Investments
GRA6534 - Lecture 1
Giovanni Pagliardi
Welcome!
• Instructor
• Giovanni Pagliardi - PhD, ESSEC Business School Paris
• Email: giovanni.pagliardi@bi.no
• Office hours: Tuesday 13.00-14.00 and Wednesday 13.00-14.00, office B4-013
• This course
• 12 lectures
• 2/3 synchronous, 1/3 asynchronous
•
•
•
•
•
One event (BI Talks) in addition to the 12 lectures: Nicolai Tangen
Pre-recorded videos. To be watched after class
Two take-home projects in groups
Digital drop-in sessions
Non-mandatory sets of individual homework assignments
Determination of final course grade
• Mid-term test (counts 30%)
• On November 4th
• 24-hour, take-home, multiple choice test to be done individually
• The multiple choice will cover theory questions or short exercises related to two
projects
•
•
•
•
Projects are not mandatory but very strongly recommended
To be done in groups of 4-5 students. Please sign up on ItsLearning
A series of mentoring drop-in sessions will be offered at the students’ request
The midterm test will be based on the projects
• You will receive a letter grade for the midterm test
Determination of final course grade
• Final written (counts 70%)
• It includes three parts:
• Multiple choice
• Exercises
• True/False with explanation of the answer in maximum 5 lines
• You will receive a letter grade from the final exam as well
• The final course grade is determined as a weighted average of the two letter grades from
the midterm and final exams, conditional on the validation of the BMC
• Example 1: The student receives A in the 70% component and B in the 30%. Grade: A
• Example 2: The student receives B in the 70% component and A in the 30%. Grade: B
What you can expect of me:
• Help outside class: do not hesitate to ask for clarifications during office hours or at break
of classes! Emails are great too, but sometimes a bit more difficult to explain
• Help in class: do not be afraid to interact and ask questions, this increases everybody’s
learning and I’m happy to repeat many times the same concept if needed
• I provide solutions to all homework assignments and at least two past exams, but try and
do them without looking at the solutions first
Some pieces of advice…
• Attend classes and read lecture notes within two days if possible
• Do the projects and the assignments with the order suggested in class
• Do not focus on inserting numbers in formulas, but on UNDERSTANDING FINANCE!
Contents – Lecture 1
1. Overview of the course
2. Pricing different investments
3. Risk – expected return trade-off
Example with lotteries
4. Two ways of pricing
1. OVERVIEW OF THE COURSE
 Four different types of investments covered in this course:
a
b
BONDS
•
Bond pricing.
•
The term structure of
first computer
assignment will come
out.
related
c
HOUSING
d
•
•
The most important question to be answered is:
Can returns be predicted?
interest rates.
➞ After this point, the
EQUITY
FUNDS
(Alternative investments)
Difference between
hedge funds, mutual
funds, and closed-end
funds.
•
There is a strong similarity between equity returns
and housing returns.
For each type of
investment, the
pricing is done in
the same way.
2. PRICING OF DIFFERENT INVESTMENTS
• There are two essential approaches to price any kind of investments:
1
•
RISK-BASED EXPLANATION
Entirely based on the rationality of the investors,
2
•
BEHAVIORAL EXPLANATION
Tells you that investors are not necessarily always
rational.
i.e., the investors can process all the info they
have well, and in the end, they base their
For this course, we will
start with the
assumption that
investors are rational
and will see how to
price the assets to
reflect their risk.
•
There are other factors, like:
decision on the trade-off between risk and
— Overreaction
expected returns.
— Optimism
Example: It’s 2011 and you have a Norwegian
government bond and a Greek government bond.
Which one would you prefer? Hint: first look at
the risk, then set the price.
Example 1: Investors’ overreactions after news.
Example 2: CUBA fund (2014)
3. RISK – EXPECTED RETURN TRADE – OFF
• The expected return of an asset is the expected gain divided by the price:
𝔼𝔼 𝑟𝑟 =
𝔼𝔼𝑡𝑡 𝐺𝐺𝑎𝑎𝑖𝑖𝑛𝑛
𝑃𝑃𝑡𝑡
= 𝔼𝔼𝑡𝑡 𝑃𝑃𝑡𝑡+1 −𝑃𝑃𝑡𝑡 = 𝔼𝔼𝑡𝑡 𝑃𝑃𝑡𝑡+1 − 1
𝑃𝑃𝑡𝑡
𝑃𝑃𝑡𝑡
If we rearrange:
1 + 𝔼𝔼 𝑟𝑟 =
𝔼𝔼𝑡𝑡 𝑃𝑃𝑡𝑡+1
𝑃𝑃𝑡𝑡
Remember our goal: To find the price of our investment, so you can decide to buy or sell.
If we solve for Pt, we get the standard pricing formula:
𝔼𝔼𝑡𝑡 𝑃𝑃𝑡𝑡+1
𝑃𝑃𝑡𝑡 =
1+𝔼𝔼 𝑟𝑟
➜ The asset’s price today is the expected value of my cash flows tomorrow, discounted at a certain interest
rate.
•
•
If you want to find the price of an investment, you need to compute two quantities: 𝔼𝔼𝑡𝑡 𝑃𝑃𝑡𝑡+1
𝔼𝔼𝑡𝑡 𝑃𝑃𝑡𝑡+1 is the expectation of a future price (i.e., what would be the cash flow tomorrow?).
and 1 + 𝔼𝔼 𝑟𝑟 .
➜ You dont’t take into account any uncertainty around that expectation. Then, where are you taking into account
the risk in your computation?
➜ At the DENOMINATOR!
⇒ Risk must be incorporated in the expected returns (𝔼𝔼 𝑟𝑟 ).
Expected returns are only
determined by risk!
EXAMPLE WITH LOTTERIES
• Let’s try to show that there must be a relationship through which expected returns are only
determined by risk.
• Let’s suppose that we have three lotteries:
Lottery
Pay-off
Probability of
pay-off
Expected pay-off
(𝔼𝔼[𝑷𝑷𝒂𝒂𝒚𝒚𝒐𝒐𝒇𝒇𝒇𝒇])
A
$100
100%
$100
B
$0
or
$200
50%
$100
or
$300
50%
C
50%
50%
$100
$200
Question: Between B and C, which lottery would you prefer?
Question: Which lottery is riskier?
Remember that:
𝑃𝑃𝑡𝑡 =
𝔼𝔼𝑡𝑡 𝑃𝑃𝑡𝑡+1
1+𝔼𝔼 𝑟𝑟
Price lottery A:
Price lottery B:
100
𝑃𝑃𝑡𝑡 =
1 + 𝑟𝑟𝑓𝑓
𝑃𝑃𝑡𝑡 =
Where 𝑟𝑟𝑓𝑓 is the risk-free rate.
Therefore, lottery A is a riskless asset.
100
1 + 𝔼𝔼[𝑟𝑟]
Let’s assume investors are willing to pay $80 for
every $100 they are promised.
⇒ 𝑃𝑃𝑡𝑡 = $80
⇒ 80 =
100
1+𝔼𝔼[𝑟𝑟]
⇒ 𝔼𝔼 𝑟𝑟 = 100 − 1 = 𝟐𝟐𝟓𝟓%
80
Question: Between lottery A and B, which one would you prefer?
Note: 𝔼𝔼 𝑟𝑟𝐵𝐵 > 𝔼𝔼[𝑟𝑟𝐴𝐴]
A statement on expected returns is always a statement
on prices
When 𝔼𝔼 𝑟𝑟 increase (decrease) ⇒ prices decrease (increase)
• We are trying to explain what is driving investors’ expectations of future returns.
Expected
returns
≠
Realized
returns
This course will
try to model
expected
returns
Price lottery C:
• Let’s assume that standard deviation (sd) is our measure of risk.
• Note that 𝑠𝑠𝑑𝑑𝐵𝐵 = 𝑠𝑠𝑑𝑑𝐶𝐶
• If B and C have the same measure of risk and risk is the only thing that drives expected returns,
⇒ Lottery B and C have the same ex-ante expected returns in equilibrium.
• If you are willing to pay $80 to get $100 (lottery B), how much are you willing to pay in order to get
$200?
Answer: $160 = 𝑃𝑃𝐶𝐶
160 =
200
1 + 𝔼𝔼[𝑟𝑟𝐶𝐶 ]
⟹ 𝔼𝔼 𝑟𝑟𝐶𝐶 =
200
160
−1
⟹ 𝔼𝔼 𝑟𝑟𝐶𝐶 = 25%
Conclusion: You should be indifferent between lottery B and C.
➜ They have the same risk and the same expected return.
4. TWO WAYS OF PRICING (for any investment)
• Remember the standard way of pricing an asset: 𝑃𝑃𝑡𝑡 = 𝔼𝔼𝑡𝑡 𝐶𝐶𝐹𝐹𝑡𝑡+1
A
Incorporate risk in the discount rate
(denominator)
𝑝𝑝×𝐶𝐶𝐹𝐹𝑡𝑡+1,1+ 1−𝑝𝑝 ×𝐶𝐶𝐹𝐹𝑡𝑡+1,2
𝑃𝑃𝑡𝑡 = 1+𝑟𝑟 +𝑅𝑅𝑃𝑃
𝑓𝑓
1+𝔼𝔼 𝑟𝑟
B
Incorporate risk in the numerator
•
In this approach, we need to adjust the expectation
𝑀𝑀𝐾𝐾𝑇𝑇 +𝑅𝑅𝑃𝑃 𝑆𝑆𝑀𝑀𝐵𝐵 +⋯
with a different probability measure to account for
risk.
Risk premium
CAPM
Multifactor models: You consider all the possible risks
that might affect your price.
Caveat: You need to estimate them and not all of them
will be priced by the market.
•
You discount at a risk-free rate. So you have:
𝑃𝑃𝑡𝑡 =
𝑞𝑞×𝐶𝐶𝐹𝐹𝑡𝑡+1,1+ 1−𝑞𝑞 ×𝐶𝐶𝐹𝐹𝑡𝑡+1,2
1+𝑟𝑟𝑓𝑓
Where 𝑞𝑞 is the risk-neutral probability measure.
“Annuities and perpetuities”
•
LECTURE 1
Video 1 – Annuity: geometric series
Video 2 – Annuity: final formula
Video 3 – Annuity due extension
Video 4 - Annuity generalization
Video 5 – Perpetuity formula
Video 6 - Perpetuity due extension
Did you know there is a trick to derive the
formulas of the annuities and perpetuities
without learning them by heart?
 To know how, watch the videos!
Test it on your own
•
How would you compute the present value (t=0) of a
perpetuity that starts in the future?
•
Would you be able to derive the formula for the
present value of a growing perpetuity?
Hw1: Ex 1