1.

Chapter 2

2. Time Value of Money

3. Effective interest rate

Consider a time interval , for . An investment of at time , accumulates
to at time

We call the effective interest rate for the interval . Note that
. It is common to take . If further then we shall consider
as the effective interest rate for a period of length 1 which could be a year, a month, a
week, a day, a two year period etc. We call such a period of length 1 a basic time unit which in most
of our cases shall be a year.

Therefore in summary the effective interest rate is the amount of money an initial investment will
earn by the end of the investment period expressed as a proportion of the initial amount. The
amount is the future value (FV) of the amount .

3.0.1. Simple vs Compound interest

Under a simple interest arrangement interest is earned on the initial investment only. The interest
payments themselves do not earn interest.

Under a compound interest arrangement interest payments are added to the initial capital invested.
Thus interest is earned on the entire accumulating investment. So here the interest payments
themselves earn further interest. For simplicity let us consider time intervals of equal length in a
cashflow timeline and assume further that the periodic effective interest rate is the same for each
subinterval of time. Lets call this effective periodic (normally annual) interest rate . Thus it is easy
to see that invested for years will accumulate to:

3.1. Effective discount rate and Present value

An accumulated value is the future value of an initial investment invested at a certain interest rate.

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We are, however, interested to know how much to invest now to provide for payments at a future
time. This initial investment is called the PV or present value.

Consider again the previous example:

We call the effective discount rate for the interval . Note that
. It is common to denote by as the effective discount rate for the period of
length 1 (day, month, year etc). For this period, is the present value (PV) of and we view as
the amount of money we need at time in order to provide a payment of at time . For
Actuaries, the present value could be a premium that the insurer must charge in order to be able
to pay a sum assured of to the insured. This can then be extended to multiple periods.

We can then summarize and say that the effective discount rate is the amount of interest required to
be paid at the beginning of the period expressed as a proportion of the final amount.

There is an obvious connection between the effective interest rate and the effective
discount rate . In order to understand this relationship easily, we consider the special case
of and .

For this particular case we have the following:

Effective rate of interest

Effective rate of discount Accumulation factor

Present value factor or discount factor .

From the second and fourth equations above we get . This is an important relationship
which you must always remember. We can then generalize and say that if ,
then For most of this course we are happy just to work with and than
their generalizations.

In the case of multiple periods and assuming that the periodic effective interest rate is and the
periodic discount rate is , then assuming compound interest, the PV (at time 0) of a payment of
made at time is . Thus invested now will accumulate to at .

here is the -year discount factor and is the -year accumulation factor

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Notice how because is paid at the beginning of the period and the lender (investor) is
"rewarded" for paying interest now than wait to pay at . The lender is given a
"discount" in interest.

A formula often used is: (discount effective rate "discounted effective interest rate").

3.1.1. Commercial or simple rate of discount

A loan is said to be issued at commercial rate of discount (or simple rate of discount) per annum
over years if the lender lends in return for a payment of
at the end of years.

Similar to effective discount rate where grows to after years.

We shall therefore be using these notations:

: Simple rate of discount, and

: compound rate of discount.

4. Remark
In the general setting, we shall denote by as the future value of a unit of currency invested
at time and withdrawn at at the effective interest rate for the period. In particular we
shall denote , so that . We then view as the accumulated value at time
of 1 invested at time 0 .

In the same way, we denote whereby . Therefore an amount invested at

time 0 will have future value at time t. In the same way, we shall denote by as the
value, at time of a unit of currency received at time at effective discount rate . In
particular we shall let . Therefore, an amount received at time , will have present
value of at time 0 .

It is easy then to argue that .

5. Chapter 3

6. Interest rates

7. Nominal interest rate

Assume the basic time unit is a year. When interest payments are made more frequently than
annually, say quarterly, we are dealing with an effective interest rate per quarter year. However,
sometimes we express this quarterly rate as an annual rate.

Thus if the effective rate is per quarter we could talk about per year. here is the nominal
interest rate per annum compounded quarterly, and in symbols we write per annum.

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The effective interest rate over a time period is the actual interest rate earned over the period where
interest is paid at the end of the time period. The nominal interest rate is usually an annual rate
where the interest payments are actually made times a year (where . We shall be using
as the notation for effective interest rate and as notation for the nominal interest rate. Note
that it does not make sense to say "the annual nominal interest rate is " without stating the
number of times per basic time unit (e.g a year) that interest is compounded. The correct statement
would be for example "the annaul nominal interest rate is compounded quarterly". In that case
we will be talking of so that the quarterly effective interest rate is .

7.1.2. Conversion from nominal to effective

A borrower who borrows 1 at time 0 (for repayment at time 1 ) must pay interest on the loan in two
possible ways (a) to pay at the end of the period or (b) to pay a total of in p equal instalments
of at the end of each th subinterval. 1 borrowed

1 borrowed

Thus instead of paying one instalment of interest at time 1 , we pay instalments throughout the
time period. Hence no arbitrage forces us to say that the total interest paid in which ever way must
be the same, i.e

Example 3.1.1 If per annum, then what is ?, i.e. what is the effective interest rate per
annum?

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Here the accumulated value after one year of an initial deposit of 1 where interest is added at the
end of every quarter year equals the accumulated value of the same deposit where interest is only
paid once, at the end of the year.

An important question will be to ask what happens to as increases, i.e from a logical point of
view what should happen to the nominal interest rate if one makes more frequent repayments to the
loan? This should be answered from the following exercise:

Exercise 3.1.2 Prove that the nominal interest rate is a decreasing function of i.e

Note that the exercise above says that the more frequent you repay your interest throughout the
year, the less annual interest should be charged! That should be logical right?

8. Nominal discount rate

In the same way we can also define the nominal discount rate as a basic time unit rate (usually
annual) payable in advance p-thly.

0 1

We can use the same mathematical argument as in the nominal interest rate case to show that
. Do this as an exercise.

We again ask what happens to the nominal discount rate if we make more frequent payments.

Exercise 3.2.1 Prove that the nominal discount rate is an increasing function of , i.e

9. Force of interest
Using the notations from Chapter 2, consider an amount at time that grows to an amount
at time . We Let equal the effective interest rate over the period, then
.

We then call (force of interest at time ) as the fraction of interest earned over tiny period to
expressed as a proportion of annualised.

Hence , where is expressed in years.

Therefore

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Therefore . We call the continuously compounded force of interest and view
is the accumulated fund value at time of a unit investment at time 0 . Note that .
Common exercises in this course consider a piecewise continuous force of interest. See exercises at
the end of this chapter to be done in class.

9.1.3. Constant force of interest

Now, assume - i.e. a constant effective interest rate of per annum over the next
years, then

This relates the constant force of interest over the period with the constant effective rate for
the same period. It is normal to express the relationship as .

Exercise 3.3.1 Prove that

9.1. Accumulation factors and discount factors re-visited

Recall that for represents the accumulated value at time of an investment at
time which is opened with an initial deposit of 1 .

9.1.1. Principle of consistency

Suppose that we partition the time interval so that . Assume that we invest
1 at time . Then at time , this amount will have accumulate to and at time it
accumulates to .A ...In the end we get the principle of consistency which says
. Therefore, no matter how interest is chargeable, the same
amount deposited at the same time MUST have the same future value. Any violation of this
principle will result in an investor creating an arbitrage opportunity. For example, assuming that
the borrowing rate equals the lending rate (a common assumption in economics), if it was possible
that an investor can borrow R100 at nominal rate (i.e interest is charged monthly) and
then immediately re-invest the same amount at nominal rate (i.e interest is charged
semi-annually), then the investor will get a "free lunch" interest of per annum on the
amount. To avoid arbitrages, the investor, if he so wishes to deposit the amount semi-annually,
should be awarded since

9.1.2. General expressions

For times and force of interest , we can easily verify that:

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 Accumulation factor:
Discount factor:
Effective interest rate:
Effective rate of discount:

9.1.3. Table of Identities

For and we have the following golden table of identities:

To use the table, for example we say and are related by the expression and and are
related by . These identities can easily be verified. You can do them as exercises. Exercise
3.4.1 Assume that , the force of interest per annum at time , (years), is given by the formula

Derive an expression for , the present value of 1 at time .

Exercise 3.4.2 In valuing future payments, an investor uses the formula

(where is a given positive constant), for the value at time 0 of 1 due at time (measured in
years). Show that the above formula implies that

1. The force of interest per annum at time will be

2. The effective rate of interest for the period to will be

Solutions to be done in the lecture

9.2. Money Market's language of interest rates (Just for extra reading)
In the Money Market, every interest rate is generally "nominal"! So the term "effective rate"
should be restricted to Actuaries and in some cases the banking sector as well. So an interest rate is
Nominal Annual Compounded which results in the following categories of interest rates

Nominal Annual Compounded Annually (NACA) is what we call the effective interest rate in
this course. So
Nominal Annual Compounded Daily (NACD) is the nominal rate compounded daily.
Examples are overnight interbank rates
Nominal Annual Compounded Weekly (NACW) is the nominal rate compounded
weekly. Examples apply in carry market
Nominal Annual Compounded Monthly (NACM) is the nominal rate compounded
monthly. Examples are call rates and credit cards rates
Nominal Annual Compounded Quarterly (NACQ) is the nominal rate compounded
quarterly. Examples are LIBOR and JIBAR

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 Nominal Annual Compounded Semiannually (NACS) is the nominal rate compounded
semi-annually. Examples are bond coupon rates.
Nominal Annual Compounded Continuously (NACC) is the force of interest.

In the Money Market environment actually most of these interest rates dont exist because a Day-
Count convention is used. By this the actual number of days between trading are counted plus or
minus certain days depending on the rule. For example the ACTUAL/365 convention means that we
count the EXACT number of days between trades and divide by 365 days in a year. In such a case a
month could have 28 days, 29 days, 30 days or 31 days. Therefore will not make sense.

Furthermore, on the Money Market instruments are either Yield Instruments, where if is the yield
rate, then R1 today accumulates to in years or they are Discount Instruments in which
case if is the discount rate then issued years from now is worth (1 Dn) today. Its just that
simple, so expect your lives to be easy after UCT on condition you work with Money Market
instruments! Exercise 3.6.1 1. Suppose that and and .
Calculate the implied interest rate of the investment as a simple and an NACQ rate. Comment on
which rate is higher, its relative magnitude, and why you think this is the case.

2. One basis point is defined as 1%%. How many basis points must you add or subtract to:

(a) to get it to equivalent NACS

(b) to get it to equivalent

(c) to get it to equivalent

3. If is the discount rate of a discount instrument between dates and and is the yield rate
of an equivalent yield instrument for the same period, prove that

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