Actuarial Statistics

ACTUARIAL STATISTICS Page 1
UNIT-1
1.) Difference between simple and compound interest:
S.NO Statement Simple Interest Compound Interest
1
Effective
interest Rate
The effective interest rate of
simple interest is lower
The effective interest rate of compound
interest is higher
2
Calculations
and formulas
Simple interest is easy to
calculate P × R× n
Compound interest calculations can
become complicated P× [1-(1 + R/t)n
]
3 Application
Simple interest is usually
preferred by borrowers to
increase certainty and decrease
their borrowing cost
Compound interest is used mostly by
lenders for growth in their investments
2. Present Value of annuity formula:
The present value of annuity formula is calculated by determining present value which is
calculated by annuity payments over the time period divided by one plus discount rate and the
present value of the annuity is determined by multiplying equated monthly payments by one
minus present value divided by discounting rate.
PV of an Annuity = C * [(1 – (1+i)-n
) / i]
Where,
• C is the cash flow per period
• i is the rate of interest
• n is the frequency of payments
3.) Annuity Certain Sum:
A deposit of $100 is placed into a college fund at the beginning of every month for 10 years. The
fund earns 9% annual interest, compounded monthly, and paid at the end of the month. How
much is in the account right after the last deposit?

SOLUTION
The value of the initial deposit is $100, so a1=100. A total of 120 monthly deposits are made in
the 10 years, so n=120. To find r, divide the annual interest rate by 12 to find the monthly
interest rate and add 1 to represent the new monthly deposit.
Substitute a1=100, r=1.0075 and n=120 into the formula for the sum of the first n terms of a
geometric series, and simplify to find the value of the annuity.
So the account has $19,351.43 after the last deposit is made.
4.) Sinking Fund:
A fund set up to receive periodic payments is called a sinking fund. The periodic
payments, together with the interest earned by the payments, are designed to produce a given
sum at some time in the future.
For example, a sinking fund might be set up to receive money that will be needed to pay
off the principal on a loan at some future time. If the payments are all the same amount and are
made at the end of a regular time period, they form an ordinary annuity.
We used sinking fund calculations to determine the amount of money that accumulates
over time through monthly payments and interest.
R is the periodic payment;
i is the interest rate per period;
n is the number of periods.

1.) Cash Flow:
Cash flow measures how much money is moving in and out of your business during a
specific period of time. Broadly speaking, businesses bring in money through sales, financing,
and returns on investments - that’s cash flowing in. And they spend money on supplies and
services, as well as utilities, taxes, loan payments, and other bills - that’s cash flowing out.
Being cash flow positive means that more money is coming into your business than is
going out of your business. Being cash flow negative means more money is leaving the business
than you have coming in.
The easiest way to think about cash flow is to think about the total amount of money that
moved into or out of your business checking account during a month. If you finish the month
with more cash in the bank than when you started, your business is cash flow positive. If you
have less cash at the end of the month, your business had negative cash flow.
Positive cash flow
Positive cash flow is defined as ending up with more liquid money on hand at the end of
a given period of time compared to what was available when that period began.
Let’s say you started with Rs.1000 in cash at the beginning of the month. You paid
Rs.500 in bills and expenses, and your customers paid you Rs.2000 for your services. Your cash
flow is positive, at Rs.2500 for the month.
Negative cash flow
Negative cash flow is when more cash is leaving the business than is coming in. When
cash flow is negative, the amount of cash in your business bank account is shrinking. This might
not be a problem if your business has plenty of cash in the bank. But, it does mean that your
business will eventually run out of money if it doesn’t become cash flow positive at some point.
Let’s say you started with Rs.1000 in the bank at the beginning of the month. You paid
Rs.1500 in bills and expenses, and even though you did plenty of work and invoiced your
customers for Rs.3000 worth of services, your customers only actually paid you Rs.200. You are
still waiting for the rest of your payments to come in. Your cash flow is negative at Rs.-300 for
the month.

2.) Simple and Compound interest sum:
Suppose Rs.10000 is deposited for 6 years in an account paying 4.25% per year
compounded annually. Find the Simple and compound amount and the amount of interest
earned?
Solution:
Simple interest:
Since 4.25% is the yearly interest rate, we know the year n = 6.
Use the formula SI = P[1+ni], with P = 5000, i = 0.0425, and n = 6.
The total interest she will pay is
SI = P[1+ni]
=10000[1+6*0.425]
=10000*1.255
SI = 12550
Compound interest:
Since interest is compounded annually, the number of compounding periods per year is t = 1.
The interest rate per period is i =4.25/100=0.0425 and the number of compounding periods is
n=6. Using the formula for the compound amount with P = 10000, i = 0.0425, and n = 6 gives
A = P[1 + i]n
= 10000 [1+0.0425]6
=10000*1.2836
A = Rs.12836
Subtract the initial deposit from the compound amount.
I = A – P
= 12836 – 10000
I = Rs.2836

3.) Repayment of Loan:
Schedule of payments
When a loan is repaid by a series of regular payments, these payments are an annuity.
Payments will normally be made in arrears so, to be more specific, they form an immediate
annuity. A schedule of payments details how much capital and interest is paid each time a
payment is made and how much of a loan remains outstanding.
If payments are annually in arrears for n years, and the loan is for C units of money, then
the annual repayment (often referred to as the premium) is found by equating P.V.s:
Let us calculate the amount of the loan still outstanding after m years, i.e., after the mth
payment has just been made. A moment’s thought shows that the amount then outstanding is just
the value at time m of C paid at time 0 minus the total value at time m of the annual repayments
made at times 1,2,. . . ,m. In fact it is more convenient to calculate these values at time 0 (i.e.,
going back m years) and then rescale by the accumulation factor (1 + i)m
. In other words,
Amount outstanding at time m
(1)
The amount outstanding can also be calculated as the value at time m of the remaining
repayments,
(2)
The expressions in (1) and (2) are equivalent. However, in practice they may give marginally
different answers because the premium P is always rounded to the nearest penny
Consolidating loans
It is possible that a person is paying back two or more existing loans and makes an
agreement that in the future he will only need to make payments on one new loan.

1.) Relationship between Effective and Nominal Rate Of Interest:
Nominal interest rates
A nominal interest rate is a rate, per unit time, of interest which applies over a different
time period. For example, “overnight money” means that a yearly rate of interest is applied daily.
A nominal interest rate of i(p)
per basic time unit is defined to mean that interest is
compounded p-thly with an interest rate of i(p)
/p in a time interval of length 1/p. Equivalently, we
have,
For example, a nominal interest rate of 12% p.a. compounded monthly (p=12) means an
interest rate of 1% per month and therefore an accumulation factor A(t, t + 1/12) = 1.01.
Note on notation:
• i(p)
does not mean i raised to the power p. The brackets are there to remind you that the
p here is just a label.
• In fact, a number in brackets to the top right of any actuarial symbol usually tells you
about the frequency of payments;
Effective Rate of Interest:
The effective rate of interest is the equivalent annual rate of interest which is
compounded annually. Further, the compounding must happen more than once every year.
By equivalent, we mean the rate which gives the same accumulation after unit time. Now,
the accumulation factor for one time unit with interest at rate i per unit time
On the other hand, for compounding p-thly we have

So, if the accumulations are the same, we require
This is a very important relationship between the effective interest rate i and the nominal interest
rate converted p-thly i(p). Similarly, the rearranged formula
is often useful.
Note:
• i = i(1)
.
• When the basic time period is 1 year, i is called the Effective Annual Rate (EAR) or the
Annual Equivalent Rate (AER).
• The AER is useful for comparing the annual cost of financial products with different periods
of compounding.
• Adverts for credit legally have to include the so-called Annual Percentage Rate (APR).
This is defined as the effective annual rate of interest on a transaction obtained by taking into
account all the items entering the total charge for credit (i.e., including fees, etc.).
Difference between Effective and Nominal Interest Rates
Although the nominal rate is the stated rate associated with a loan, it is typically not the
rate that the consumer pays. Rather, the consumer pays an effective rate that varies based on fees
and the effect of compounding. To that end, annual percentage rate (APR) differs from the
nominal rate, as it takes fees into account, and annual percentage yield (APY) takes both fees and
compounding into account.
Relationship between Nominal and Effective Rate of interest:
Nominal rates are quoted, published or stated rates for loans, credit cards, savings
accounts or other short-term investments. Effective rates are what borrowers or investors actually
pay or receive, depending on whether or how frequently interest is compounded. When interest is

calculated and added only once, such as in a s0imple interest calculation, the nominal rate and
effective interest rates are equal. With compounding, a calculation in which interest is charged
on the loan or investment principal plus any accrued interest up to the point at which interest is
being calculated, however, the difference between nominal and effective increases exponentially
according to the number of compounding periods. Compounding can take place daily, monthly,
quarterly or semi-annually, depending on the account and financial institution regulations.
2.) Types of Annuities Certain:
An annuity is a series of payments made at regular time intervals. We restrict ourselves
here mainly to level annuities where the payments are all equal.
There are two sorts of annuities: annuities-certain and life annuities. For an annuity
certain the number of payments is certain and specified in the contract. In contrast, the payments
may depend on the survival of one or more human lives, then we say life annuity. In that case,
the number of payments is uncertain. For example, pensions are life annuities.
We will derive the P.V.s, for annuities-certain where one unit of money is paid per unit
time. Obviously for annuities where payment is C units of money per unit time, the P.V. is
obtained by multiplying the corresponding expression by C.
The main mathematical result we will need is the well-known formula for the sum of a
geometric progression:
Immediate annuity
Consider n payments of one unit of money to be made at intervals of one unit of time
with the first payment due one unit of time from now (i.e., the first payment is at time 1 and the
last at time n).
This situation is known as an immediate annuity, the symbol for its present value is

Annuity-due
Now consider the same series of payments as in the previous paragraph but with payments made
in advance, i.e., at the beginning of each time period (so that the first payment is at time 0 and the
last at n - 1). This situation is known as an annuity-due, the symbol for its present value is
Deferred annuities
Suppose a series of n unit payments starts at time m + 1, the last one due at time m + n.
This may be considered as an immediate annuity deferred m time periods. The symbol for the
present value is
Increasing annuities
Annuities where the payments are not equal are called varying annuities. In particular, an
annuity which pays k units of money at the end of the kth
time period is called an increasing
immediate annuity. Its present value is
Similarly, an annuity paying k units of money at the beginning of the kth
time period for n time
periods is an increasing annuity-due and has present value

UNIT-2
1.) Mortality Table:
Life Table is also known as Mortality Table. Life table is a mathematical sample which
gives a view of death in a country and is the basis for measuring the average life expectancy in a
society. It tells about the probability of a person dying at a certain age, or living upto a definite
age.
According to Bogue, “The life table is a mathematical model that portrays mortality
condition at a particular time among a population and provides a basis for measuring longevity.
It is based on age specific mortality rates observed for a population for a particular year.”
2.) Assurances Benefits:
➢ A closer look at a business through the eyes of a skilled business professional
➢ Ensures users are more confident that the information presented to them is reliable
➢ Gives more credibility to business reporting processes
➢ Facilitates the running and management of the business
➢ Advice and recommendations will be given during the process
➢ Improves business processes
➢ Aids the process of obtaining finance
➢ Assists management in understanding and reducing business risks
➢ Better controls can increase profit
➢ Gives more confidence in internal controls
➢ More valid conclusions can be drawn from the accounts
3.) Life Assurance:
i. Life assurance pays a lump sum on death for monthly or annual premiums that
depend on age and health of the policy holder when the policy is underwritten.
The sum assured may be decreasing in accordance with an outstanding mortgage.

ii. Life assurance = An agreement between a life assurance company and a
policyholder; in return for a payment (premium) from the policyholder, the
company commits to pay someone or something (the beneficiary) upon the death
of the person whose life is being covered (the life assured).
4.) Net Premium:
The net premium (or pure premium) is the premium amount required to
meet the expected benefits under a contract, given mortality and interest assumptions.
1.) Construction of Life Tables:
Life tables are constructed on the basis of a single cross-sectional time data for a generation.
There is also a longitudinal life table method which takes a real cohort of persons that start life at
a specific age interval and follow it throughout life until they die.
Further, a complete life table may be constructed on the basis of single years of ages. An
abridged life table can also be constructed wherein ages are grouped in 5 or 10 years of interval,
taking the initial year as 0-1.
Symbol Definition Equation
X Specific Age l0 =100000
lx The number of persons surviving to exact age x. lx = lx-1 * (1 - qx-1)
dx The number of deaths between exact ages x and x+1. dx=lx – lx+1 or lx.qx
fx The number of persons surviving at age x to x + n fx = lx - dx
qx Probability of death per person in the specific age x qx = dx / lx
Px 1 – Probability of surviving per individual person or 1 – qx
Px = 1 - qx or
px = lx-1/ lx
Lx Number of years lived by the cohort in the age x to x + n Lx = (lx + lx=n)/2
Tx The number of person-years lived after exact age x.
Lx + Lx+1 + Lx+2 +
…

ex
The average number of years of life remaining at exact age
x.
ex = Tx / fx
The above life table provides the column wise information which is generally provided and
followed by all life tables.
❖ x = Specific Age, If the age at birth is x then the age at one year is x + 1. Similarly the
age at 15 years is x + 15.
❖ dx = Number of deaths, at any particular age. i.e., at the age x, 13000 deaths occur out of
100000 births, then at age x + 1 , 87000 persons will be alive.
❖ fx = The number of persons surviving at age x to x + n i.e., at the age x + 1 = 100000-
13000 = 87000
❖ qx = Probability of death per person in the specific age i.e., total deaths occurred.
Probability = 13,000÷ 1,00,000 = 0.13
Similarly, at the age x + 1, 1300 persons died out of 87,000 live population then
❖ Px = 1 – Probability of surviving per individual person or 1 – qx, i.e., at age x + 1, 1-
.01494 = .98506.
❖ Lx = Number of years lived by the cohort in the age x to x + n or fx of any two age groups
÷ 2
Suppose,
Lx = 163640 ÷ 2 = 81820
❖ Tx – Total number of years lived by the cohort after exact age x. This can be found out
from the reverse side of life table, i.e., At the age of 94 Lx = 525 and at age 93 Lx = 925
then at age x + 93, total number of years lived by Cohort = 525 + 925 = 1450 and at age x
+ 92, it will be 525 + 925 + 1400 = 2850.
❖ ex = Tx ÷fx. This gives average life expectancy.
2.) Life Table sum:

We find records about a group of 10000 from when they were all aged 20, which reveal
that 8948 were still alive 5 years later, 7813 another 5 years later, 6613 after 5 years later, 5563
next 5 years later and 4328 another 5 years later.
Solution:
Mortality table
X lx dx Px fx qx Lx Tx ex
20 10000 1052 0.895 8948 0.105 7164 30290 3.385
25 8948 1135 0.873 7813 0.127 6638 23126 2.960
30 7813 1200 0.846 6613 0.154 6071 16488 2.493
35 6613 1050 0.841 5563 0.159 5471 10417 1.873
40 5563 1235 0.778 4328 0.222 4946 4946 1.143
45 4328 - - - - - -
3.) Net Premium for Assurance Plan:
We will use the following notation for the regular net premium payable annually
throughout the duration of the contract:

In each case, the understanding is that we apply a second principle which stipulates that the
premium payments end upon death, making the premium payment cash-flow a random cashflow.
We also introduce
To calculate net premiums, recall that premium payments form a life annuity (temporary
or whole-life), so we obtain net annual premiums from the first principle of equal expected
discounted values for premiums and benefits, e.g.
1.) Types, Uses and Assumption of life tables:
Types of Life Tables
There are two types of life tables.
1. Cohort or Generation Life Table
2. Period Life Table.
The Cohort or Generation Life Table “summarizes the age specific mortality experience of a
given birth cohort (a group of persons all born at the same time) for its life and thus extends over
many calendar years.”
The “Period Life Table summarizes the age specific mortality conditions pertaining to a given or
other short time period.”

Uses of life tables
❖ Life table is used to project future population on the basis of the present death rate.
❖ It helps in determining the average expectation of life based on age specific death rates.
❖ The method of constructing a life table can be followed to estimate the cause of specific
death rates, male and female death rates, etc.
❖ The survival rates in a life table can be used to calculate the net migration rate on the
basis of age distribution at 5 or 10 year interval.
❖ Life tables can be used to compare population trends at national and international levels.
❖ By constructing a life table based on the age at marriage, marriage patterns and changes
in them can be estimated.
❖ Instead of a single life table, multiple decrement life tables relating to cause specific
death rate, male and female death rates, etc. can be constructed for analyzing socio-
economic data in a country.
❖ Life tables are particularly used for formulating family planning programmes relating to
infant mortality, maternal deaths, health programmes, etc. They can also be used for
evaluating family planning programmes.
❖ Now a days, life tables are used by life insurance companies in order to estimate the
average life expectancy of persons, separately for males and females. They help in
determining the amount of premium to be paid by a person falling in a specific age group.
❖ Besides, if an insured person dies before the policy matures, the life table provides
economic support to the insurance company without facing financial loss and it is able to
give the insured amount to the legal heirs of the deceased.
Assumption of life tables:
❖ A hypothetical cohort of life table usually comprises of 1,000 or 10,000 or 1,00,000
births.
❖ The deaths are equally distributed throughout the year.
❖ The cohort of people diminishes gradually by death only.
❖ The cohort is closed to the in-migration and out-migration.
❖ The death rate is related to a pre-determined age specific death rate.
❖ The cohort of persons dies at a fixed age which does not change.

❖ There is no change in death rates overtime.
❖ The cohort of life tables are generally constructed separately for males and females.
2.) Life Annuity and Temporary Annuity:
Life Annuity
The annual life annuity is paid once each year, conditional on the survival of a life (the
annuitant) to the payment date. If the annuity is to be paid throughout the annuitant’s life, it is
called a whole life annuity. If there is to be a specified maximum term, it is called a term or
temporary annuity.
Whole-life annuity-due
Consider a series of annual payments, each of one unit of money, made in advance to a
life of age x. The payments are only made while x is alive. This situation is a whole-life annuity-
due. The present value of these payments (when the policyholder is exact age x) is, of course, a
random variable. Its expectation is denoted by a¨x and will be calculated below.
First we notice that if K(x) is the curtate further lifetime of (x) (i.e., the number of
complete years still to be lived) then payments will be made at ages x, x + 1, x + 2, . . . , x + K(x).
Hence there are K(x) + 1 payments in total and

Ax = 1 − d
Nx
Dx
, d = 1 − v (2)
It’s also straightforward to find a relationship between the variance of the P.V. of a
whole-life annuity due and the variance of the P.V. of a whole-life assurance

Obviously, if the annual payments are P rather than 1, the E.P.V. and the variance of the P.V. are
scaled by P and P2
respectively.
Whole-life immediate annuity
Let us now consider a series of annual payments, each of one unit of money, made in
arrears to a life of age x. The payments are only made while (x) is alive. This situation is a
whole-life immediate annuity. The expected present value of the payments is denoted by ax.
In fact, this is just the same as the previous case except that the payment at age x is not
made. Since the P.V. of 1 due at age x is just 1, and the payment is guaranteed because the
person is certainly still alive at age x, we must have
Temporary Annuity
Term annuity-due

Now suppose we wish to value a term annuity-due of 1 per year. We assume the annuity
is payable annually to a life now aged x for a maximum of n years. Thus, payments are made at
times k = 0, 1, 2, . . . , n - 1, provided that (x) has survived to age x + k. The present value of this
annuity is Y , say, where

Term immediate annuity
The EPV of a term immediate annuity of 1 per year is denoted ax:n . Under this annuity
payments of 1 are made at times k = 1, 2, . . . , n, conditional on the survival of the annuitant.
The random variable for the present value is

UNIT – 3
1.) Office Premium:
The office premium (or gross premium) is the premium required to meet all the costs
under an insurance contract, usually including expected benefit cost, expenses and profit
margin. This is the premium which the policy holder pays.
2.) Policy Value:
Generally, when a policy is issued the future premiums should be expected to be
sufficient to pay for the future benefits and expenses. (If not, the premium should be increased!)
However, it is usually the case that for a policy which is still in force t years after being issued,
the future premiums (from time t) are not expected to be sufficient to pay for the future benefits
and expenses. The amount needed to cover this shortfall is called the policy value for the policy
at time t.
3.) Valuation:

An important element in the financial control of an insurance company is the calculation
at regular intervals, usually at least annually, of the sum of the policy values for all policies in
force at that time and also the value of all the company’s investments. For the company to be
financially sound, the investments should have a greater value than the total policy value. This
process is called a valuation of the company.
4.) Life Contingent Financial Instruments
The mathematical principles that under life contingent insurance products such as
• Life Insurance
• Pensions
• Lifetime Annuities
1.) Surrender value and Paid-up value
Surrender value
The policyholder wishes to cancel the policy with immediate effect. In this case, it may
be appropriate for the insurance company to pay a lump sum immediately to the policyholder.
This will be the case if the policy has a significant investment component – such as an
endowment insurance, or a whole life insurance. Term insurance contracts generally do not have
an investment objective. A policy which is cancelled at the request of the policyholder before the
end of its originally agreed term, is said to lapse or to be surrendered, and any lump sum
payable by the insurance company for such a policy is called a surrender value or a cash value.
We tend to use the term lapse to indicate a voluntary cessation when no surrender value
is paid, and surrender when there is a return of assets of some amount to the policyholder, but
the words may be used interchangeably.
Paid-up value
The policyholder wishes to pay no more premiums but does not want to cancel the policy,
so that, in the case of an endowment insurance for example, a (reduced) sum insured is still
payable on death or on survival to the end of the original term, whichever occurs sooner. Any

policy for which no further premiums are payable is said to be paid-up, and the reduced sum
insured for a policy which becomes paid-up before the end of its original premium paying term is
called a paid-up sum insured.
A whole life policy may be converted to a paid-up term insurance policy for the original
sum insured.
2.) Gross premium, Net premium Policy Value and policy value basis:
Gross premium policy value
The gross premium policy value for a policy in force at duration t (≥ 0) years after it was
purchased is the expected value at that time of the gross future loss random variable on a
specified basis. The premiums used in the calculation are the actual premiums payable under the
contract.
Net premium policy value
The net premium policy value for a policy in force at duration t (≥ 0) years after it was
purchased is the expected value at that time of the net future loss random variable on a specified
basis (which makes no allowance for expenses). The premiums used in the calculation are the net
premiums calculated on the policy value basis using the equivalence principle, not the
actual premiums payable.
Policy value basis
The numerical value of a gross or net premium policy value depends on the assumptions
– survival model, interest, expenses, future bonuses – used in its calculation. These assumptions,
called the policy value basis, may differ from the assumptions used to calculate the premium,
that is, the premium basis.
3.) DISTRIBUTION OF SURPLUS
Cash bonuses
It was shown in the previous section that the surplus accumulates at the rate ᵞ1(t) in state i
at time t. Hence, the surplus may be distributed by simply paying the policyholder an annuity
ᵞ1(t) while the policy is in state t. These dividend payments may then supplement annuity

benefits or partly offset premiums paid under the terms of the policy. The present value at time 0
of the total bonuses paid during [0, t]
Terminal bonuses
In this subsection we discuss a distribution method according to which the surplus is
distributed to the policyholders only when the policies expire. No additional benefits are paid
during the term of the policy, except at the maturity date. Hence, terminal bonuses may be used
to enhance the maturity value of the policy.

1.) Surrender value and Paid-up value Sum:
Robert Lee takes out a whole-life assurance on his 50th birthday with a sum assured of
Rs.20,000 (payable at the end of the year of death), with premiums payable annually in advance.
Suppose that he is still alive at the age of 60. Find the surrender value and paid-up value of his
policy at that time. Use select values with a 4% p.a. effective interest rate. To use values of Ax
and a¨x directly from the AMC00 life table.
Solution:
Note that the select period is only 2 years, so for all variables after age 52 ultimate tables
should be used. The first step is to calculate the net annual premium (P) payable for the contact.
This is found from:

So, the annual premium is Rs.338.60.
Ignoring expenses, the surrender value is given by:
So the surrender value is Rs.3534.19.
Letting the death benefit be Rs.S then the Equation of Value is
3534:19 = SA60
and hence
So the surrender value would purchase Rs.8250.90 of fully-paid up life assurance. As
expected, this is considerably less than the original death benefit of Rs.20000.

2.) Methods of Valuation:
Comparison Method of Valuation
Comparison Method of Valuation is the most commonly used and accepted method in
ascertaining the market value of properties. Under the Comparison Method, the valuation
approach entails comparing the subject property with similar properties that were sold recently
and those that are currently being offered for sale in the vicinity or other comparable localities.
Investment Method of Valuation
This method of valuation is usually applied for investment properties. In the Investment
Method, the annual rental income presently received or expected over a period of time for the
lease of the property is estimated and deducted there from the expenses or outgoings incidental
to the ownership of the property to obtain the net annual rental value. This net annual income is
then capitalized by an appropriate capitalization rate or Years’ Purchase figure to arrive at the
present Capital Value of the property.
The relevant capitalization rate is chosen based on the investment rate of return expected
(as derived from comparisons of other similar property investments) for the type of property
concerned taking into consideration such factors as risk, capital appreciation, security of income,
ease of sale, management of the property, etc.
Residual Method of Valuation
The Residual Method of Valuation is normally used for development land or projects.
This approach entails estimating the gross development value of the development components
and deducting therefrom the development costs to be incurred, i.e. preliminary expenses,
statutory payments, earthworks, infrastructure and building construction costs, professional fees,
contingencies, project management fees, marketing and legal fees, financing costs, developer’s
profits and other costs (if any) to arrive at the residual value. This residual value appropriately
discounted for the period of development and sale is deemed to be the present market value of
the subject property.

Cost Method of Valuation
It is normally used for individually designed properties or specialized properties for
which comparisons are not available or in appropriate. In this approach, the value of the land is
added to the replacement cost of the building and other site improvements.
The value of the site is determined by comparison with similar lands that were sold recently and
those that are currently being offered for sale in the vicinity with appropriate adjustments made
to reflect improvements and other dissimilarities and to arrive at the value of the land as an
improved site.
Profits Method of Valuation
The Profits Method of Valuation is used to determine the market value of properties with special
licensing requirements. It entails the use of the trading accounts derived from the business
operation of the subject property. The gross receipts are adjusted to cover payments for
purchases and stocks to determine the gross profit. The operating expenses are then deducted
there from to assess the net trading profit. This figure of net trading profit less the remunerative
interest on the tenant’s capital is the divisible balance. A percentage of the divisible balance is
deemed to be the estimated net annual rental value of the subject property. This estimated net
annual income is then capitalized by an appropriate capitalization rate or Years’ Purchase figure
to capitalize the income to the present Capital Value of the property.
Discounted cash flow method
This method estimates the value of an asset based on its expected future cash flows,
which are discounted to the present (i.e., the present value). This concept of discounting future
money is commonly known as the time value of money. For instance, an asset that matures and
pays $1 in one year is worth less than $1 today. The size of the discount is based on
an opportunity cost of capital and it is expressed as a percentage or discount rate.
For a valuation using the discounted cash flow method, one first estimates the future cash
flows from the investment and then estimates a reasonable discount rate after considering the
riskiness of those cash flows and interest rates in the capital markets. Next, one makes a
calculation to compute the present value of the future cash flows.

UNIT – 4
1.) Commutation functions
The life assurance fund earns interest at a constant effective rate of i per annum. Hence
the P.V. of one unit of money due in t years is vt
, where v = 1/(1 + i).
2.) Reserves:
An insurance company will want to have cash available to pay its customers’ benefits.
This available cash is called reserves. In general, reserves will be established to eliminate any
future losses (i.e. such that the expected present value of future reserves is zero)
3.) Retrospective Reserves:
We define the retrospective reserve to be the difference between the retrospective
accumulated premiums and benefits. Thus, for whole life assurance payable at end of year of
death the retrospective reserve is given by,

4.) Force of a single decrement
1.) Risk Model
There are two types of Risk Models.
• Individual risk model
• Collective risk model
Individual risk model
The insurer, of course, is interested in the total benefits paid on an entire portfolio of
policies. An obvious way to handle this is simply to obtain the present value of the total amount

paid on all policies in the portfolio, as the sum of the individual random variables. This is known
as the individual risk model.
Collective risk model
There is another method for estimating the total amount paid on a group of policies,
known as the collective risk model. The collective risk model is particularly useful for casualty
insurance such as automobile, home, or health policies.
The collective risk model identifies two main factors that inﬂuence the total claims. One
is the claim frequency, that is, the number of claims that will occur over a certain period. This
will be a discrete random variable taking nonnegative integers as values. The second factor is the
amount that will be paid, given that a claim has occurred. This is known as the severity of the
claim. We have observed that in any fixed period under a life insurance policy, the claim severity
is normally just a constant, but under other types of insurance, it will vary substantially.
2.) Multiple decrement models
• Multiple decrement model - expressed in terms of multiple state model Multiple
Decrement Tables (MDT)
1. Several causes of decrement
2. Probabilities of decrement
3. Forces of decrement
• The Associated Single Decrement Tables (ASDT)
• Uniform distribution of decrements
1. in the multiple decrement context
2. in the associated single decrement context
Examples of multiple decrement models
Multiple decrement models are extensions of standard mortality models whereby there is
simultaneous operation of several causes of decrement.
A life fails because of one of these decrements.

Examples include:
• Life insurance contract is terminated because of death/survival or withdrawal
(lapse).
• An insurance contract provides coverage for disability and death, which are
considered distinct claims.
• Life insurance contract pays a different benefit for different causes of death (e.g.
accidental death benefits are doubled).
• Pension plan provides benefit for death, disability, employment termination and
retirement.
Introducing notation
3.) Problem:

Solution
1.) Endowment Insurance
Endowment insurance provides a combination of a term insurance and a pure
endowment. The sum insured is payable on the death of (x) should (x) die within a fixed term,
say n years, but if (x) survives for n years, the sum insured is payable at the end of the nth year.
Traditional endowment insurance policie were popular in Australia, North America and
the UK up to the 1990s, but are rarely sold these days in these markets. However, as with the
pure endowment, the valuation function turns out to be quite useful in other contexts. Also,
companies operating in these territories will be managing the ongoing liabilities under the
policies already written for some time to come. Furthermore, traditional endowment insurance
is still relevant and popular in some other insurance markets.
We first consider the case when the death benefit (of amount 1) is payable immediately
on death. The present value of the benefit is Z, say, where

In the situation when the death benefit is payable at the end of the year of death, the present
value of the benefit is

Finally, when the death benefit is payable at the end of the 1/mth year of death, the present value
of the benefit is

2.) Single and Multiple Decrement Models:
Single-decrement model:
For a simple model we show the creation of a mortality table from observed data. Let us
assume that we follow a group of students on a two year post-high school course.
At the start of the course they are aged 18 and we observe the following:
– the number of students at the start of the first year was 550,
– during the course of the first year 5 students died,
– during the course of the second year 7 students died.
Using these data we can construct part of a mortality table.
In the model shown we have the base state (0 – alive) and one possibility of
Decrement to state (1 – dead). This is then a single-decrement model, as illustrated in
Figure 1.
Figure 1. Single-decrement model of a post-high school course
In the mortality table we show the following observed quantities:
.

Multiple-decrement model
Given that a student can leave the course for reasons other than death, we will
extend the previous model to include the following possibilities: voluntary withdrawal from the
course and expulsion from the course due to poor results, where the various reasons can be
stated as decrements. We therefore have three decrements:
− decrement (1) – death,
− decrement (2) – voluntary withdrawal from the course,
− decrement (3) – expulsion from the course due to poor results.
This situation is represented in Figure 2.
Figure 2. Three-decrement model of a post-high school course

UNIT-5
1.) Risk Measure:
Calculating reserves for policies with significant non-diversifiable risk requires a methodology
that takes account of more than just the expected value of the loss distribution. Such
methodologies are called risk measures. A risk measure is a functional that is applied to a
random loss to give a reserve value that reflects the riskiness of the loss.
There are two common risk measures used to calculate reserves for non-diversifiable
risks: the quantile reserve and the conditional tail expectation reserve.
2.) Return
Return refers to expected rate of return from an investment. Return is an important
characteristic of investment. Return is the major factor which influences the pattern of
investment that is made by the investor. Investor always prefers high rate of return for his
investment.
3.) Diversification
Portfolio risk can be reduced by the simplest kind of diversification. Portfolio means
the group of assets an investor owns. The assets may vary from stocks to different types
of bonds. Sometimes the portfolio may consist of securities of different industries. When
different assets are added to the portfolio, the total risk tends to decrease. In the case of
common stocks, diversification reduces the unsystematic risk or unique risk. Analysts
opine that if 15 stocks are added in a portfolio of the investor, the unsystematic risk can be
reduced to zero. But at the same time if the number exceeds 15, additional risk reduction
cannot be gained. But diversification cannot reduce systematic or undiversifiable risk.
4.) Efficient Frontier
The risk and return of all portfolios plotted in risk-return space would be dominated
by efficient portfolios. Portfolio may be constructed from available securities. All the
possible combination of expected return and risk compose the attainable set.

1.) Capital Asset Pricing Model Assumption:
1. An individual seller or buyer cannot affect the price of a stock. This assumption is the
basic assumption of the perfectly competitive market.
2. Investors make their decisions only on the basis of the expected returns, standard
deviations and co variances of all pairs of securities.
3. Investors are assumed to have homogenous expectations during the decision-making
period.
4. The investor can lend or borrow any amount of funds at the riskless rate of interest.
The riskless rate of interest is the rate of interest offered for the treasury bills or Government
securities.
5. Assets are infinitely divisible. According to this assumption, investor could buy any
quantity of share i.e. they can even buy ten rupees worth of Reliance Industry shares.
6. There is no transaction cost i.e. no cost involved in buying and selling of stocks.
7. There is no personal income tax. Hence, the investor is indifferent to the form of return
either capital gain or dividend.
8. Unlimited quantum of short sales is allowed. Any amount of shares an individual can
sell short.
2.) Security Market Line
The Capital Asset Pricing Model also has implications for the returns on individual
assets. Consider plotting the covariance of an asset with the market against the asset’s expected
return. Combining M and risk-free allows movement along a line through the two points these
assets determine. The covariance of the risk-free asset with the market is zero and the assets
return is ᵞf. The covariance of the market with the market is 𝜎𝑀
2
. Hence the points (0, ᵞf) and
(𝜎𝑀
2
,𝛾𝑀
̅̅̅̅ .) can be linearly combined to determine the Security Market Line. In equilibrium, all
assets must offer return and risk combinations that lie on this line. If there was an asset (or
portfolios) located above this line, all investors would buy it. Equally, if there was an asset that

lay below the line, no investor would hold it. Trading these assets must ensure that in
equilibrium, they will lie on the line.
3.) Portfolio Analysis
An investor owns a portfolio composed of five securities with the following
characteristics:
Securit
y
Beta
Random error term
standard
deviation (per cent)
Proportio
n
1 1.35 5 0.10
2 1.05 9 0.20
3 0.80 4 0.15
4 1.50 12 0.30

5 1.12 8 0.25
If the standard deviation of the market index is 20 per cent, what is the total risk of
the portfolio?
Solution
The total portfolio risk may be expressed as:
=20.81

1.) Security Market Line
The risk-return relationship of an efficient portfolio is measured by the capital market
line. But, it does not show the risk-return trade off for other portfolios and individual securities.
Inefficient portfolios lie below the capital market line and the risk return relationship cannot be
established with the help of the capital market line. Standard deviation includes the systematic
and unsystematic risk. Unsystematic risk can be diversified and it is not related to the market. If
the unsystematic risk is eliminated, then the matter of concern is systematic risk alone. This
systematic risk could be measured by beta. The beta analysis is useful for individual securities
arid portfolios whether efficient or inefficient.
When an additional security is added to the market portfolio, an additional risk is also
added to it. The variance of a portfolio is equal to the weighted sum of the co-variances of the
individual securities in the portfolio.
If we add an additional security to the market portfolio, its marginal contribution to the
variance of the market is the covariance between the security’s return and market portfolio’s
return. If the security i am included, the covariance between the security and the market
measures the risk. Covariance can be standardized by dividing it by standard deviation of market
portfolio coy im/σm. This shows the systematic risk of the security. Then, the expected return of
the security i is given by the equation:

Security A
Beta = 1.10 238
E(R) =7+1.10(8) = 15.8
Security B
Beta = 1.20
E(R) = 7 + 1.20(8) = 16.8 = 16.6
Security C
Beta = .7
E(R) = 7 + .7(8) =12.6
The same can be found out easily from the figure too. All we have to do is, to mark the beta on
the horizontal axis and draw a vertical line from the relevant point to touch the SML line. Then
from the point of intersection, draw another horizontal line to touch the Y axis. The expected
return could be very easily read from the Y axis. The securities A and B are aggressive
securities, because their beta values are greater than one. When beta values are less than one,
they are known as defensive securities. In our example, security C has the beta value less than
one.
2.) Markowitz Diversification:
Most people agree that holding two stocks is less risky than holding one stock.
For example, holding stocks from textile, banking, and electronic companies is better
than investing all the money on the textile company’s stock. But building up the optimal
portfolio is very difficult. Markowitz provides an answer to it with the help of risk and
return relationship.

Assumptions
The individual investor estimates risk on the basis of variability of returns i.e. the
variance of returns. Investor’s decision is solely based on the expected return and variance
of returns only.
For a given level of risk, investor prefers higher return to lower return. Likewise, for a
given level of return investor prefers lower risk than higher risk.
The Concept
In developing his model, Markowitz had given up the single stock portfolio and
introduced diversification. The single security portfolio would be preference if the investor
is perfectly certain that his expectation of highest return would turn out to be real. In the
world of uncertainty, most of the risk averse investors would like to join Markowitz rather
than keeping a single stock, because diversification reduces the risk.

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