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ACTL 1101 Introduction to Actuarial Studies
创建日期：2024-05-30 16:37:56
所属分类：数学mathematics
标签： ACTL 1101
Introduction to Actuarial Studies

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ACTL 1101 Introduction to Actuarial Studies

Week 3-4:
Financial Mathematics 1
1Readings: Sherris 4.2.1-4.2.4, 4.2.61/33
1 Introduction and Motivation
2 Compound Interest
3 Annuities and Actuarial Notation
4 Typical Exam Questions
2/33
1 Introduction and Motivation
2 Compound Interest
3 Annuities and Actuarial Notation
4 Typical Exam Questions
3/33
Motivation
From the Actuaries Institute website:
‘Actuaries evaluate risk and opportunity - applying mathematical,
statistical, economic and financial analyses to a wide range of business
problems.’
A key word here is financial: most of an actuary’s work relates in
some way or another to questions of money: insurance claims,
superannuation benefits, investments returns, etc.
Actuaries must have a good knowledge of the time value of money:
1,000$ today don’t hold the same value as 1,000$ in 20 years...
3/33
Overview
Financial mathematics are usually not an end in itself, but a
fundamental tool actuaries need.
This week covers valuation of known future cash-flows.
Another difficulty in actuarial studies is that cash flows related to
losses are by nature uncertain.
In-depth coverage in ACTL2111: Financial Mathematics
As an historical note: actuaries developed the application of the
mathematics of finance to insurance problems as early as the 1700’s.
4/33
Some Applications
Personal loans and housing loans
▶ personal loan - usually with level repayments, "flat" fixed interest rates
(3 to 5 years)
▶ housing loan - level repayments of loan principal and interest (20 to 30
years), variable interest rates in Australia, flexible repayments
Fixed and floating interest securities (bonds)
▶ repayments of interest (coupons) and face value on maturity
▶ government bonds, corporate bonds
5/33
Terminology
principal maturity repayments
(principle) (of principal & interest)
6/33
Interest Rates
Interest is due at different frequencies during the year - monthly,
quarterly, semi-annual or annual
⇒ It matters because once interest is paid, there is interest on the
interest (that is where the compounding effect happens)
Payments due at the end of the period - payable in arrears
Payments due at the start of the period - payable in advance
Interest rates can be fixed or variable (floating rate)
7/33
1 Introduction and Motivation
2 Compound Interest
3 Annuities and Actuarial Notation
4 Typical Exam Questions
8/33
Compound Interest
‘Money makes money. And the money that money makes, makes
money.’
- Benjamin Franklin
‘The most powerful force in the universe is compound interest.’
- Albert Einstein ???
(Urban legend? ‘this perspective on the power of compound interest is a fairly modern
invention, one which has been retroactively placed into the mouth of a prominent dead
person to give it more punch’ Snopes.com)
8/33
Nominal Interest Rate
Interest rates are normally QUOTED as per annum percentage
nominal rates
Number of periods, denoted by m, is also called the compounding
frequency of the interest rate
▶ e.g. m = 12 corresponding to monthly, m = 4 corresponding to
quarterly
The nominal interest rate is
▶ the rate that is effective per period multiplied by the number of periods
m in a year
10/33
Effective Interest Rate per Period
Let j (m) denote the per annum nominal interest rate with m periods
The effective interest rate per period is
r =
j (m)
m
11/33
Effective Interest Rate per Period - Example
Example 4.1: A 10-year loan is payable with quarterly cash flows and with
a constant nominal interest rate of 12% p.a. payable quarterly. Calculate
effective interest rate per period for this loan.
Solution:
The nominal interest rate is j (4) = 12%, therefore the effective interest rate
per period is
r =
j (4)
4
=
12%
4
= 3%
12/33
Annual Effective Interest Rate
j (m) is the annual nominal interest rate with m periods
Let j denote the annual effective interest rate, then
1+ j =
(
1+
j (m)
m
)m
Note that j = j (1)
13/33
Annual Effective Interest Rate - Example I
Example 4.2 (1): Calculate the annual effective rate corresponding to 6%
p.a. nominal assuming a monthly compounding frequency.
Solution:
The frequency is m = 12.
Therefore, the annual effective interest rate is
j =
(
1+
0.06
12
)12
− 1
= 0.0617
or 6.17% p.a. effective
14/33
Annual Effective Interest Rate - Example II
Example 4.2 (2): Calculate the annual effective rate corresponding to 6%
p.a. nominal assuming a semi-annual compounding frequency.
Solution:
The frequency is m = 2.
Therefore, the annual effective interest rate is
j =
(
1+
0.06
2
)2
− 1
= 0.0609
or 6.09% p.a. effective
15/33
Continuous Compounding
The continuously compounded interest rate is the nominal interest
rate obtained when the compounding frequency is increased to infinity.
Let δ denote the continuously compounded interest rate, i.e.
δ := lim
m→∞ j
(m)
We have:
1+ j = lim
m→∞
(
1+
j (m)
m
)m
= eδ
Therefore,
j = eδ − 1 or δ = ln [1+ j ]
16/33
Continuous Compounding
The ‘continuously compounded interest rate’ is often called force of
interest.
The famous mathematical constant e was first discovered by Jacob
Bernoulli in 1683 as he was trying to find the effective annual rate of
interest corresponding to a 100% nominal rate, if the frequency of
payments is infinite.
Details in this video from a popular YouTube channel.
17/33
Continuous Compounding - Example
Example 4.9: Calculate the continuously compounded interest rate
equivalent to an annual effective rate of 10% p.a.
Solution:
δ = ln [1+ j ] = ln [1.1] = 0.09531.
This is 9.530%, very close to 10%. Do you find that surprising?
18/33
1 Introduction and Motivation
2 Compound Interest
3 Annuities and Actuarial Notation
4 Typical Exam Questions
19/33
Time Certain Annuity
A time certain annuity is a stream of level payments, happening at
regular intervals.
Assuming a constant interest rate, the symbol an represents the PV of
n payments of 1, payable in arrears (at end of each period).
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