ACTL 1101 Introduction to Actuarial Studies
Xiao Xu
© University of New South Wales (2023)
School of Risk and Actuarial Studies, UNSW Business School
x.xu@unsw.edu.au
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
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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...
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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.
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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
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Terminology
principal maturity repayments
(principle) (of principal & interest)
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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)
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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)
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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
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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
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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%
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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)
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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
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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
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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 ]
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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.
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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?
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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).
Let i be the interest rate effective per period, then we have...
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Computation of an
an =
n∑
j=1
(
1
1+ i
)j
=
(
1
1+ i
) n−1∑
j=0
(
1
1+ i
)j
=
1
(1+ i)
1−
(
1
1+i
)n
1−
(
1
1+i
)

=
1−
(
1
1+i
)n
i
=
1− vn
i
, where v :=
1
1+ i
.
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Time Certain Annuity - Example
Example 4.7: Calculate a10 at rate of interest i = 0.01 per period.
Solution:
a10 =
1− vn
i
=
1− ( 11.01)10
0.01
= 9.4713
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Annuity Due - Payments in Advance
Actuarial notation: place “double-dots” over the annuity symbol to
indicate n payments made in advance: a¨n
This symbol a¨n (pronounced “a double dot n”) then represents the
Present Value of n payments of 1, if the first payment is made
immediately
a¨n =
n−1∑
j=0
(
1
1+ i
)j
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Relationship between a¨n and an
We have:
a¨n =
1−
(
1
1+i
)n
1−
(
1
1+i
)
 =
1−
(
1
1+i
)n(
1+i−1
1+i
)

= (1+ i)
1−
(
1
1+i
)n
i

= [1+ i ] an
But did we really need those calculations?
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Annuity Due - Example
Example 4.8: Calculate the present value of a 5 year annuity due at 6% p.a.
Solution:
The value is
a¨5 = [1.06]
[
1− ( 11.06)5
0.06
]
= 4.4651
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1 Introduction and Motivation
2 Compound Interest
3 Annuities and Actuarial Notation
4 Typical Exam Questions
25/33
Loan - Example
Josephine has a 30 years mortgage on her house with a balance of
$1,000,000. The interest rate is 4% (annual nominal) payable monthly in
arrears.
1 What is the amount of the monthly repayments (principal and
interest)?
2 What is the interest part of the first monthly repayment, and what will
the balance of the mortgage be just after that first monthly repayment
is made?
3 Assume that interest rates do not change for the first 15 years. What
is the balance of the mortgage after 15 years, right after the monthly
repayment? What was the interest part of that repayment?
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Loan - Solution I
1 The duration of the loan is 30 years with monthly repayments, so
there will be 360 repayments. The effective interest rate per month is
4%/12 = 1/3%. Because the Present Value of the loan at time 0 is
$1,000,000, we have
R × a360 1/3% = $1,000,000
Hence, the amount to repay each month is
R =
1,000,000
a360 1/3%
= 4,774.15
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Loan - Solution II
2 The first repayment has an interest component of
I = 1,000,000 · 1/3% = 3,333.35,
and a principal repayment component of
P = R − I = 1,440.80.
The balance right after this repayment is then
OB1 = 1,000,000− P = 998,559.20.
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Loan - Solution III
3 The balance will be
OB180 = Ra180 1/3% = 645,427.60,
because this must be the present value of all remaining payments
(which will reimburse the balance, no more, no less).
The interest part of the repayment would have been
1/3% · OB179 = 1/3% · Ra181 1/3% = 2,160.14
which is obviously much less than after the first month.
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Taken from Quiz 2020
You have spent too much time on Amazon and now your credit card is
maxed out. You negotiate with the credit card company to repay your total
balance of 8,000$ over a period of 2 years, with monthly level repayments
R , starting one month from now. The annual effective rate of interest
charged to you is 26.824%.
1 (1pt) Show that the rate of interest effective per month is 2%.
2 (2pt) Find the amount R of the individual level repayments.
3 (1pt) Consider the very last of those repayments. What amount of
that repayment is repaying the principal? Express your result as a
function of R .
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Taken from Quiz 2020
4 (1pt) Imagine that instead of monthly repayments, you repay the debt
with one payment at the end of two years (with the same annual
effective rate of interest). In that case, would the total amount repaid
be bigger, the same, or smaller than with monthly repayments?
Explain.
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Solution I
1 It is simply a matter of checking that:
(1.02)12 = 1+ 0.26824.
2 We must have ‘Initial balance of loan = PV of payments’, i.e. solve:
8000 = Ra24
R =
8000 · i
1− v24
=
8000 · 0.02
1− (1.02)−24 ≈ 422.97.
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Solution II
3 The Opening Balance of the loan exactly one period before the last
payment (and right after the payment-before-last is made) is precisely
the amount of principal that will be repaid in the last payment.
Indeed, the last payment must reimburse the exact amount of the
outstanding loan, no more and no less.
Hence, we must find this Opening Balance at time t = 23. This is not
hard to do, as it must equal the Present Value of all remaining
payments (just one here), i.e.
OB23 =
R
1.02
.
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Solution III
4 The total amount repaid would be bigger, because the balance of the
loan would accumulate for two straight years without being reduced by
any repayment. Said otherwise, the interest on interest would be
bigger. One can back this claim with calculations (but it is not
necessary to get full marks):
8000 · 1.0224 = 12,867.50 > 24 · R = 10,151.25