Mathematics Advanced: Financial Mathematics
Module 2 for Mathematics Advanced (HSC)
Table of Contents
Simple and Compound Interest
 Simple interest $(I)$ is calculated with $\color{orange}I=PRn$, an Arithmetic Progression
 $P$ is the principal (initial) sum,
 $R$ is the rate of interest per unit of time
 $n$ is the number of time intervals which have passed
 If the question asks for the total amount, add $P$ to $I$ at the end
 Compound interest is found by $\color{orange}A_{n}=P(1+R)^{n}$, a Geometric Progression
 $A_n$ is the amount of interest after $n$ units of time
 To find the interest (without the initial amount), subtract $P$ from $A_n$
 Depreciation is a form of compound in terest, where the value decreases over time
 Depreciation is expressed as $\color{orange}A_{n}=P(1R)^{n}$ (also a Geometric Progression)
 $R$ is the rate of depreciation per unit time
 To find the interest (without the initial amount), subtract $P$ from $A_n$
Annuities
 Annuities are compound interest investments, from which equal payments are recieved on a regular basis, for a fixed period of time
Practice Question
Minho deposits 200$ per month at the start of each month into an annuity which pays 6% p.a. for 20 years. How much will the account hold after the full 20 years?
Toggle Answer

After 1 month, the account has $200(1+0.005)$ dollars

After 2 months, $200(1.005)^{2}+200(1.005)$

After $n$ months, we have $200(1.005^{n}+1.005^{n1}+…+1.005)$

The geometric progression in the brackets is: $$S_{(20\times 12)}=\frac{1.005(1.005^{240})1}{1.0051}=464.3511$$

Therefore, $464.3511\times 200=92870.22$ $ after 20 years
Present and Future Values
 The Future value $(FV)$ is the total value of an investment at the end of its term, including all interest
 The Present value $(PV)$ is the single lump of money that could be initially invested to yield a given future value over a given period
 Present values are calculated using the compound interest formula
 Future value is calculated using a variant of the compound interest formula: $$\color{orange}FV=PV(1+r)^{n}$$
Loan Repayments
 Loans are usually repaid through regular installments, with compound interest charged on the balance owed
 $\color{orange}A_n = \text{principle + interest  installments + interest}$
 The loan is paid off when $A_{n}=0$
Practice Question
Michael takes out $10000 to buy a car. He will repay the loan in 5 years, paying 60 equal monthly instalments, beginning 1 month after he takes out the loan. Interest is 6% p.a. compounded monthly. How much is the monthly installment?
Toggle Answer
Method 1:
Let $M$ be the monthly installment:
 $A_{1}=10000(1.005)M$
 $A_{2}=(10000(1.005)M)(1.005)M$
 $\therefore A_{2}=10000(1.005)^{2}1.005MM$
 $A_{60}=0=10000(1.005)^{60}M(1+1.005+…+1.005^{59})$
GP inside the brackets is $\frac{10000(1.005^{60})}{\frac{1.005^{60}1}{0.005}}=\$193.33$
Method 2 (Speed Hack):
 $A_{n}=10000(1.005)^{n}M(1+1.005+…+1.005^{n1})$
 $10000(1.005)^{60}=M(1+1.005+…+1.005^{59})$
GP inside brackets is $S_{60}=\frac{1.005^{60}1}{0.005}=69.77$
 $\therefore M=\frac{10000(1.005)^{60}}{69.77}$
 $=\$193.33 $