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Mathematical Finance Presentation Handout
Abstract
This presentation will give a broad overview of financial mathematics with topics including,
measuring risk and return, methods of valuation for various assets, and portfolio optimization. I
will discuss the background of how these concepts are derived from a mathematical basis and
how they are implemented by investment professionals.
Background Information and Basic Concepts
What is finance and how is different from accounting?
Finance studies and addresses the ways in which individuals, businesses and organizations raise,
allocate and use monetary resources over time, taking into account the risks entailed in their
projects. (Wikipedia) Accounting is financial record keeping.
Time Value of Money
A basic assumption in finance is that a dollar received today is worth more than a dollar received
tomorrow. Most financial problems rely on this idea. The difficulty is often in how to apply this
concept.
Probability and Statistics
Because future events are uncertain, investors try to analyze past data to make predictions about
the future. This is somewhat effective, but has its limitations. Namely, market events can be
chaotic.
Presentation Outline
I.
II.
III.
IV.
V.
VI.
Introduction
Outline
Stocks
a. What is a stock?
b. Return
c. Risk
d. Risk vs. Return
e. Valuing a Stock
Bonds
a. What is a bond?
b. Pricing a bond
Financial Derivatives
a. What are derivatives?
b. How are derivatives valued?
Discussion
Helpful Formulas
n
1/ n
Average Return (geometric average) = (1  ri )
Arithmetic Average =
1 n
 ri
n i 1
i 1
n
1/ n
 1  ( ri )
i 1
APR = periodic rate * number of periods in a year
APY = (1 + periodic rate) ^ number of periods in a year - 1
Sharpe Ratio = (return – risk free rate) / standard deviation.
CAPM: E(R) = Rf + β(Rm-Rf)
β = Cov(Rm,R) / (σR)2 = Cor(Rm,R) * σRm / σR
2 asset portfolio variance: 2p = W1212 + W2222 + 2W1W212 where 12 = 12*1*2
n
P
i 1
C
M

t 1
v
(1  r ) (1  r )
(1  r ) (1  r ) n 1
v
Exercises
1. Use the data to answer the following questions.
Month
Price ($) % Change
1
100 100.00%
2
200 -50.00%
3
100 -50.00%
4
50 100.00%
5
100
a. What is the geometric mean of the monthly returns?
b. What is the arithmetic mean of the monthly returns?
c. Calculate the value in the fifth month letting the percentage change for each
month be the geometric mean.
d. Calculate the value in the fifth month letting the percentage change for each
month be the arithmetic mean.
2. Convert the following
a. 4% monthly rate to APR and APY.
b. 18% APR to APY assuming monthly compounding.
3. Price a 5 year bond with $100 face value, a semiannual coupon of 10% and a yield of 8%.
Comments, Criticisms, Suggestions, etc.