H5
Geometric Average Versus
Arithmetic Average
Foundations of Finance
Prof. Eduardo Davila
The Question Suppose you invest $435 in a zero coupon bond for one year and earn a
return of 8%. You then reinvest the proceeds at 12% for a second year. How do you

H21
Note on Forward Rates
Foundations of Finance
Prof. Eduardo Davila
Setting
Suppose that
A 1-year ZCB has a Y T M of y (1) = 2%.
A 2-year ZCB has a Y T M of y (2) = 3%.
What is the forward rate for the second period? Assuming a face value of $1000,
th

H23
Duration: Formulas and Calculations
Foundations of Finance
Prof. Eduardo Davila
1
Definition
Pn
D=
2
Ct
t=1 (1+r)t
Pn
Ct
t=1 (1+r)t
t
.
Explicit Sample Calculations
a. For an 8% coupon (annual pay) four-year bond with a yield to maturity of 10% we ha

H24
Example of Immunization
Foundations of Finance
Prof. Eduardo Davila
1
The Problem
The idea behind immunization is to eliminate interest rate risk. Most companies balance
sheets are subject to interest rate risk because the duration of their assets and

H26
Numerical Example of Arbitrage When Calls
Violate the Minimum Value Bound Prior to
Expiration
Foundations of Finance
Prof. Eduardo Davila
1. Prior to expiration, the minimum value is
C max 0, S Xert .
(1)
2. Suppose S = $101, X = $100, r = 6%, t = 1

H27
Simple Numerical Arbitrage Example
Demonstrating Put-Call Parity
(Assuming r = 0 for simplicity)
Foundations of Finance
Prof. Eduardo Davila
1. Put-call parity states
C P = S Xert .
(1)
2. Assume S = $110, X = $100, r = 0, t can be any positive number

Foundations of Finance Technical Note Class 3
Eduardo Dvila
Manipulating FV = PV (1 + R)
T
Future value FV = PV (1 + R) T
Present value PV =
FV
(1+ R ) T
Yield/Interest rate R =
Time to maturity T =
= FV (1 + R)T
FV
PV
1
T
1
FV
log( PV
)
log(1+ R)
FV

H28
Cash-Futures Arbitrage
Foundations of Finance
Prof. Eduardo Davila
1
Terminology
If an investor enters into a long future position, he assumes the obligation to take delivery
of the underlying on the settlement date (date T ) at a price agreed upon wh

Foundations of Finance Technical Note Class 4
Eduardo Dvila
1
Return measures
1.1
Effective annual rate
1 + EAR =
1+
APR
m
m
The APR is the quoted rate.
Examples:
24% (annual) rate compounded yearly 0.24/1 = 0.24 is a 24% per year, EAR = 24%, m = 1
24%

Foundations of Finance Technical Note Class 19
Eduardo Dvila
1
Relation between yields and forward rates
We showed in class that
(1 + yt (n 1)n1 (1 + f t (n 1) = (1 + yt (n)n ,
where yt (n 1) is the yield today of the zero coupon bond that pays at n 1, yt

H20
An Example of Financial Engineering
Constructing a Forward Loan
from Spot Securities
Foundations of Finance
Prof. Eduardo Davila
Objective
You wish to guarantee that you will be able to lend the $1,000,000 you will inherit at the
beginning of next yea

H19
Summary of Yield Measures
Foundations of Finance
Prof. Eduardo Davila
1
Zeros
1. Yield to maturity (t = years to maturity):
a. Annual compounding = (F/P )1/t 1.
h
i
b. Semi-annual compounding = 2 (F/P )1/(2t) 1 .
c. Effective annual rate:
EAR =
semi-a

H14
Note on Internal Rate of Return
Foundations of Finance
Prof. Eduardo Davila
Setting Suppose we have the following cash flows (draw a timeline):
Time
Cash flow
Description
t=0
$39.63
Buy 1 Coca-Cola share
t=1
$1.12
Receive dividend
t=2
$45.42
Receive a

H3
Continuous Compounding:
Some Basics
Foundations of Finance
Prof. Eduardo Davila
Because you may encounter continuously compounded growth rates elsewhere, and because you will encounter continuously compounded discount rates when we examine the
Black-Sc

H6
Calculating Loss Probabilities
from Means and Standard Deviations:
An Example from the S&P 500
Foundations of Finance
Prof. Eduardo Davila
1. Modern portfolio theory identifies the risk and return on a portfolio with the mean and
standard deviation of

H7
Note on Covariance and Correlation
Foundations of Finance
Prof. Eduardo Davila
1. Theoretical Definitions
The covariance of two random variables, R1 and R2 , is defined as:
Cov (R1 , R2 ) = E [(R1 1 ) (R2 2 )] .
The covariance can be calculated as foll

H10
Portfolio Variance with
Many Risky Securities
Foundations of Finance
Prof. Eduardo Davila
Case 1: Non-systematic risk only
Recall that when the correlation between two securities equals zero, the portfolio variance
is given by:
p2 = 12 12 + 22 22 .
A

H8
Numerical Example of Creating a Zero Risk
Portfolio with Two Risky Securities Whose
Returns are Perfectly Negatively Correlated
Foundations of Finance
Prof. Eduardo Davila
Security
Good year
(P rob = 0.5)
Bad year
(P rob = 0.5)
16%
-2%
2%
12%
A
B
Table

H12
Excess Returns and Beta:
Deriving the Security Market Line
Foundations of Finance
Prof. Eduardo Davila
1. A First Risk-Reward Relationship
We showed that market forces combined with a search by investors for efficient portfolios would produce the foll

H15
Equity Valuation Formulas
Foundations of Finance
Prof. Eduardo Davila
1
The Dividend Discount Model
Suppose a stock with price P0 pays dividend D1 one year from now, D2 two years from
now, and so on, for the rest of time. P0 is then equal to the disco

H18
Calculating the Annual Return
(Realized Compound Yield)
on a Coupon Bond
Foundations of Finance
Prof. Eduardo Davila
Objective
To show that the annual return actually earned on a coupon-bearing bond will equal its yield
to maturity only if you can and

H11
Three Special Portfolios
Foundations of Finance
Prof. Eduardo Davila
We discussed in class how to form efficient portfolios of many risky assets. This note
serves to formulate the mathematical counterpart to this problem. In other words,
how do you go

H17
Arbitrage Handout
Foundations of Finance
Prof. Eduardo Davila
1
Introuction
Our working definition so far was: Arbitrage is the transaction of selling something at a
high price and simultaneously buying that same thing at a low price, without cash out

Foundations of Finance Technical Note Class 5
Eduardo Dvila
1
Statistics
Example with the probability distributions of returns of IBM and Dell:
Scenario
Probability
IBM
Dell
Recession
0.10
0.00
-0.10
Normal
0.50
0.10
0.10
Expansion
0.40
0.20
0.30
We can