Brandon Groeger
April 6, 2010
Mathematics in Finance
Outline
I.
II.
III.
IV.
Stocks
a.
b.
c.
d.
e.
What is a stock?
Return
Risk
Risk vs. Return
Valuing a Stock
a.
b.
What is a bond?
Pricing a bond
a.
b.
What are derivatives?
How are derivatives valued?
Bonds
Financial Derivatives
Discussion
What is a stock?
Stock represents a share of ownership in a
company.
 Companies issue stock to raise money to run
their business.
 Investors buy stock with expectation of future
income from the stock. This is why stock has
value.
 Stocks can be publicly or privately traded.

Primerica Initial Public Offering (IPO)
Primerica, a financial services company had
their IPO this past Thursday, April 1st.
 The company sold 21.4 million shares for $15
a piece. The market price closed at around $19
per share.
 The market price of a stock is determined like
the price at an auction: it is a compromise
between the buyers and the sellers.

Measuring Stock Growth
Stock returns can be measured daily, weekly,
monthly, quarterly, or yearly.
 The geometric mean is used to calculate average
return over a period. n
1/ n
1/ n
n

Average Return =
(1  ri )
i 1

 1  ( ri )
i 1
The arithmetic mean is used to calculate the
expected value of return given past data.
1 n
ri

n i 1
Monthly Returns for Google
Date
Open Mnth Rtrn Index
Total Rtrn
4/1/2009 343.78
14.91%
1.15
14.91%
5/1/2009 395.03
6.00% 1.21802
21.80%
6/1/2009 418.73
1.31% 1.23393
23.39%
7/1/2009 424.2
5.79% 1.30531
30.53%
8/3/2009 448.74
2.44% 1.33713
33.71%
9/1/2009 459.68
7.25% 1.43406
43.41%
10/1/2009
493
8.94% 1.56228
56.23%
11/2/2009 537.08
9.51% 1.71077
71.08%
12/1/2009 588.13
6.60% 1.8237
82.37%
1/4/2010 626.95
-14.73% 1.55506
55.51%
2/1/2010 534.6
-1.01% 1.53936
53.94%
3/1/2010 529.2
7.96% 1.66196
66.20%
4/1/2010 571.35
Average Monthly Return
Expected Value of Monthly Return
4.32%
4.58%
Exercise 1
1a. Geometric Mean = [(1+1)(1+-.5)(1+.5)(1+1)]^(1/4) - 1 = 1 – 1 = 0
 1b. Arithmetic mean = [1 + -.5 + -.5 +1] / 4 =
1/4 = 25%
 1c. $100 * (1 + 0)^4 = $100 = true value
 1d. $100 * (1 + .25)^4 = $244.14

Converting Rates Between Periods
APR or annual percentage rate = periodic rate
* number of periods in a year
 APY or annual percentage yield = (1 + periodic
rate) ^ number of periods in a year - 1
 APR ≠ APY
 Example: 1% monthly rate has a 12% APR and
a 12.68% APY

Exercise 2

2a.
 APR
= 4% * 12 = 48%
 APY = (1.04)^12 - 1 = 60.1%

2b.
 18%/12
= 1.5% monthly
 (1.015)^12 - 1 = 19.56%
Risk
Risk is the probability of unfavorable
conditions.
 All investments have risk.
 There are many types of risk.

 Specific
 Risk
associated with a certain stock
 Market
 Risk
risk
risk
associated with the market as a whole
Measuring Risk


Standard Deviation (σ) of
returns can be used to
measure volatility which is
risky.
Standard Deviation 10.78%
Arithmetic Mean
2.49%
 Geometric Mean
1.93%

Standard Deviation



Assume a normal distribution.
68% confidence interval for monthly return: Mean ± σ =
(-8.29%, 13.27%)
95% confidence interval: Mean ± 2σ = (-19.07%, 24.05)
Risk vs. Return




Investors seeks to
maximize return while
minimizing risk.
Sharpe Ratio =
(return – risk free rate)
/ standard deviation.
Can be computed for an
individual asset or a
portfolio.
The higher the Sharpe
ratio the better.
Capital Asset Pricing Model (CAPM)
E(R) = Rf + β(Rm-Rf)
 E(R) = expected return for an asset
 Rf = risk free rate
 β = the sensitivity of an asset to change in the
market. It is a measure of risk
 Rm = the expected market return
 Rm - Rf = the market risk premium

Calculating Beta

β = Cov(Rm,R) / (σR)2 = Cor(Rm,R) * σRm / σR
 σR
is the standard deviation of the asset’s returns
 σRm is the standard deviation of the market’s
returns
 Cov(Rm,R) = covariance of Rm and R
 Cor(Rm,R) = correlation of Rm and R

In practice beta can also be calculated through
linear regression.
Implications of CAPM
Higher beta stocks
have higher risks
which means that the
market should
demand higher
returns.
 Assumes that the
market is efficient or
equivalently perfectly
competitive.

Portfolio Diversification

Two assets are almost always correlated due to
market risk.

Equivalently, 2p = W1212 + W2222 + 2W1W212
where 12 = 12*1*2
 This statistical result implies that the variance for
a two or more assets is not equal to the sum of
there variances which implies the risk is not equal
to the sum each assets risk.
 In general two assets have less risk than one
asset.

Diversification example
Suppose stock A has average returns of 5% with a
standard deviation of 6% and stock B has average
returns of 8% with a standard deviation of 10%.
The correlation between stock A and B is .25.
What is the expected return and standard
deviation (risk) of a portfolio with 50% stock A and
50% stock B.
 E(Rp) = .5 * .05 + .5 * .08 = 6.5%
 V(Rp) = .52 * .062 +.52 * .12 + 2 * .5 * .5 *.25 *
.1 * .06 = .00415
 σRp = V(Rp) ^.5 = 6.44%

Diversification Graphs
Methods of Valuation for stock
Discounted Cash Flow Analysis uses the
assumptions of time value of money and the
expected earnings of a company over time to
compute a value for the company.
 Relative Valuation bases the value of one company
on the value of other similar companies.
 For a publicly traded company the market value of
the company is the market stock price multiplied
by the number or shares outstanding.

Relative Valuation Example
Royal Dutch Shell and Chevron Corporation are
comparable companies.
 Royal Dutch Shell is trading at $59.26 per share
and has a P/E ratio of 14.52
 Chevron Corporation has earnings per share of
$5.24
 What would you expect for the price of a share of
Chevron Corporation?
 P/E * EPS = 14.52 * $5.24 = $76.08
 Chevron is actually trading around $77.55

What is a bond?
A Bond is “a debt investment in which
an investor loans money to an entity (corporate
or governmental) that borrows the funds for a
defined period of time at a fixed interest rate.”
(Investopedia)
 Bonds can be categorized into coupon paying
bonds and zero-coupon bonds.
 Example: $1000, 10 year treasury note with 5%
interest pays a $25 coupon twice a year.

Bond Terminology





Principal/Face Value/Nominal Value: The amount of
money which the bond issuer pays interest on. Also the
amount of money repaid to the bond holder at maturity.
Maturity: the date on which the bond issuer must repair
the bond principal.
Settlement: the date a bond is bought or sold
Coupon Rate/Interest Rate: the rate used to determine
the coupon
Yield: the total annual rate of return on a bond,
calculated using the purchase price and the coupon
amount.
Bond Pricing

Based on time value of money concept.

C = coupon payment
n = number of payments
i = interest rate, or required yield
M = value at maturity, or par value
Exercise 3

Price a 5 year bond with $100 face value, a
semiannual coupon of 10% and a yield of 8%.

Price = 5 * [1 - [1 / (1+.08/2)^10]] / (.08 / 2)
+ 100 / (1 + .08/2)^10 = $108.11
Other Considerations for Bond Pricing


The previous examples have assumed that the bond is
being priced on the issue date of the bond or a coupon
pay date, but a bond may be bought or sold at any time.
Pricing a bond between coupon periods requires the
following formula where v = the number of days between
settlement date and next coupon date.
n
C
M
P

v
t 1
v
n 1
(
1

r
)
(
1

r
)
(
1

r
)
(
1

r
)
i 1

Some bonds use a convention of 30 days per month,
other bonds use the actually number of days per month.
Types of Bonds that Affect Valuation
Municipal bonds pay coupons that are often
exempt from state or local taxes. This makes them
more valuable to residents but not to others.
 Some bonds have floating interest rates that
makes them impossible to accurately price.
 Some bonds have call options which allow the
bond issuer to buy back the bond before maturity.
 Some bonds have put options which allow the
bond holder to demand an early redemption.

What is a Financial Derivative?
A derivative is a financial instrument that
derives it value from other financial
instruments, events or conditions. (Wikipedia)
 Derivatives are used to manipulate risk and
return. Often to hedge, that is to say generate
return regardless of market conditions.
 Derivatives are bought and sold like any other
financial asset, relying on market conditions to
determine pricing.

Examples of Derivatives

Options



Future


A contract between two parties to buy and sell a commodity at a
certain price at a certain time in the future.
Swaps


A call gives the buyer the right to buy an asset at a certain price in
the future.
A put gives the buyer the right to sell an asset at a certain price in
the future.
Two parties agree to exchange cash flows on their assets.
Collateralized Debt Obligations (CDO’s)

Asset backed fixed income securities (often mortgages) are
bundled together and then split into “tranches” based on risk.
Valuing Derivatives
Valuing derivatives can be difficult because of
the many factors that effect value and because
of high levels of future uncertainty.
 Stochastic Calculus is used frequently.
 For very complex derivatives Monte Carlo
simulation can be used to determine an
approximate value.
 Many people are critical of complex derivatives
because they are so difficult to value.

Discussion
What else would you like to know about finance
or the mathematics of finance?
 Is it possible to find a “formula” for the stock
market?
 What regulations should financial markets
have? Why?
