Introduction to actuarial calculus
2012/02/10
Lesson1
Introduction :
Why does an interest exist?
-because there is a preference for the time being (the present time)
-because lending money is the same as renting money:
The interest is equivalent to the renting cost.
What justifies the rate of interest?
-It is impossible to take advantage of other investments producing better financial returns.
There is a rate of privation which remunerates the delay necessary to resume consumption.
-There is a risk that the principal will not be recovered if the debtor is defaulting.
The rate incorporates a risk premium.
-There is a risk that the purchasing power of the money depreciates.
The rate anticipates inflation.
In this class, we don’t explain how to define a rate of interest and how to calculate it. Also, we
will not explain why different rates of interest exist and why they move as time goes by.
At first, we will define how to calculate a simple interest.
A- Interest
A-1-Simple interest
In this paragraph the interest rate is annual and fixed.
A-1-1Simple interest:
The value of a simple interest is proportional to loan capital (C ), to interest rate (i )( decimal number)
and to investment period (n) (number of years) :
I = C.i.n
Remark 1:
When we use this formula, the interest rate (i) is an annual interest rate and (I) is called simple
interest.
Remark2:
If we use (i ) expressed by a % instead of a decimal number, then (I) is expressed in % of capital.
Remark 3:
If a capital is invested for a period inferior to one year then (n) is a fraction number of days for the
investment period divided by the number of days to be considered in one year. To count the number
of days for the investment period, count all days, including the first day and excluding the last day.
The number of days in one year depends on the market and also on the country, i.e. 360days or
365days or 366 days. For example, for the French monetary market, the number of days in one year
is 360 days (called account year), for the bond French market the number of days is 365 days (or 366
days).
A-1-2 Day count conventions
-
Don’t confuse date and period.
For instance, date 0 is the date of the beginning of an investment. Date t is a given date such as
2011/09/19 when the time is per day. Date t can be also 2011 when the time is per year.
A period is a length of time between two successive dates. The length of time between two
successive dates depends on the unit (days months years….) and day count convention.
For instance, the time is per month, and the investment produces a simple interest every 6 months
(period), during 4 years, beginning in January 2011 (date0). Interests are paid in June2011, January
2012, and so on.
The period between two payments has a length of 6 months.
For instance, the time is per day, and the investment produces a simple interest every 6 months
(period), during 4 years, beginning in January 2011 (date0). Interests are paid in June 1, 2011,
January1, 2012, and so on.
The period between two payments has a length of the number of days during 6 months.
This number depends on day count convention.
-
The number of periods depends on day count convention
Keep in mind that in the bond market the practice for calculating the number of days between two
dates depends on day count convention. The convention differs by country and the type of security.
Day count conventions are used to calculate the number of days in the numerator and denominator
of the interest.
French day count conventions:
Example 1 for the French monetary market:
Calculate the interest on a capital of 1 500 monetary units invested at a 10% interest rate for the
period from October 12th ,2011 to January1st , 2012.
Answer: 33.75 unit of currency.
Example 2 for the French bond market (annual coupon) :
A bond has the following information: nominal 1000 units of currency, nominal rate 1.12%, excoupon on February 15th , maturity in 2015.
1 - Calculate the value of the coupon at the date of January 12th, 2012,
2- Then calculate the value of the coupon at the date of March 15th, 2012.
Answer:1 calculate with 365 days per year ; 2- calculate with 366 days per year.
USA day count conventions:
In USA bond market, considering a bond and assuming semiannual payments, the accrued interest
(AI) is calculated as follows:
AI=
The number of days used depends on the day count convention for the particular security.
Specifically, day count conventions differ for Treasury securities, government agency securities,
municipal bonds and corporate bonds.
For coupon-bearing Treasury securities, the day count convention used is to determine the actual
number of days between two dates. This is referred to as the “actual/actual” day count convention.
Example3:
Consider a coupon-bearing Treasury security whose previous coupon payment was March 1. The
next coupon payment would be on September 1. Suppose this Treasury security is purchased with
settlement date of July 17. The actual number of days between July 17 (settlement date) and
September 1 (the date of the next coupon payment) is
July 17 to July 31: 14 days
August
September 1
31 days
1 day
46 days
Note that the settlement date (July 17) is not counted.
The number of days in the coupon period is the actual number of days between March 1 and
September 1, which is 184 days. The number of days between the last coupon payment(March1)
through July 17 is therefore 138 days (184 days -46 days).
For coupon-bearing agency, municipal and corporate bonds, a different day count convention is
used.
It is assumed that every month has 30 days, that any 6-month period has 180 days and that there are
360 days in a year. This day count convention is referred to as the “30/360” day count convention.
Example 4:
Consider once again a Treasury security purchased with a settlement date of July 17, the previous
coupon payment on March, and the next coupon payment on September 1. If the security is an
agency, municipal or corporate bond, the number of days until the next coupon payment is
July 17 to July 31
13 days
August
30 days
September1
1 day
44 days
Note that the settlement date, July 17, is not counted. Since July is treated as having 30 days, there
are 13 days (30 days minus the first 17 days in July).
The number of days from March 1 to July 17 is 136, which is the number of days counted in the
interest period.
Return on invested capital:
The return (decimal number) on the capital which is invested with simple interest rate is
r = I/C.
Remark4:
We will see later in the course, another calculus for return.
Remark5:
Return has no monetary unit. It can also be given in percentage of the invested capital : r*100 .
Example 5:
In example 1, the return on capital is 33,75/1500 = 0, 0225 or 33,75/1500 *100=2.25%.
Mean rate of return:
If we invest at different dates, different amounts of money and during different periods, the mean
rate of return is the arithmetic weighted average of the rates of interest. The rates of interest are
weighted by their relative capital shares.
 C *n
r  k 1  p k k

 k 1 Ck * nk

p
 t
* k
 100

where p is the number of investments
Ck is the amount of the investment k
nk is the investment period for the investment k
t k is the interest rate for the asset k.
It also represents the total return expressed as follows:
p

r
C
k 1

k
* nk *
p
k 1
tk
100
C k * nk
Remark 6:
If we don’t divide the rates of interest ( t k ) by 100 then the mean rate of return is in percentage of
the total capital invested.
Example 6: Calculate the mean rate of return (%)for the following assets:
1000 monetary units invested during 29 days with a 6% interest rate,
7050 monetary units invested during 6 days with a 10% interest rate,
540 monetary units invested during 47 days with a 8% interest rate.
Answer:8,275%