1.

Basics of Insurance, Role of Statistics and Interest

1.1. Introduction to Insurance Companies

According to Kumsa (1992), there are some traditional associations which are used to share risks
among the members. People contribute either money or labor to assist each other by the
association whenever a member faced financial difficulties or needs assistance. Some of these
associations are “Edir” and “Equb” which have some similarities with modern insurance.

 “Edir” is a gathering such as people form an association whereby each member

contributes a fixed amount monthly, to common fund from which predetermined
compensations is paid to members upon occurrence of unforeseen events such as death
of family members or relatives.
 „‟Ekub‟‟ is another kind of traditional association, each member contribute a fixed
sum of money for a special time and the winner receives the money and uses it for a
project if he/she has one, or sells to another at a premium. Both “Edir” and “Equb”
have their trustees who are elected among the members to administer the processes for
a certain period of time.

Insurance companies are a modern form of financial organization which provide insurance
policies to firms or individuals that transfer the burden or risks of the insured to the insurer. They
charge premium in exchange for the insurance that they provide and invest the funds that they
receive in the form of premium until the funds are needed to cover insurance claims.

Premium is the amount of money that the insured pays once or regularly to the insurance policy
so as to be insured during his/her injuries or damage. All the insured add the premiums together
towards a fund, and out of which the persons/business facing a specific risk is paid, which is
facilitated by insurer. In other words the insurers reimburse the financial cost of a particular
event against the premium they collect from the insured/policy holders. It is usually comprised of
multiple insurance agents; and it can specialize in one type of insurance, such as life insurance,
health insurance, or auto insurance, or offer multiple types of insurance.

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Afza & Asghar (2012) stated that insurance companies pool the savings of the policyholders and
invest those in capital markets which ultimately contribute positively towards the development of
the country. Gashaw (2012) also stated that insurance companies have double responsibility; in
one hand they are required to be profitable to have higher rate of return for new investment, in
other hand they need to be profitable to be solvent to make other industries where they were
before even after the risk occurred. Insurance promote finance and social stability; mobilize and
channels savings, supports trade, commerce and entrepreneurial activity and improve the quality
of the lives of individuals and the overall wellbeing in a county.

Insurers help individuals as well as businesses to minimize the consequences of risk and put the
insured in the positions where they were before the risk occurs. In order to carry all risks,
insurers should keep their profitability and focus on the appropriate factors of profitability to
maximize their profit.

Zeleke (2007) stated that insurance cannot eliminate the occurrence of the loss, but finance the
loss in such a way that the policyholder would be restored to approximately where his/her
financial position before the occurrence of the risk, except the case of life and personal accident
insurance. Insurance companies play a vital role to the economic development of a given
country, by intermediation, saving mobilization and by protecting policy holders from adverse
events.

Akotey et al. (2011) stated that the insurance companies are specialized financial services range
from the underwriting of risks inherent in economic entities and mobilization of large amount of
funds through premiums for long term investments. He also explained that the importance of
insurance companies as the risk absorption role of insurers promotes financial stability in the
financial markets and provides a “sense of peace” to economic entities. Ahmed et al. (2011)
explained that insurance role as insurers provide economic and social benefits in the society such
as prevention of losses, reduction in anxiousness, fear and increasing employment.

As all types of activities are subject to risks of loss or damage due to unforeseen/unexpected
events which are beyond control of individual or business, insurance industry would help to take
risks on behalf of the insured. Therefore, insurance companies transfer risk from insured to
insurer; by doing this they promote financial stability and economic growth, rising contribution

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to GDP and increase employment. There are many types of insurance such as health, life,
auto/motor, property and the like; but in general terms, the insurance companies service is
mainly divided into general (non-life) and life insurance, the general insurance provide non-life
services whereas life insurance provides life related insurance services.

1.1.2 Insurance Company as Financial Institutions and Its Contribution to Economy

Financial institutions are organizations that are involved in providing various types of financial
service to their customers, and are controlled and supervised by the rules and regulations defined
by the government authorities. It collects funds from the public and place the fund in financial
assets. Competent financial market is a key factor in producing high economic growth.

Financial institutions are vital contributor to the overall performance of an economy in any
country by serving the economy as intermediary as well as risk taker. Insurance companies
provide financial coverage of the loss that a business or an individual is expected to suffer, by
doing so they reduce the impact of a certain events. Workie (2012) stated that financial
institutions serve as a medium of exchange and facilitate business activities, support mobilization
of resources through savings and allocating resources to activities with highest returns, follow up
investments and exert corporate governance, and offer a diversity of financial instruments.
Gashaw (2012) explained that the current business world without financial institutions such as
insurance is unsustainable, in practice some economic units are in surplus whereas the others
remain in deficit, in other way risky businesses do not have capacity to retain all types of risk in
the uncertain environment.

One of the major financial institution sectors is insurance, which provide a unique financial
service by serving the societies in managing risk, protecting policyholders from adverse events
and playing a vital role for economic growth of a country. Insurance sector‟s significant role in
economic development is through reducing uncertainty, and encouraging innovation and
investment. In other words, their contribution to the economic development of a given country is
articulated by reducing the impact of large loses as a result encouraging new investments and
healthy competition. Whatever the definition, and regardless of the size of the economy, the
importance of insurance companies is crucial to the economic growth.

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According to NBE annual report (2013-2014), the major financial institutions operating in
Ethiopia are banks, insurance companies and micro-finance institutions.
Boadi et al. (2013) stated that some developed countries have seen significant improvements in
their economy that leads the emergence of insurance industry and mobilizing of funds has been
exercised and has made huge investments that have facilitated the development of nations. Malik
(2011) stated that insurance companies‟ play a crucial role in fostering/promoting commercial
and infrastructural businesses that promotes financial and social stability like mobilizes and
channels savings, support trade, commerce and entrepreneurial activity and improves the quality
of the lives of individuals and the overall wellbeing of the society.

1.1.3 Development of Insurance Company in Ethiopia

According to Association of Kenya Insurers 2012 annual report, the African insurance market
has strong growth potential, especially in sub-Saharan African. However, Africa continues
registered the lowest contribution to the global insurance premium at 1.6% and most African
countries penetration level is below 1%. This report indicates that the insurance industry in
Africa is expected to do much to increase its market penetration in this potential area.
Charumathi (2012) stated that well developed insurance market facilitates efficient resource
allocation through transfer of risk and saving mobilization; hence, insurance industry plays
fundamental role in the economic growth.

Zeleke (2007) stated that the history of modern forms of insurance industry in Ethiopia was
introduced in 1905 together with the banking industry by Europeans, Emperor Minelik II and a
representative of the British owned National Bank of Egypt. The first provider of modern
insurance in Ethiopia was Bank of Abyssinia, it served as an agent of a foreign insurance
company, which gave fire and marine policies. In 1951 the first domestic insurer, Imperial
insurance company was opened and until 1951 except Imperial Insurance Company, all the
remaining were branches or agents of foreign insurance companies.

NBE annual report (2013-14) states that the number of insurance companies operating in
Ethiopia reached to 17, among these, one is state owned and the remaining sixteen are privately
owned companies. The number of total branches available in the year 2013 increased from 273
to 332, which showed increase of 59 branches within a year. But all this growth is not as it is

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supposed to be. Shifa (2014) stated that the insurance market is growing at the rate above 20%
annually during the last 10 years, reached almost 5 billion in 2014 from birr 641 million 10 years
ago. He also stated that although improved over time, the insurance sector still contributed less
than 1% to country‟s GDP.

However, the insurance sector in Ethiopia does not grow as expected and could not take a major
part for the economic development. Shifa (2014) stated that although the insurance industry
keeps on improving over time, it contributes less than 1% to country‟s GDP. Gashayie (2013)
stated that the Ethiopia‟s life insurance sector is less mature as compared to international life
insurance market, and its contribution to the country‟s GDP is very low. Due to Ethiopia‟s
current strong economic performance, the potential of insurance industry is promising; as a result
the number of insurance firms increasing from time to time somehow, but not as expected. The
Ethiopian government is also working on improving rules and regulations to protect the society
at large. According to Shifa (2014) to accelerate the development of the insurance industry NBE
introduced risk based supervision in the year 2012 such as prohibiting of credit sales which help
the industry for better management of their risk. The first insurance proclamation was issued in
1970 by proclamation number 281/1970 and followed by insurance regulations in 1971 by legal
notice number 393/1971.

After 1974 all private insurance enterprises were nationalized, and became one after merging all
the privately-owned insurance companies, which is owned by the government and a name given
as Ethiopian Insurance Corporation. According to NBE 2013/14 report the numbers of insurance
companies currently in operations are one government and 16 privately owned insurance
companies. The National Bank of Ethiopia remained the sole regulator of insurance supervision
in Ethiopia.

According to Asnakew (2011), the newly emerging insurance companies between 1994 and 1997
were encouraging; but the range of insurance products offered is limited indicating that the sector
is still at an early stage of development.

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1.2 The Role and Application of Statistics in Insurance Companies

Insurance business has always been based on actuarial principle from the beginning. The
actuarial principle and methods for assessing risk under conditions of uncertainty are as much
applicable to general insurance as they are to life insurance. The services of actuaries are being
utilized increasingly in general insurance also in areas like pricing, claims reserving, reinsurance
placement, investment and in fact in most area of general insurance.

If we look back into history when insurance was in its infancy, we find that the judgment and
skill of the underwriter in assessing the diverse risk and underwriting the same without statistical
data was “key factor”. But then the risk profile in those days was simple and the business was
generally profitable and hence the need to base underwriting decision on past data was never felt
and consequently the need to develop a system for collection of relevant statistical data also
never arose. The fact of the matter is that insurance started without first evaluating the risk in the
sense we understand it today.

However, the 20th century (especially after World War-II) brought risk of previously
unimaginable magnitude and complexity. Rapid development of new technologies brought new
risk with insufficient experience. 20th Century also saw:

 High inflation, which meant that, claims settlement amount would be higher than claims
cost provided in the premium.
 Fierce competition between insurer and hence rate-cutting.
 Development of consumerism which meant changing of insurance companies for
slightest of benefit

In view of these developments suddenly pricing and underwriting became very crucial for
insurance companies to survive and remain in the business. A need therefore was felt to develop
ways and means to meet these challenges. Fortunately for us, 20th century also saw the
development and growth of statistical theory and sophisticated computers with huge storage
capacities that also enabled us to manipulate data and draw meaningful inferences which in turn
can help in making more informed decisions. Actuarial principles basically make use of them

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andthey provide us with necessary tools to manage modern day general insurance business with
all its challenges and complexities. They are used for:

- Working out premium rates

- Provisions for outstanding claims
- Prudential supervision (solvency concern, protection of policyholders interest, etc.)
- Retention limit / reinsurance plan
- Investment management

Before we actually see how actuarial / statistical techniques are used for the above jobs, it is
better to have a look at some of the relevant statistical tools that can help us in our task as
follows.

- Random variable and theory of probability

- Law of large numbers
- Various distribution models
- Measure of central tendency (average)
- Measure of dispersion or spread of data – range – deviation from central tendency
- Measure of symmetry by skewness
- Measure of flatness or peakedness by kurtosis
- Point and interval estimate of claims (number of claims reported) and claim size.
- Simulation techniques
- Interpolation & extrapolation

A variable is a measure that can assume any value within a given range of possible values. A
variable is said to be a random variable if the chance of its assuming any value in a given range
is equally likely. It is random in the sense that the outcome is uncertain and we do not know the
reasons why the variable assumes one out come in preference to the other options.

A random variable can either be a discrete random variable or a continuous random variable. It is
discrete if it is restricted to point values and cannot assume all values in any interval. On the
other hand, some variables, by their nature are continuous and may assume any value over a
continuous interval.

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For our purpose, thus, the number of claims incurred by an insurance company is always a
discrete random variable and the claim size on the other hand is always a continuous random
variable. Random variables are amenable to statistical/ mathematical treatment/manipulation to
study their behavior pattern and to make projections and hence the usefulness.

There is similarity between insurance and game of chance; and therefore, understanding the
concept of probability and its application to general insurance is of importance to us. When we
toss a coin, we say that the probability of getting a head is ½. There are two possibilities – head
or tail – out of which one possibility i.e. head is favorable and therefore we say that probability
of getting head is ½ or 50%. This is the theoretical way of calculating probability called “a
priori” i.e. prior to experience. In contrast, we cannot deduce theoretically the probability of a car
being stolen within the year. For handling problems of this nature, we need to have data about
the total number of cars and the proportion that is stolen. There is another way of looking at
probability of getting head in tossing of coin which is more relevant for our purpose. If we go on
tossing the coin and note the number of heads coming and if ideally this tossing is continued for
infinite number of times we will find that the proportion of head coming is ½. Every time we are
tossing, we are in effect generating experience.

The fact that probability involves long run concept is important in the general insurance contract.
Further head & tail are mutually exclusive events.The idea of mutual exclusivity can apply for
example to calculating the probability of an injured employee being male or female, injured or
killed, damages being above or below certain level, etc. We may say that if the event is certain to
happen the probability is one and if the happening is absolute impossibility the probability is
zero. If the probability of happening a claim is one i.e. a certainty no insurance company will
assume such a risk except perhaps by charging the premium which is more than the sum insured

If the happening is impossibility i.e. probability is zero, nobody would like to insure it. Between
these two extremes, lies the various risk that come for insurance. The higher the probability of
claims happening, the higher should be the premium. Probability thus attaches a numerical value
to our measurement of the likelihood of an event occurring. We shall now examine the law of
large number and the concept of probability distribution.

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We shall also see how these probability distributions help us in estimating the number of claims
that will be reported in future during a given period of time and what will be the size of these
claims. The law of large numbers in simple terms means that the larger the data, the more
accurate will be the estimate made. In other words, the larger the sample, the more accurate will
be the estimates of the population parameters. In general insurance, it would mean that the larger
the past data about claims, the better will be the estimate of the prediction about claim frequency
and size. It is assumed that the claim will occur in future as they have occurred in the past.

For our purpose, the probability distribution can be considered to be a mathematical model
which can describe the actual probability distribution. Of course, the actual probability estimated
from the available data will rarely coincide with those generated by the theoretical distribution.
But the law of large number says that it will tend closer & closer if we have sufficiently large
database. Even if the data available is not extensive, we can make use of various theoretical
distributions to make meaningful inferences about the behavior of data relating to a particular
insurance portfolio. The fact that this theoretical distribution can be completely summarized by a
small number of parameters is of great help. The shape of distribution is determined by its
parameters. Parameters are numerical characteristics of population. If we have set of data
relating to say claims size, we cannot make best use of them in their raw form. We may be
interested to know about the average size of the claim. We have a whole set of measures called
the measure of central tendency. Similarly, to properly understand the significance of the data, it
is essential to know the variability of data around the central tendency. In case the variance is too
high, may be one has to decide about the required reinsurance support.

Yet another aspect to properly understand the given set of data is the “Skew ness aspect”.The
distribution may be very symmetric or it may be skewed having long trail to the right (positively
skewed). Many of the distribution we encounter in general insurance is skew with long tail to the
right. We have a measure of this skew ness which is zero for symmetric distribution. Positive for
positively skewed distribution and negative for negatively skewed distribution (long tail to the
left).

In addition, the knowledge and measure of flatness or peakedness of a distribution is important

for us. So, we have what is called a measure of kurtosis.

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These aspects if known properly can help in better claims management. Fortunately for us, there
are theoretical distribution models which approximate the existing claims data relating to various
risk categories. The actuaries make use of these models.

Normal Distribution – It is a continuous bell-shaped symmetrical distribution. It helps us to

find the claim size. We can find out the probability that a particular claim is between X & Y (X
&Y are claim sizes). Whatever may be the distribution of claims size in respect of a particular
class of business, the distribution of the total payout over a large number of years will
approximately be normal distribution. This is what is called central limit theorem and its
importance for us is very obvious.

Binomial Distribution: If a coin is tossed n times, the probability of getting x heads can be
determined using this distribution. For our purpose whenever a policy is issued, having a claim
in the policy can be likened to obtaining a head and not having claim to obtaining tail. It can thus
help us to estimate the frequency of claims.

Poisson Distribution: This is a discrete distribution. It is generally employed for analyzing and
estimating the incidence of claims. This is a non-negative integer based distribution. It is usually
reasonably safe to assume that the number of claims on a policy in a given period follows the
Poisson distribution. Unlike life insurance, general insurance policies are subject to multiple
claims and hence more amenable to Poisson distribution. It helps in finding out the chances of a
certain number of claims being reported during a particular period.

Log Normal Distribution: This distribution model, which is continuous in nature, is a useful
model for the claim size distribution as it is positively skewed and this is an important feature of
claim size distribution. It has range from zero to infinity. The distribution tails are important for
Reinsurance purpose and we must not under estimate it. If log normal distribution is appropriate,
then we can calculate estimate of the tail probability. Classes of insurance in which claim tend to
take a long time to settle are known as “longtail” e.g. liability claims but this should not be
confused with tail probability of log normal distribution.

Pareto Distribution: However, for Reinsurance purpose Pareto distribution is more satisfactory
than log normal. The Pareto probability distribution function tapers away to zero much more

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slowly than long normal. Hence, it is more appropriate for estimating reinsurance premium in
respect of very large claims.

The Gamma Distribution: This is also a continuous variable and finds application in the study
of claims size distribution and analysis of heterogeneity. Hence the alternative left is we
formulate a model and make use of the theoretical distribution. But in a particular situation
which model to use, is the domain of the actuary. A model can be used only when an adequate fit
is obtained. For example, if the assumption made is “number of claims reported follow Poisson
distribution.” Only one parameter defines this distribution and if we estimate the same on the
basis of observed data, many of our questions can be answered e.g. What is the chance of getting
more than 5 claims in a policy? In other distribution models, there may be more than one
parameters. These parameters determine locations spread and shape of the distribution. The
parameters of a distribution are unknown and we need to estimate them from the available
statistics (data). If we are interested in estimating the claim frequency, then perhaps we are more
interested in a “point estimate.” But for claim size it will more relevant if we can calculate the
probability of claim size falling with a given limit rather than a specific point value. It is
therefore useful to obtain an interval within which we are reasonably confident that the true
values will lie.

The next logical question would be “How much reliability can be placed in these estimate.”
Fortunately for us we have appropriate statistical tools for testing the reliability of these
estimates as also how much confidence can be placed in them. These relate to what are called
“statistical hypothesis testing and confidence limit.” At this point it will not be out of place to
mention here that in certain situations computer aided simulation can be a useful technique to
solve difficult problems in general insurance. The simulation helps in imitating the uncertainty
involved in the happening of events. We should also be aware of the sensitivity of the estimates
to the various assumptions made and should update the estimates as further data come to hand. If
we have set of claim data over a period of time, then we can make estimate of the value on some
intermediate time point through a technique called interpolation. If we want to make estimate at
some future point of time, the techniques of extrapolation is used. Incidentally the various
techniques discussed are also used by risk management department in organization for deciding

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risk retention. Investigation of claims experience is very important in general insurance and we
have seen the various theoretical distribution models and techniques that help us in the matter.

1.3 Theory of Interest

A typical part of most insurance contracts is that the insured pays the insurer a fixed premium on
a periodic (usually annual or semi–annual) basis. Money has time value, that is, $1 in hand today
is more valuable than $1 to be received one year hence. A careful analysis of insurance problems
must take this effect into account. The purpose of this section is to examine the basic aspects of
the theory of interest.

1.3.1 Simple and Compound Interest

While theoretically there are numerous ways of calculating the interest, there are two methods
which are commonly used in practice. These are the simple-interest rate method and the
compound-interest rate method. For the simple-interest method the interest earned over a period
of time is proportional to the length of the period. The interest incurred from time 0 to time t, for
a principal of 1unit, is r×t, where r is the constant of proportion called the rate of interest. The
most commonly used base is the year, in which case the term annual rate of interest is used.

Suppose amount A is deposited at interest rate per period for t time units and earns simple
interst, the amount at the end of the period is .

Example1: A person borrows $2,000 for 3 years at simple interest. The rate of interest is 8% per
annum. What are the interest charges for years 1and2? What is the accumulated amount at the
end of year 3?

Solution: The interest charges for years1and 2 are both equal to 2,000×0.08=$160. The
accumulated amount at the end of year 3 is 2,000(1+0.08×3)=$2,480.

Example 2: If the principal of 5000 Birr is invested with 5% interest rate per annum, How long
should it stay in the banks to make the account 7500 Birr?

To understand how compound interest works, suppose an amount A is invested at interest rate
per year and this interest is compounded annually. After 1 year, the amount in the account will

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be , and this total amount will earn interest the second year because for the
compound-interest method the accumulated amount over a period of time is the principal for the
next period. Thus, after n years the amount will be . The factor is
sometimes called the accumulation factor. If interest is compounded daily after the same n years,

the amount will be ( ) . In this last context, the interest rate i is called the nominal

annual rate of interest. The effective annual rate of interest is the amount of money that one
unit invested at the beginning of the year will earn during the year, when the amount earned is
paid at the end of the year. In the daily compounding example the effective annual rate of interest

is( ) . This is the rate of interest compounded annually would provide the same

return. When the time period is not specified, both nominal and effective interest rates are
assumed to be annual rates. Also, the terminology „convertible daily‟ is sometimes used instead
of „compounded daily.‟ This serves as a reminder that at the end of a conversion period
(compounding period) the interest that has just been earned is treated as principal for the
subsequent period.

Exercise 1: What is the effective rate of interest corresponding to an interest rate of 5%

compounded quarterly?

Two different investment schemes with two different nominal annual rates of interest may in fact
be equivalent, that is, may have equal dollar value at any fixed date in the future. This possibility
is illustrated by means of an example.

Example 1: Suppose I have the opportunity to invest $1 in Bank A which pays 5% interest
compounded monthly. What interest rate does Bank B have to pay, compounded daily, to
provide an equivalent investment? What is the effective rate of interest?

Situations in which interest is compounded more often than annually will arise frequently. Some
notation is needed to discuss these situations conveniently. Denote by the nominal annual
interest rate compounded m times per year which is equivalent to the interest rate compounded
annually. This means that

( )

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Exercise 2: Compute total amount you will earn at the first year if the interest is compounded
monthly having nominal interest 0.05.

1.3.2 Continuous Compounding

An important abstraction of the idea of compound interest is the idea of continuous

compounding. If interest is compounded n times per year the amount after t years is given

by( ) . Letting n → ∞ in this expression produces , and this corresponds to the notion

of instantaneous compounding of interest. In this context denote by δ the rate of instantaneous

compounding which is equivalent to interest rate . Here δ is called the force of interest. The
force of interest is extremely important from a theoretical standpoint and also provides some
useful quick approximations.

Example: If $4000 is invested in an account paying 3% interest compounded continuously, what

is the balance after 7 years?

P =4000, r =0.03, t=7

A=4000 e0.03(7) =4000 e0.21 =4000(1.23367806) = $4934.71

Exercise 1: Show that .

Exercise 2: Find the force of interest which is equivalent to 5% compounded daily.

1.3.3 Present and Future Values of the Payments

The converse of the problem of obtaining the amount after n years at compound interest is as
follows. Suppose the objective is to have an amount An after n years. If money can be invested
at interest rate , how much should be deposited today in order to achieve this objective? The
amount required is . This quantity is called the present value of A. The factor
is often called the discount factor and is denoted by . The notation is used if the
value of i needs to be specified.

Example 2: Suppose the annual interest rate is 5%. What is the present value of a payment of
$2000 payable 10 years from now? The present value is .

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The notion of present value is used to move payments of money through time in order to
simplify the analysis of a complex sequence of payments. In the simple case of the last example
the important idea is this. Suppose you were given the following choice. You may either receive
$1227.83 today or you may receive $2000 10 years from now. If you can earn 5% on your
money (compounded annually) you should be indifferent between these two choices. Under the
assumption of an interest rate of 5%, the payment of $2000 in 10 years can be replaced by a
payment of $1227.83 today. Thus, the payment of $2000 can be moved through time using the
idea of present value. A visual aid that is often used is that of a time diagram which shows the
time and amounts that are paid. Under the assumption of an interest rate of 5%, the following
two diagrams are equivalent.

Two Equivalent Cash Flows:

̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅

The advantage of moving amounts of money through time is that once all amounts are paid at the
same point in time, the most favorable option is readily apparent.

Exercise 5: What happens in comparing these cash flows if the interest rate is 6% rather than
5%?

Notice too that a payment amount can be easily moved either forward or backward in time. A
positive power of v is used to move an amount backward in time; a negative power of v is used
to move an amount forward in time.

In an interest payment setting, the payment of interest of at the end of the period is equivalent to
the payment of at the beginning of the period. Such a payment at the beginning of a period is
called a discount. Formally, the effective annual rate of discount is the amount of discount paid
at the beginning of a year when the amount invested at the end of the year is a unit amount. What
relationship between and must hold for a discount payment to be equivalent to the interest
payment?

Exercise 6: Denote by the rate of discount payable m times per year that is equivalent to a
nominal annual rate of interest . What is the relationship between and ? Hint: Draw
the time diagram illustrating the two payments made at time 0 and 1/m.
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1.3.4 Fixed vs. Variable Interest Rates

Fixed Interest Rates: An interest rate that will remain at a predetermined rate for the entire
term of the loan, no matter what market interest rates do. This will result in payments remaining
the same over the entire term.

When someone applies for a loan with a fixed interest rate, the rate they will receive is typically
determined at the time of approval, and it does not change for the entire life of the loan. When
lenders determine price points for their fixed interest rate products, they base them on market
rates available at that point in time.

 Lenders who offer credit-based pricing will offer a range of rates on their fixed rate
product, based on creditworthiness. In that case, the better the applicant‟s credit score is
(or that of the cosigner/co-applicant), the better their chances for a lower rate.
 The market rate, on the other hand, depends largely on the length of the loan and other
features, and can vary based on market conditions. This means that lenders may change
the fixed rates they offer to new applicants as market conditions change – consumers
should review the lender‟s current product offer before applying for a loan.

There are a variety of financing options with different market rates that lenders may use to fund a
fixed interest rate product. Usually the market rate is based on financing vehicles that have a
similar length as the average life of the loan product – for example, if a loan product has an
average life of 5 years, the market rate may be based on the 5-year US Treasury Bond. Fixed
interest rates are almost always higher than variable rates at the time the loan is originated.

Variable Interest Rates:

All the discussion up to now assumes that the interest rate stays constant. However, this is not a
realistic assumption. In this section, we consider situations in which the interest rate varies.
Variable interest rate is an interest rate that moves up and down based on the changes of an
underlying interest rate index. When someone applies for a variable rate loan, the interest rate is
also usually determined at the time of approval – however, the interest rate will fluctuate over
time. Variable rates consist of two components: An index (which is publicly available and not
controlled by the lender), plus a credit-based margin determined by the lender.

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Again, the applicants/co-applicants with the best credit scores would qualify for the lowest
margins. The starting rate on a variable rate loan is usually lower than the rate on a fixed rate
loan. The index rate will vary over time based on economic conditions. The margin, however, is
locked in at the time of credit approval, meaning it will not change until the loan is paid off.

Suppose you invest 2000 dollars in an account that pays 4% interest in the first year, 5% in the
second year, and 6% in the third year. How much do you have at the end of the third year?
Answer. You have dollars after the first year,
dollars after the second year, and dollars after the third
year. The following computation is not valid: the average rate is 5%, and $2000 at 5% over three
years accumulates to dollars. Indeed, though the result is close
to the correct answer, it is not the same.

Now we formulate the generalization of these ideas to the case of non-constant instantaneously
varying, but known or observed, nominal interest rates δ(t), for which the old-fashioned name
would be time-varying force of interest. Here, if there is a compounding-interval [kh, (k + 1)h) of
length h = 1/m, one would first use the instantaneous continuously-compounding interest-rate
δ(kh) available at the beginning of the interval to calculate an equivalent annualized nominal
interest-rate over the interval, i.e., to find a number im(kh) such that

δ ⁄ δ
( ) ( ) ( )

In the limit of large m, there is an essentially constant principal amount over each interval of
length 1/m, so that over the interval [b,b + t), with instantaneous compounding, the unit principal
]
amount accumulates to

| |

( ∑ δ ) ∫

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