Discounting and Present Values (Week 1)

Question 1

Opportunity Cost

Answer 1

• Opportunity cost is the forgone benefit that would have been derived from an option not chosen.
• To properly evaluate opportunity costs, the costs and benefits of every option available must be considered and weighed against the others.

Question 2

How do you value investment opportunities? (simple terms)

Answer 2

• Consider the benefits
• Consider the costs (including opportunity costs)
• If the BENEFITS > COSTS then the investment will increase the firm’s value
• However, they need to be compared in the same terms and using the same metrics. They might not happen at the same time, therefore we need to align them.
• Lecture example: cannot compare USD and GBP directly. Convert both to SEK, then compare.

Question 3

Time Value of Money

Answer 3

• Time value of money means that a sum of money is worth more now than the same sum of money in the future. This is because money can grow only through investing.
• The formula for computing the time value of money considers the amount of money, its future value, the amount it can earn, and the time frame.

Question 4

Example: Put £100 in a bank with a 3% interest rate p.a.. What is the time value of money and why?

Answer 4

• (100 + (100*0.03) = £103
• TVM = £3.
• This is the interest the bank pays us for letting them have our money for 1 year.
• This is also compensation for our opportunity cost, as we cannot use that £100 for 1 year.

Question 5

Interest Rate/ Discount Rate

Answer 5

• How we convert money from one point in time to another (e.g. today vs. one year from now).

Question 6

Risk-free interest rate

Answer 6

• Interest rate at which money can be borrowed (or lent) without risk over a certain period.
• e.g. US government bonds: the government cannot default its debt

Question 7

How to calculate the Net Present Value:

Answer 7

NPV = PV (Benefits) - PV (Costs)

Question 8

Example of a NPV Calculation:
Discount rate = 10% p.a.
Cash flow Y1 = $300
Project Cost today = $200
Should we go ahead with the project?

Answer 8

NPV = -$200 + $300/(1+10%)
= -$200 + $300/1.1
= -$200 + $272.73
= $72.73
NPV > 0
Therefore - accept the project!

Question 9

Arbitrage

Answer 9

• The practice of instantaneously buying and selling equivalent goods in different markets to take advantage of price differences.
• No risk is taken on, no investment outlay is made.
• The resulting profits are riskless and called arbitrage profits.
• aka buy cheap, sell for more
• must be instantaneous and risk-free. e.g. poker is not arbitrage, holding and selling shares is not arbitrage

Question 10

Law of One Price

Answer 10

• If equivalent investment opportunities trade simultaneously in different competitive markets, then they must trade for the same price in both markets.
• Otherwise, there are arbitrage opportunities.

Question 11

Arbitrage bond example:
Assume a bond promises a risk-free payment of $1000 in one year. If the risk-free interest rate is 5%, what can we conclude about the price of this bond in a normal market?
Assume the price of the bond is actually $940. What is the arbitrage profit?

Answer 11

• PV($1000) = $1000/1.05 = $952.38
• Get a bank loan of +$952.38
• Purchase the bond for $940
• Arbitrage profit = $12.38
• In reality, the price of the bond will rise to $952.38 because of the opportunity for arbitrage

Question 12

Value Additivity Principle:

Answer 12

Price(C) = Price (A + B) = Price(A) + Price(B)

Question 13

Value Additivity Example:
Asset A: Price = $8.70, CF Y1 = $8, CF Y2 = $2
Asset B: Price = $8.30 , CF Y1 = $2, CF Y2 = $8
Asset C: CF Y1 = $10, CF Y2 = $10
What is the arbitrage-free price of Asset C?

Answer 13

• C’s cash flows for Y1 and Y2 = the sum of A and B
• Assets (A+B) = C
• the value additivity principle says that the arbitrage free price of C is therefore A+B
  = $8.70 + $8.30 = $17.00

Question 14

Value Additivity Example:
Asset A: Price = $8.70, CF Y1 = $8, CF Y2 = $2
Asset B: Price = $8.30 , CF Y1 = $2, CF Y2 = $8
Asset C: CF Y1 = $10, CF Y2 = $10
What if Asset C is trading for $16?

Answer 14

• It is relatively undervalued
• We should buy asset C, and sell A and B separately.
• Profit = $1

Question 15

Value Additivity Example:
Asset A: Price = $8.70, CF Y1 = $8, CF Y2 = $2
Asset B: Price = $8.30 , CF Y1 = $2, CF Y2 = $8
Asset C: CF Y1 = $10, CF Y2 = $10
What if Asset C is trading for $20?

Answer 15

• It is relatively overvalued.
• We should sell asset C, and buy assets A and B.
• The arbitrage profit is $3.