Introduction - Week 1

Question 1

Some pearls of wisdom about theories and models:

Answer 1

• The purpose of theory is to predict and explain. A theory is not tested by the reasonableness of its assumptions but by its ability to predict accurately and explain.

Attribution Unknown

• A model is to be used, not to be believed.

Milton Friedman

Question 2

How can I know if a theory is good or bad ?

Answer 2

• A theory is good if it matches the real world well
• A theory is bad if it matches the real world poorly

It is necessary to have a fair degree of skepticism regarding theories in general: they are not the truth and what works today might not work tomorrow..

Question 3

Random variables and probabilities :

Answer 3

Risks and gambles are central to theoretical and practical finance. We will, therefore, rely on tools that describe gambles.

Question 4

Random variables and possibilities outcomes Setup:

Consider a random variable Xi ;

Answer 4

Question 5

Summation: if you want to write the sum of Z_i where i runs from 1 to K, we have;

Answer 5

Question 6

Product: similarly, if you want to write the product of Z_i where i runs from 1 to K, we define;

Answer 6

Question 7

What is The expected value or mean of X ?

(Def. + Formula)

Answer 7

Its probability-weighted average

The mean measures the “average” outcome from a gamble or variable.

Question 8

Why is the expected value or mean important for companies and investors?

Answer 8

• Investors are concerned with mean returns from portfolios.
• Companies care about expected values of cashflows from investment projects.

Question 9

The variance of a random variable measures …

Answer 9

The (probability weighted) dispersion of the outcomes around the mean.

The more (less) spread out a variable’s outcomes are around its mean, the larger (smaller) the variance.

Question 10

Formula of the variance:

Answer 10

Question 11

Why is the variance important for investors?

Answer 11

Investors care about the variance of portfolio returns (as this is a measure of portfolio risk).

• The less spread out a variable’s outcomes are around its mean, the smaller the variance. Therefore, the ROI amount will be around the expected value of the mean. Investors know what they’ll get. Fewer risks.
• The more spread out a variable’s outcomes are around its mean, the larger the variance. Therefore, the investors can expect higher returns (far above the mean) but also smaller return or losses (far under the mean). More risks.

Question 12

What is the Discount Rate?

Answer 12

Interest rate used to compute present values of future cash flows

Question 13

What is the Discount Factor?

Answer 13

Present value of a $1 future payment

Question 14

What is the Present Value?

Answer 14

Value today of a future cash flow

Question 15

What is the Future Value?

Answer 15

Amount to which an investment will grow after earning interest.

Question 16

A discount rate is…

Answer 16

The reward that investors demand for accepting delayed rather than immediate gratification.

Question 17

The discount rate is also called:

Answer 17

• Interest rate
• Required rate of return
• Opportunity cost of capital.

Question 18

What is the discount rate in real life?

Answer 18

• If you lend someone money for a year, you demand interest as you cannot instantly spend the money you have lent on consumption goods.
• If you lend money to a less trustworthy person/company, you require a greater interest rate as you’re less confident that you’ll get your money back.
• The discount rate is also called opportunity cost of capital because it is the return foregone by investing in a capital project rather than investing in freely-available securities.

Question 19

How many varieties of interest calculation are there?

Answer 19

2

Simple and compound interest.

Question 20

What is the simple interest?

Answer 20

The interest earned or paid is just the original balance of the deposit/loan (X) times the interest rate (r).

So over T periods, the total balance of your deposit/debt will grow to be: X + r × X × T =

X (1 + rT ).

Question 21

What is the Compound Interest?

Answer 21

Interest must be paid on previously charged/earned interest.

So over T periods, the total balance of your deposit/debt will grow to be:

X(1+r)^T

r=interest rate

T=time

Question 22

Compound versus simple interest at 10% per annum :

Answer 22

Clearly: compound interest is good for savers (bigger sum earned) but worrying for borrowers (higher monthly/annual payments)

Question 23

What type of question asks for the future value of X, assuming an interest rate of r and an investment period of T ?

Answer 23

We are given a cashflow of X today. What value will this cashflow grow to if invested at interest rate r for T periods?

FV (r,T) = X (1 + r )^T

• Obviously, the future value is larger if r is larger
• The future value is obviously also greater when T is greater or when X is greater.

Question 24

Future value: example

Answer 24

Question 25

How can we explain the Present Value ?

Answer 25

• Assume that you’re due to receive a payment of X in T years.
• What current cashflow is equivalent to that future cashflow?
• Compute the amount that you would have to place in a bank today such that it would grow to be exactly worth X in T years.
• Equivalently, how much would a bank lend you today if you promise to give them a repayment of X in T years.

Question 26

How to compute the Present Value?

Answer 26

Or X x (1 + r)^-T

Question 27

When computing present values we often make use of an object called a …

Answer 27

Discount Factor

• A discount factor is just the present value of £1.
• Discount factors vary with the interest rate and with the investment horizon.
• Higher interest rates lead to lower discount factors and longer investment periods lead to lower discount factors

Question 28

We calculate a discount factor as :

Answer 28

Question 29

Present values and discount factors: using the definition of the discount factor we can rewrite the present value as:

Answer 29

PV_(r,T) = DF_(r,T) × X

Question 30

Positive cashflows are :

Answer 30

Receipts

Question 31

Negative cashflows are :

Answer 31

Payments

Question 32

Until now we’ve thought about single cashflows (current or future) only.

Most investments or applications involve multiple cashflows received and paid at different points in time.

How do you compute present values for these more complicated streams of money?

Answer 32

This is called the Net Present Value or NPV.

Take each individual cashflow and compute its present value.

Sum present values across all of the cashflows.

(Receipts will contribute positively to NPV and payments will contribute negatively.)

Question 33

How to Calculate the NPV ?

Answer 33

(Assume a constant interest rate of r per period. Assume a investment project which will deliver a cashflow of C1 at the end of 1 period, C2 at the end of two periods, continuing until it finishes by delivering a cashflow of CK after K periods)

Question 34

The NPV rule:

Consider a financial or physical investment project for which you have calculated the NPV

Answer 34

• If the NPV is positive you should invest in the project.
• If the NPV is negative, you should turn down the investment opportunity.

Question 35

Comments on the NPV:

Answer 35

• The discount rate used in the NPV calculation should reflect the project’s risk: more risky projects require a greater return and so you should use a larger discount rate.
• If you don’t know the cashflows associated with the project precisely, use the expected value of each cashflow instead.

Question 36

Why is it optimal for individuals to invest in projects with positive NPV and discard projects with negative NPV?

Answer 36

It turns out that NPV is optimal (under some assumptions) in the sense that use of the rule leads to investors maximising their expected wealth.

This is true regardless of how patient or impatient an investor is, and thus the rule can be used for all investors (they will all agree on which investments to choose and which to discard).