Lecture notes
Financial Economics and Capital Markets Autumn Term Lecture Notes 21/22
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- The University Of York (UOY)
Financial Economics and Capital Markets Autumn Term Lecture Notes 21/22
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Financial Economics and Capital Markets1. Arbitrage and Financial Decision Making
Valuing decisions
The valuation principle states that the benefits and costs of decisions should be evaluated using
competitive market prices and, when the value of the benefits exceeds the value of the costs, the
investment decision should be made.
The value of the benefit exceeds the value of the costs when the Net Value (NV) is positive.
In order to identify costs and benefits we may need help from other areas (e.g. Marketing,
Economics, Organisational Behaviour, Strategy, Operations, etc.).
We must always use competitive market prices to determine cash values; a competitive market is a
market in which goods can be bought and sold at the same price.
The time value of money
The time value of money principle is the idea that a pound today is worth more than a pound
tomorrow. The rate at which we can exchange money today fro money in the future is determined
by the current interest rate.
The risk-free interest rate or discount rate, rf, is the interest rate at which money can be borrowed or
lent without risk.
The interest rate factor is equal to 1 + rf, which means that £1 today is worth £1 + rf in the future
(and vice versa). The discount factor is equal to 1/(1 + rf), which means that £1 in the future is worth
£1/(1 + rf) today (and vice versa).
EXAMPLE 1 You have an investment opportunity with the following certain cash flows. The cost
today is £100 and the benefit in one year is £105. Suppose that the current risk-free interest rate is
7%. Should you take this opportunity?
£100 today is worth £100 + (0.07 x £100) = £107
The net future value is £105 - £107 = -£2 < 0
£105 in the future is worth 105/(1 + 0.07) = £98.10
The net present value is £98.10 - £100 = -£1.90 < 0
As both the net future value and net present value are less than zero, we should not make this
investment. If an investment has a negative net future value, it will also have a negative present
value, and vice versa. We will focus on calculating the net present value as we will need to specify
which point in the future we are referring to.
Net present value decision rule
When making an investment decision, we always take the alternative with the highest net present
value (NPV). The NPV of an investment is the present value of its expected cash inflows minus the
,present value of its costs. The NPV of not doing a project is always zero. When deciding whether to
accept or reject a project, we should accept it only if the NPV is positive.
Arbitrage and Law of One price
The law of one price states that if equivalent investment opportunities trade simultaneously in
different competitive markets, then they must trade for the same price in both markets. This is to
avoid arbitrage.
Arbitrage is the practice of buying and selling equivalent goods in different markets to take
advantage of a price difference. An arbitrage opportunity occurs when it is possible to make a profit
without taking any risk or making any investment; arbitrage opportunities should quickly evaporate
in the markets.
A normal market is a competitive market in which there are no arbitrage opportunities.
No-arbitrage and security prices
The no arbitrage condition allows us to compute the price of any security; the price of a security is
equal to the present value of all cash flows paid by the security. If this is not the case, we will see
arbitrage opportunities and there will be imbalances in the market, until the price of the security is
equal to the present values of the security’s cash flows. This implies that, in a normal market, the
NPV of buying or selling a security is zero.
EXAMPLE 2 A bond promises a risk-free payment of £1,000 in one year. If the risk-free interest rate
is 5%, what is the price of this bond in a normal market?
£1000/(1 + 0.05) = £952.38
If the price of the bond is £952.38 then the law of one price is satisfied and there are no arbitrage
opportunities.
What happens if the price of the bond is £940?
The opportunity for arbitrage will force the price of the bond to rise until it is equal to £952.38.
What happens if the price of the bond is £960?
The opportunity for arbitrage will force the price of the bond to fall until it is equal to £952.38.
Arbitrage and security prices
,Consider two securities, A and B. Suppose a third security, C, has the same cash flows as A and B
combined. In this case, security C is equivalent to a portfolio, or combination, of the securities A and
B. This is the idea of value additivity: if PV(C) = PV(A + B) then Price(C) = Price(A) + Price(B).
EXAMPLE 3 Bond A gives £100 tomorrow, bond B gives £50 tomorrow and bond C gives £150
tomorrow. Then in a competitive market Price(C) = Price(A) + Price(B).
Price of risk
EXAMPLE 4 You can invest in a bond that pays £1,100 next year or on a market index that pays
£1,400 if next year the economy will be strong and £800 otherwise. There is an equal probability
(0.5) of either a weak economy or strong economy. If the price of the two assets is the same, which
investment do you prefer?
The expected cash flow is calculated by E(C) = Σi p(i)C(i) where p(i) is the probability state of i and C(i)
is the cash-flow state of i.
The market index is calculated by E(C) = 0.5 x £800 + 0.5 + £1400 = £1100.
Both investments have the same expected value, but the market index has a greater amount of risk
(chance of strong or weak economy, whereas bond will always be £1100). This relates to the idea of
risk aversion: investors prefer to have a safe income rather than a risky one of the same average
amount.
If the price of the bond is £1058 and the price of the market index is £1000, how do returns of the
two investments compare?
The bond return is calculated by rf = (£1100 - £1058)/£1058 = 4%.
For the market index
• Market return:
o Strong economy (£1400 - £1000)/£1000 = 40%
o Weak economy (£800 - £1000)/£1000 = -20%
• Expected return (can be computed in two ways)
o Expected return E(R) = Σi p(i)C(i) = 0.5 x 40% + 0.5 x (-20%) = 10%
o Return of the expected cash flow: E(Rm) = (E(C) – P0)/P0 = (£1100 – £1000)/£1000 =
10%]
The expected return on the market index (10%) higher than the investment in the stock because the
latter has a risk-free interest rate
The expected return on the market index (10%) is higher than that of the stock (4%) because the
stock has a risk-free interest rate, and investors are rewarded (risk premium) for taking risk with the
market index.
The risk premium is the additional return that investors expect to earn to compensate them for a
security’s risk. When a cash flow is risky, to compute its present value we must discount the cash
flow we expect on average at a rate rs that equals the risk-free interest rate plus an appropriate risk
premium: rs = rf + (risk premium for investment s). If an investment has much more variable returns,
it must pay investors a higher risk premium.
, The price of risk principle is the idea that the cost of losing £1 in bad times is greater than the benefit
of an extra £1 in goods times. The risk of a security must be evaluated in relation to the fluctuations
of other investments in the economy. A security’s risk premium will be higher the more its returns
tend to vary with the overall economy and the market index. If the security’s returns vary in the
opposite direction of the market index, it offers insurance and will have a negative risk premium.
EXAMPLE 5 A risky stock gives a cash flow of £1500 when the economy is strong and £800 when the
economy is weak. Each state of the economy has an equal probability of occurring. An 8% risk
premium is appropriate for this particular stock. If the risk-free interest rate is 2%, what is the price
of the stock today?
The appropriate discount rate is rs = rf + rp = 2% + 8% = 10%. The expected cash flow of the stock in
one year is 0.5 x £1500 + 0.5 x £800 = £1150. The price of the stock today is £1150 x 1/1 + 0.1 =
£1045.45.
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