Page 1The University of Sydney
ECON5007
The Economics of Financial Markets:
Financial Mathematics
Sinan Deng
The University of Sydney
Page 2The University of Sydney
References and Readings
2022 CFA level 1:
• SS1 Quantitative Methods: R1 The Time Value of Money
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Overview
Financial
Mathematics
Time Value of
Money
Interest Rates
Holding Period Return
Simple and Compound Interest
Effective Annual Rate
Present Value and Future Value
NPV Decision Rule
Annuities
Ordinary Annuity
Perpetuity
Financial
Prices and
Returns
Prices and Returns
The Random Walk Model
Page 4The University of Sydney
Overview
Time Value of
Money
Interest Rates
Holding Period Return
Simple and Compound Interest
Effective Annual Rate
Present Value and Future Value
NPV Decision Rule
Page 5The University of Sydney
Is $100 today the same as $100 tomorrow?
• Sinan wants to give you $100, and you can choose when to get it. Would
you prefer to get it now or one year later?
• In reality, we value the present more than the future, so we prefer to get
it now than in the future.
• E.g. Suppose the interest rate is 5%. If you can get $100 from Sinan
today, you will receive $100*(1+5%) one year later.
$100(1+5%)
present value future value
$100 $105
$105÷ (1+5%)
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Interest Rate
The interest rate is
• the amount of interest due per period, as a proportion of the amount lent,
deposited, or borrowed.
Required rate of return is
• affected by the supply and demand of funds in the market;
• the minimum rate of return an investor must receive to accept the
investment.
Discount rate is
• the interest rate we use to discount payments to be made in the future.
• usually used interchangeably with the interest rate.
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Interest Rate
Opportunity cost is
• also understood as a form of interest rate. It is the value that investors
forgo by choosing a particular course of action.
Decompose required rate of return
• Required interest rate on a security = nominal risk-free rate + default
risk premium + liquidity risk premium + maturity risk premium
• Nominal risk-free rate = real risk-free rate + expected inflation rate
Risk-free investments:
• Bank deposits
• U.S. T-bills (<1 year), but not T-notes (1-10 years), T-bonds (>10 years)
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Holding Period Return
• Definition: the holding period return is simply the percentage change in
the value of an investment over the period it is hold.
• Formula: HPR = !!"!"#$%!!" or, HPR = %&"!&!&
Example 1:
• Sinan purchased a stock at $10, and after 90 days, the price increased
to $15. She also received $1 dividend right before she sold the stock.
• What’s the holding period return?
• = '(")()( = *+#*"*,*, = 60%
Page 9The University of Sydney
Holding Period Return
Example 2:
• Sinan purchased a stock at $10, and received $1 dividend right after she
bought the stock. After 90 days, the price increased to $15.
• What’s the holding period return?
• = '(")()( = *+"*,"*** = 36%
Example 3:
• Product A matures in 30 days with HPR=3%.
• Product B matures in 30 years with HPR=10%
• Can we conclude that B is better than A because 10%>3%?
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Simple Interest
Simple interest is a method of interest that always applies to the original
principal amount, with the same rate of interest for every time cycle.
• If the interest rate is 10% p.a., and principal is $100:
• how much will Sinan receive if she decides to save it for 1 year?
• how much will Sinan receive if she decides to save it for 2 years?
• how much will Sinan receive if she decides to save it for 3 years?
0 1 2 3
100 100+10 100+20 100+30
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Compound Interest
Compound interest is calculated on the principal amount and the accumulated
interest of previous periods, and thus can be regarded as “interest on
interest.”
• If the interest rate is 10% p.a., and principal is $100:
• How much will Sinan receive if she decides to save it for 1 year?
• After 1 year, Sinan deposits both principal and interest into the bank, how
much will she have after another 1 year?
• If she repeats this process, how much will she have at the end of year 3?
0 1 2 3
100 100×(1 + 10%) 100× 1 + 10% ! 100× 1 + 10% "

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Effective Annual Rate
• Suppose Sinan deposits $100 into the bank, and the interest rate of the
deposit is 5% per annum.
• If the bank pays interest annually, the end-of-year value of the deposit is $100×(1 + 5%) = $105.
• If the bank pays interest semi-annually, then the end-of-year value of the
deposit is $100× 1 + +%. . = $105.0625.
• In general, if the bank pays interest times a year, then the end-of-year
value of the deposit is $100× 1 + +%/ /
• What is the end-of-year value of the deposit if the 5% annual interest
rate is paid continuously, when → ∞? lim/→1$100× 1 + 5% / = $100×+%
• The continuous compounding will yield the end-of-year value $105.1271
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Effective Annual Rate
• The effective interest rate, effective annual interest rate, annual
equivalent rate or simply effective rate is the percentage of interest on a
loan or financial product if compound interest accumulates over a year
during which no payments are made.1 + = 1 + /
• If semi-annually compounding, then = 2
• If quarterly compounding, then = 4
• If continuously compounding, then 1 + = r
• Features:
✓ The more frequency of compounding, the larger the EAR.
✓ The largest EAR exists if it is continuously compounding.
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The Compound Effect
• “Compound interest is the eighth wonder of the
world. He who understands it, earns it; he who
doesn’t, pays it.” – Albert Einstein
• "Money makes money. And the money that
money makes makes money" - Benjamin Franklin
• A key feature of compound interest is that you
make interest on interest. While the interest
"rate" stays the same, your principal each period
changes.
• https://www.youtube.com/watch?v=qDxDCtZ9U
kE
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Time Value of Money
• Finance is the analysis of the world of return and risk. In finance, we have
to value assets and projects that give you cashflows in the future.
• One key idea is: Cash today is different from cash tomorrow. A safe
dollar is worth more than a risky dollar.
• Reasons are:
1. Impatience
2. Positive Interest Rate
3. Inflation
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Time Value of Money
Financial decisions often require combining cash flows or comparing
values. Three rules govern these processes.
Rule 1 Only values at the same point in
time can be compared or combined
Rule 2 To move a cash flow forward in
time, you must compound it.
Rule 3 To move a cash flow backward
in time, you must discount it.
• How do we compare money across time?
• The present value is the current value of future cash flows given a
specified rate of return.
• The future value is the value of a current asset at a future date
based on an assumed rate of return.
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NPV Decision Rule
• Now we understand how to calculate the time value of money.
• The Net Present Value (NPV) of a project or investment is the difference
between the present value of its benefits and the present value of its costs.
NPV = PV( Benefits ) − PV( Costs )
NPV = PV( All project cash flows )
• To make a financial decision: Accept those projects with positive NPV.
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NPV Decision Rule
• You have saved up $25, 000 to buy your girlfriend a new handbag. The
interest rate is 4%. The shop is offering the following two options:
• Option 1: $25, 000 in one year.
• Option 2: $23, 500 now.
• Option 1: = .+,,,*.,3 = 24038. The cost in today’s dollars is $24, 038. This
is greater than the cash price today.
• Taking the deal now, and you saved
$24, 038.46 − $23, 500 = $538.46 today
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Overview
Annuities
Ordinary Annuity
Perpetuity
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Annuity
What’s annuity?
is a finite set of level sequential cash flows.
• Equal intervals.
• Equal amount of cash flows.
• Same direction.
Elements of annuity:
• N = number of periods
• I/Y = interest rate per period
• PV = present value
• PMT = amount of each periodic payment
• FV= future value
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Future Value of an Ordinary Annuity
• An ordinary annuity is an annuity where payments are made at the end
of each period. Examples: interest payments, salaries, etc.
• An annuity due is an annuity with payment due or made at the beginning
of the payment interval. The first cash flow occurs immediately (at t=0).
Examples: rental fees, tuition fees, living expenses, etc.
• Example of an ordinary annuity: Sinan wants to save up for a handbag,
and she deposits $100 at the end of every year. The interest rate is 5%
p.a.. How much money will she have after four years?

• Is the future value $400?
0 1 2 3 4
$100 $100 $100 $100
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Future Value of an Ordinary Annuity
• * = 100 1 + 0.05 4 = 115.76
• . = 100 1 + 0.05 . = 110.25
• 4 = 100 1 + 0.05 * = 105
• 3 = 100
• = 115.76 + 110.25 + 105 + 100 = 431.01
• The future value of an annuity is simply the sum of the future value of
each payment. = 1 + , + 1 + * +⋯+ 1 + /"*
Ø A: the annuity payment
Ø i: the compound interest rate per time period
Ø n: the number of time periods.
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Future Value of an Ordinary Annuity
• Instead, we can do it all in one step using a single equation. Here is the
formula for calculating the future value of an ordinary annuity:
= ( 1 + / − 1 )
• While we call it the future value, what you are essentially calculating is
the accumulated value, which is just how much you would have at the end
of the annuity period, if you reinvested all your cash flows along the way.
• *#5 #"*5 is called annuity future value factor
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Present Value of an Ordinary Annuity
• Example: Sinan offers a four-year product which pays $100 at the end
of every year and the interest rate is 5% p.a.. What is your willing to
pay for this product?
• Is the present value $400?
0 1 2 3 4
$100 $100 $100 $100
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Present Value of an Ordinary Annuity
• * = *,,*#+% = 95.24
• . = *,,*#+% $ = 90.7
• 4 = *,,*#+% % = 86.39
• 3 = *,,*#+% & = 82.27
• = 95.24 + 90.7 + 86.39 + 82.27 = 354.6
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Present Value of an Ordinary Annuity
• The present value of an annuity is simply the sum of the present value of
each payment.
= 1 + * + 1 + . +⋯+ 1 + /
Ø A: the annuity payment
Ø i: the compound interest rate per time period
Ø n: the number of time periods.
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Present Value of an Ordinary Annuity
• Instead, we can do it all in one step using a single equation. Here is the
formula for calculating the present value of an ordinary annuity:
= (1 − 1 + "/ )
Ø A: the cash flow each period
Ø i: the interest rate each period
• The present value of an annuity is the sum of the present value of each
annuity payment.
• *" *#5 '#6 is called annuity discount factor
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Present Value of a Perpetuity
• A perpetuity is a special case of annuity where the cash flows are
expected to continue forever. Perpetuity is simply an annuity that doesn’t
end, → ∞.
• We can think of perpetuities as a conceptual device used to value long-
lived assets, like stocks. In fact, most of the fundamental valuation
techniques for stocks rely on calculating the present values of perpetuities.
= 1 − 1 + "/ , → ∞
=
Ø A: the cash flow each period
Ø i: the interest rate each period
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Perpetuities Features
• To use the perpetuity formula, the cash flows need to satisfy the following
conditions:
1. Infinite life / Forever
2. Regular payment intervals / Payments are equally spaced
3. Constant dollar value of payment / Level cash flows
• Example: Sinan offers an investment that pays a cash flow of $1000
every year forever (starting in one year’s time) and the interest rate is
10% p.a.. What is the present value of the investment?
= = 10000.1 = 10000
Page 31The University of Sydney
Overview
Financial
Prices and
Returns
Prices and Returns
The Random Walk Model
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Gross and Net Returns
• The gross return of stocks at time t+1 is defined as
7#* = 7#* + 7#*7
where 7#* is the price of the stock at time t+1 and 7#* is the dividend at
time t+1.
• The arithmetic net return is
7#* = 7#* − 1 = 7#* + 7#* − 77
• If we assume the stock does not pay dividend, then
7#* = )()!)( and 7#* = 7#* − 1 = )()!")()(
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Log Returns
• The log return is defined as
7#*8 = ln 7#* = ln 7#*7 = 7#* − 7
• Note that 7#*8 = ln 1 + 7#* ≈ 7#*
when 7#* is small.
• Therefore, the log return is approximately net return, when the net return
is “small”.
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Multi-Period Returns
• Suppose we hold the asset for k periods from time t to t + k, the k-period
gross return is 7→7#9 = 7#97
• Then it follows that 7→7#9 = 7#97 = 7#97#9"*×7#9"*7#9".×⋯×7#*7= 7#97#9"*…7#* =P:;*9 7#:
• The k-period gross return 7→7#9 is the product of the k one-period gross
return: The multi-period return is often called a compound return.
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Financial Prices
• Financial prices are not predictable.
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Financial Returns
• Financial returns are predictable.
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The Random Walk Model
• Given a conditioning set F which contains known information, we can
define the conditional distribution of a random variable X conditional on
F. Conditional moments are based on the conditional distribution.
• Let ℱ7 = 7, 7"*, … = 7, 7"*, … to be the information set at time t.
• We form the forecasts of future prices or returns conditional on the
information set ℱ7. For example, one step ahead price forecast is (7#*|ℱ7). Two step ahead return forecast is (7#.|ℱ7).
• A famous model for asset returns is the random walk model. Let 7 be log
return and 7 be log price.7#* = 7 + 7#*
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The Random Walk Model
• Depending on the dependence structure in 7 , we have different
versions of random walk models. The most restrictive one is to assume 7 is an i.i.d. sequence. The least restrictive one is to assume 7 is a
white noise. While the most famous one is to assume 7 is a “martingale
difference sequence” (m.d.s.),(7#*|ℱ7) = 0, ∀
• In this case, the log price sequence 7 is called a “martingale”. This
model is consistent with the Effienct Market Hypothesis, as(7#* − 7|ℱ7) = 7#* ℱ7) − 7 = 0 7#* ℱ7) = 7
• The best forecast of tomorrow’s price is today’s price.