
• We have defined accumulation function for simple interest only for integral values of t.
• Naturally, we want to extend the definition to non-integral 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).
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Rigorous mathematical approach
• 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.
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Rigorous mathematical approach – cont’d
• Assuming a(t) is differentiable, from basic definition of derivative, we have:
a constant.
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• 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.
Rigorous mathematical approach – cont’d
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Example I: Simple interest
• 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.
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Example 2: Simple interest
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:
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Shortfall of simple interest
• 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 re-invest ¥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.
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Compound interest
• 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.
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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.
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Effective rate of compound interest
• Let i be the rate of compound interest and let in be the effective rate of interest for the n-th period.
• Then using the formula:
we can show that in = i, which is independent of n.
• Although defined differently, a rate of compound interest and an effective rate of interest are identical.
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Accumulation function
• 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.
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Mathematical approach
• Assuming that a(t) is differentiable, from the definition of derivatives, we have:
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Mathematical approach – Cont’d
• It follows that and
• Since loge a(0) = 0, so if we let t = 1 and remember that a(1) = 1 + i, we have
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Exponential accumulation function
• 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.
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Remarks: simple and compound interest
• 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.
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Further remarks
• 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.
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Example: compound interest
• 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
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Example: 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.
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Accumulation and discount factor
• 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.
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Discount function
• In general, the investment required to produce an amount of 1 at the end of one period is a1(t).
• We will call a1(t) the discount function.
• We obtained the following results for t ≥ 0:
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Present value
• 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.
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Worked examples
• 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:
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Future value
t
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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?
Example9: Future & present values
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.
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Solution of Example9: Future & present values
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Example10: 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
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The time value of money is the value of money figuring in a given amount of interest earned over a given amount of time.
Time value of money
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.
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Interest and Discount
• 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.
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Effective rate of discount 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.
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Some observations on Discount
• 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.
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• Effective rate of discount can be calculated over any particular period.
• Let dn be the effective rate of discount for the n-th period. A formula analogous to that for in is:
• Note again that In can be called either the “amount of interest” or “the amount of discount”.
Some observations
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Compound discount
• In general, dn may vary over different periods.
• 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.
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Concept of equivalency
• 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
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What is the equivalent effective rate of interest for an account that earns interest at an effective discount rate of 3.7%?
So the equivalent effective interest rate is 3.84%.
Example of equivalency
Solution: = 1 + i
or i = d / (1 - d) = 0.037 / ( 1- 0.037) = 0.0384
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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.
Application of “Discount”: US Treasury bills
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
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Some formulas
• From the previous slide, we have
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Relationships between d and v 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.
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Relationships between d and v 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.
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Relationships between d and v 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.
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a(1) =
a(0) = 1
Graphical relationship between d and v
Time
Accumulation function
effective rate of discount
discount factor
i = effective rate of interest, equivalent to discount rate d
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Simple discount
• 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.
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Simple interest and simple discount
• 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.
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Some worked examples
• 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.
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Nominal rates （名义利率）• 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.
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Symbols for nominal interest rates
• A rate of interest or discount is called nominal 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).
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An example
• 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.
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Some general formulae
• 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
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Example
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?
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Symbols for nominal discount rates
• 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.
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Relationship between d (m) and d
• 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
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More relationships between d (m) and d
• Rearranging we have
and
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Relationships between d (m) and i (m)
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Relationships between d (m) and i (m)
nn
mm
n
iivd
m
d ]1[)1(1]1[)(
1)(
91
Worked example - 1
• Find the accumulated value of ¥500 invested for five years at 8% per annum convertible quarterly.
• The answer is
• 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.
2054
)02.1(5004
08.01500
92
Worked example - 2
• 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
• 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%.
.)97.0(10002
06.011000 12
62
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Worked example - 3
• 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 .
12
06.01
41
124)4(
i
.)995.0(4
1 3)4(
i
].1)995.0[(4 3)4( i
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Forces of interest – 1( 利息强度）
• 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
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Forces of interest - 2
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Definitions
• 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.
97
Annual force of interest at time n
• 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 haverate for sub-
period
or annual rate
)(
)()1
()(
nA
nAm
nA
m
i m
)(
)()1
()(
nA
nAm
nAmi m
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Annual force of interest at time n, ni
• 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
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Annual force of interest at time n, ni
• Force of interest at time n is therefore
)(
)(
)(
)(
)(
)()1
(lim
na
nadnd
nA
nAdnd
nA
nAm
nAm
m
in
)](log[)](log[ nadn
dnA
dn
din
100
Accumulation function
• Recall that the Force of interest is defined as
• Integrate both sides from time 0 to t results in
)])((log[)](log[ naddnnadn
d in
in
101
Accumulation function
• 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.
102
Interest earned over t years
• 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].
103
An account grows with a force of interest of 0.0334 per year. What is the interest rate?
Solution:
Example
104
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
105
Annual force of discount at time n, nd
• Force of discount at time n is therefore
• From now on, we will use n instead of ni and n
d
)1
(
)()1
(lim
mnA
nAm
nAm
m
dn
in
dn na
nadnd
nA
nAdnd
)(
)](log[
)(
)](log[
106
Alternative Definition
Similarly to the interest force , the force of discount can be defined as:
• In fact, force of interest and force of discount are equivalent
)(
)(
1
1
ta
tadtd
dt
it
itd
t ta
tata
ta
tadtd
ta
ta
tadtd
)(
)()(
)(
)()(
)(
)(
1
2
1
2
1
1
107
Force of interest when interest rate is constant
• In general, n can vary at each instantaneous moment.
• Suppose that n = in = i for all n, then
108
Force of interest under simple interest
• A constant rate of simple interest implies a decreasing force of interest:
109
Force of interest under simple discount
• A constant rate of simple discount implies an increasing force of interest:
110
Example
• 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 e0.5
111
Varying Interest – first type
• The first type - a continuously varying force of interest.
• Recall the basic formula:
• If n is readily integrable, then a(t) can be derived easily.
• If n is not readily integrable, approximate methods of integration are necessary.
112
Varying interest – second type
• The second type – changes in the effective rate of interest over a period of time.
• Let in be the effective interest rate for the n-th period.
• Then for t ≥ 1, we have
• If i1 = i2 = = it = i, then we have a(t) = (1 + i)t.
113
Present value
• Present value with varying effective rates of interest can be handled similarly.
• Let in be the effective interest rate for the n-th period.
• Then for t ≥ 1, we have
• If i1 = i2 = = it = i, then we have a1(t) = (1 + i)t.
114
Example
• 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:
.1
)1(log000 1
1
nteeee
nnn
t dtt
dt
115
Example
• 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
116
Example 23
117
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?
Example 24
(1+i)^5 = (1-0.06/12)^(-24) * exp(0.05*3)
The annual effective rate i is determined by
118
Factors affecting decision
• 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
119
Graphical illustration
discount factor
effective rate of discount
i = effective rate of interest