EC371 Economic Analysis of Asset Prices
Topic #1: Asset Markets and Asset Prices
R. E. Bailey
Department of Economics
University of Essex
Outline
Contents
1
Capital Markets
2
2
Asset Price determination: an introduction
3
2.1
Asset prices, continued . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.2
The role of expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
3
Indicators
6
4
Performance risk, margins and short selling
8
5
Arbitrage
5.1
6
11
Arbitrage examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Asset market efficiency
12
Reading:
Necessary: Economics of Financial Markets, chapter 1
Additional reading:
Keynes, J. M. The General Theory of Employment, Interest and Money, chapter 12. This chapter,
strange though it may seem, merits repeated reading. It raises many important issues in financial
markets and portfolio decision-making – issues that are too commonly neglected. You will find no
‘answers’, but that’s not the purpose, which is rather to raise your critical awareness to the points of
weaknesses in much else that you will read in finance.
Shiller, R. J. Irrational Exuberance, chapters 1, 3.
Beware: Weasel Words1
Three words that you should treat very carefully in finance: arbitrage (e.g. ‘arbitrage opportunity’), rational (e.g. ‘rational investors’, ‘rational expectations’), efficiency (e.g. ‘efficient markets’). These words are so lazily used that you should make sure you always define them precisely
to avoid ambiguity. Always.
1
Weasel word: “an equivocating or ambiguous word which takes away the force or meaning of the concept being
expressed” OED
1
1
Capital Markets
The main capital markets
1. Equity (stock) markets.
• Primary - new issue market
• Secondary - trade existing shares
The assets traded in stock markets are the shares of individual companies, e.g. for ‘Tesco’.
Much of the discussion is, however, is about stock market indexes, such as the FTSE100
or the S&P500. There are many such indexes, which are essentially averages of individual
companies’ share prices. Many of them can be interpreted as expressing the market value
of a portfolio of shares, with the composition of the portfolio reflecting the weights for the
constituent companies in the index.
If you are unfamiliar with the construction of stock market indexes, read Appendix 1.1,
pp. 24–28 of The Economics of Financial Markets.
2. Bond markets (mostly government and corporations)
3. Money markets (short-term borrowing & lending)
4. Commodity markets (commonly as futures markets)
5. Physical asset markets (e.g. real estate)
6. FOREX markets (currency markets)
7. Derivative markets
• Bundles of assets (e.g. mutual funds and ETFs)
These ‘bundles’ are essentially pre-packaged portfolios, which take different forms according to the rules governing their operation:
– Closed-end mutual funds (or Investment Trusts) are companies that exist to hold
the securities – typically equities and corporate bonds – of other companies: they
themselves issue equity and debt.
– Open-end mutual funds exist to hold marketable securities – typically equities and
corporate bonds – but they shares they issue are not traded in secondary markets:
investors in open-end funds trade directly with the fund managers (who sell and
redeem shares in the fund at their market values).
– Exchange Traded Funds (ETFs): similar to closed-end mutual funds except that it is
possible to purchase or redeem ‘creation units’ from the fund.2 ‘Creation units’ are
bundles of assets that exactly replicate the portfolio of each ETF. Hence, the market
value of ETFs in the secondary market closely matches the value of the underlying
portfolio that each ETF comprises (otherwise it would be profitable to trade creation
units with the ETF). This is not the case for closed-end mutual funds, which are not
obliged to buy/sell creation units: hence the value of their equities can – and does
– diverge from their net asset values (the equities of most closed-end fund typically
trade at a significant discount).
ETFs are particularly attractive for professional investors as a result of their liquidity (active secondary markets, with low commission charges) and tax advantages
in many jurisdictions.
2
Index-tracker ETFs, for example, are funds designed to match the portfolio implicit in a stock-market index, e.g.
FTSE-100.
2
• Forward agreements (most importantly as futures markets)
• Options markets
2
Asset Price determination: an introduction
Asset price determination: overview
Price
6
Supply
p∗
Demand
Q
Stock
Figure 1: Asset price determination for a single asset
• Prices are determined by supply and demand for STOCKS.
• At each date, total stock is fixed.
• Quantity held depends on price.
• Price adjusts so that investors, in the aggregate, are prepared to hold the existing stock.
2.1
Asset prices, continued
Asset price determination, continued
What determines demand to hold the asset?
1. preferences
2. price
3. budget (wealth) constraint.
Multiple asset markets:
• Stocks are constant at each instant.
• Prices adjust so that stocks are willingly held.
• As time passes, so do asset stocks, beliefs and preferences.
3
2.2
The role of expectations
The role of expectations
Payoff minus Price
Price
By convention, rates of return are almost always quoted as percentages per annum, that is measured as if the return is to be received for twelve months. This is irrespective of whether the asset
is held for one day, one month, or ten years, let alone a year. Thus, for example, ‘5%’ should be
interpreted as the growth from 100 to 105 if the asset is held for one year, irrespective of how long it
is actually held.
Rate of return ≡
Adjusting for periods different from a year can require some messy calculations, but introduces
no issues of principle, hence is ignored in most of this module (which is not intended to teach you
arithmetic).
Measuring rates of return:
EC371 uses the simplest method of measuring rates of return, as described, above. Other methods – neither better nor worse – are available. The commonest of these (widely used in finance) is
that of ‘continuous compounding’. To see what this means, write the rate of return, defined above,
as r:
v−p
r=
p
where v is the asset’s payoff (at a date ‘one period’, say a year) in the future and p is its price today.
Implicitly this assumes that the whole of the increment v − p is received at the very last instant of
the period. But the increment could have accumulated in an infinity of different patterns during the
period. One possibility is to assume that the increment accumulates continuously at a constant rate.
This rate, known as the ‘force of interest’ – denote it by ρ – is defined as follows:
Force of interest: ρ ≡ ln(v) − ln(p)
where ln(·) is the natural logarithm operator. Thus, the relationship between r and ρ is: ρ = ln(1+r),
or 1 + r = eρ . If r is small, the two are close, e.g. for r = 4%, ρ ≈ 3.922%.
For extensions and details, see Economics of Financial Markets, pp. 29–31.
Note: unless you are told otherwise, in EC371 the ‘rate of return’ should be interpreted to mean
r = (payoff − price)/price = (v − p)/p.
For investors, rates of return are forward looking: depend on future payoffs, hence uncertain.
Observed market prices play two roles:
• prices reflect opportunity cost
• prices convey information
Information ⇒ Beliefs ⇒ Expectations
Rational expectations supposedly reflect the “true” economic model.
4
But who can know the “true” model?
Beware of “rational expectations” – it’s not much help
“Rational expectations” is a concept that should be treated with great care. Why? Because it
is ambiguous (used differently by different people) and potentially misleading. Always, always be
sure to explain what you mean if you use the term in any written work.
Almost always, “rational expectations” is interpreted to imply that decision makers (e.g. investors, or ‘economic agents’ more generally) do not make systematic errors.
This is a good start but many sorts of behaviour (‘models’) are consistent with not making
systematic errors. In other words: not making systematic errors from what benchmark (or: ‘model’s
prediction’)?
An infinity of models could be dreamt up. How to choose among them? A common assumption
is to assume that decision makers use the ‘true’ model. But what’s ‘true’? Sometimes it is supposed
that the true model represents Nature – how the world works. But this does not address the question
of how anyone, let alone model builders, can possibly know Nature’s model.
Inevitably, it is necessary to propose (with supporting arguments) criteria that any model should
satisfy. Very often the crucial criterion is that the model should ‘fit the facts’ (be consistent with the
evidence). On this criterion, some models are undoubtedly favoured over others, though very seldom
does any single model dominate the rest. Even then, new evidence may undermine its dominant
status. The ‘best’ model cannot be guaranteed to retain its status.
Another criterion, especially associated with ‘rational’ behaviour is that decision makers behave
according to particular principles that define ‘rationality’. This the so-called ‘a priori’ approach,
usually invoked in studies of ‘efficiency’ for example. The problem here is that there are many ways
to characterise rationality. Don’t be fooled by anyone who insists on specific criteria for rationality.
You’ll be able to find other proponents of equally plausible criteria. There is no escape from examining each set of assumptions (principles, or criteria) on its merits, according to the problem under
analysis.
Behavioural finance allows for many sorts of unusual behavior – usually interpreted to represent
departures from ‘rationality’.
Keynes’s illustration about expectation formation:
. . . professional investment may be likened to those newspaper competitions in which
the competitors have to pick out the six prettiest faces from a hundred photographs,
the prize being awarded to the competitor whose choice most nearly corresponds to the
average preferences of the competitors as a whole; so that each competitor has to pick,
not those faces which he himself finds prettiest, but those which he thinks likeliest to
catch the fancy of the other competitors, all of whom are looking at the problem from
the same point of view. It is not a case of choosing those which, to the best of one’s
judgement, are really the prettiest, nor even those which average opinion genuinely
thinks the prettiest. We have reached the third degree where we devote our intelligences
to anticipating what average opinion expects average opinion to be. And there are some,
I believe, who practise the fourth, fifth and higher degrees.
(Keynes, General Theory, p. 156)
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3
Indicators
Summary measures of asset prices and returns
• Among the common indicators of asset prices or returns are:
1. Net Present Value – to predict asset prices
2. Price/Earnings ratio – simple to understand and compute
3. VIX – a fear index, measuring the variability of asset prices
Net Present Values – NPV
• It is often supposed that asset prices should equal the Net Present Value of payoffs on the asset.
• Simplest form: asset with a single payoff, one-period in the future:
Asset value =
Payoff
vt+1
=
(1 + ‘interest rate’)
(1 + r)
– Here the ‘asset value’ can be understood as a prediction of the asset’s market price or
what the asset price ‘should’ be, although the sense of ‘should’ here is ambiguous.
– ‘Present Discounted Value’ (PDV) is synonymous with NPV here: r is the rate at which
1
future payoffs are ‘discounted’. The ratio (1+r)
is commonly called the ‘discount factor’,
corresponding to the ‘discount rate’ r.
– How can the NPV (PDV) relationship be justified? There are several possibilities:
1. An identity:3 Re-arrange the definition of rate of return, where the ‘interest rate’
equals the rate of return, r, and the asset value equals the asset’s market price, pt :
r=
vt+1 − pt
pt
=⇒
pt =
vt+1
(1 + r)
Although suggestive, a single expression (no matter how expressed) cannot be used
to determine both r and pt : either market price is given and the expression defines
rate of return; or, the ‘interest rate’ is given and the expression determines the asset’s
value.
2. Arbitrage: the approach in EC371 interprets the NPV to be an implication of the
arbitrage principle (Absence of Arbitrage Opportunities). In this interpretation, the
arbitrage principle predicts that r is the same for all assets whose future payoffs are
known. Given r, then each asset’s price is determined to satisfy the NPV relationship.
3. Cost of capital: in this interpretation the ‘asset value’ is calculated for a given r,
the cost of capital, without reference to the Arbitrage Principle. But, without the
Arbitrage Principle, some other justification is needed to link the ‘asset value’ with
the asset’s market price, pt .
4. Internal rate of return: the objective here is to take market price (or any other measure of ‘asset value’) as given, then to infer a measure of the asset’s rate of return,
without any necessary relationship with a market interest rate. This approach is
commonly used for bonds, in order to calculate the bond’s ‘yield to maturity’.
3
A tautology, true by definition and with no predictive value
6
• More generally, with a sequence of N payoffs:
Asset value =
vt+3
vt+N
vt+1
vt+2
+
+ ··· +
,
+
2
3
(1 + r) (1 + r)
(1 + r)
(1 + r)N
Here vt+1 , vt+2 , vt+3 . . . , vt+N denote future payoffs. Note that the discount rate, r, is assumed constant over time: rates could be allowed to differ across time (the notation is more
complicated, that’s all).
Where the asset is corporate equity (ordinary shares, or common stock) usually N −→ ∞,
because most companies are of indefinite lifespan, hence, in principle, could exist forever –
for equity the stream of payoffs is then the stream of dividends paid to shareholders. While
this is a common assumption, it does raise serious issues, to be considered later in EC371.
• The challenge: how are vt+i and r to be estimated?
Quite apart from the question of how NPV is to be justified in principle, any application requires estimation of an asset’s future payoffs and the discount rate. Except in the simplest
cases, these are not known in advance. Hence, a theory (model, set of assumptions) is needed
to explain their determination.
Price/Earnings ratio
• Measured either as Price/Earnings, or Earnings/Price where ‘Price’ = asset price & ‘Earnings’
≈ Profits
• Attraction: estimation of the PE ratio is straightforward.
• But: how can the PE ratio be justified (interpreted)?
• The ratio Earnings/Price a simple indicator of ‘rate of return’.
One justification for the PE ratio is to apply NPV, assuming constant payoffs, v, and discount
rate r, with N −→ ∞, so that:
1
1
1
v
Asset value = v
+
+
+ ... = ,
2
3
(1 + r) (1 + r)
(1 + r)
r
applying the rule for the sum of an infinite geometric sequence.
Now interpret price as ‘asset value’, v as earnings and r as ‘rate of return’, and re-arrange to
give the Earnings/Price ratio. Clearly, the assumptions needed for this justification are strong
– i.e., unrealistic.
Another justification for the PE ratio also applies the NPV: assume now that the payoff,
vt+1 , is interpreted as the sum of ‘dividends’ (= Earnings), dt+1 , plus the asset’s market price
at t + 1. More crucially assume that the price at t + 1 equals today’s price p = pt = pt+1 .
Hence:
pt =
dt+1 + p
vt+1
Earnings
dt+1 + pt+1
=⇒ p =
=⇒ Rate of return =
=
(1 + Rate of return)
(1 + Rate of return)
p
Price
Again, to assume that the asset’s price does not change is strong, i.e., unrealistic.
In summary: the PE ratio is a handy indicator, that’s all.
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VIX – a ‘fear’ index
• The VIX index is a measure of the variability of stock market prices – hence an index of ‘fear’.
A common measure of the magnitude of fluctuations in asset prices is the variance (or its
square-root, the standard deviation) in price. It is a ‘second moment’ in contrast to indicators
such as NPV and PE which measure average values – ‘first moments’. The VIX is an index of
‘fear’ in that investors tend to regard variability of prices as a ‘bad’ to be avoided.
The VIX is just one of these measures based on Standard & Poor’s 500 index of US stock
prices. It was devised for, and is reported at, the Chicago Board Options Exchange. For details
see: http://www.cboe.com/micro/vix/vixintro.aspx
Indicators that use the VIX methodology are now available for various other stock market
indexes.
• Forward looking – intended to reflect expected future volatility of asset prices.
The VIX index, and others like it, are constructed as forecasts of the variability in asset prices
over a fortcoming period, typically three months, rather than estimates of variability in past
months or years.
• Calculated from observed prices of options to buy (or sell) the asset for a stipulated strike price
at a future date
Given that future values of asset prices are unobserved, measures of volatility based on their
values are necessarily indicators of what happened in the past, which may or may not be
relevant for the future.
Option prices, however, reflect investors’ expectations of asset price variability during the
life of the option (before it expires). An option holder has essentially purchased ‘protection’
against future asset price fluctuations. Hence, the price of this protection – the option premium – can be interpreted to reflect expected asset price variability, i.e., the option premium
is a function of volatility (among other things such as the current asset price, the time to the
option’s expiry date, the option strike price and the market interest rate. Options are studied in
detail in EC372, not EC371.
• The index is a measure of implicit volatility because it is based, not on asset prices themselves
but is implicit in option prices.
Measures of asset price variability based on option prices are implicit because the estimates
are indirect, unlike measures based on asset prices themselves, which are explicit estimates of
volatility. Note that both implicit and explicit volatility seek to measure the variance of asset
prices, not of option prices themselves.
4
Performance risk, margins and short selling
Performance risk and margins
• Two categories of risk:
1. price risk: future prices are unknown.
2. performance (counterparty) risk: default on contract
• Margins: good faith deposits (collateral).
8
• “The margin”: proportion of an investment paid for in advance
• Purpose: a mechanism for minimising performance risk.
• Margins also allow leveraged asset purchases or sales
• Counterparty default was a central feature of the Credit Crunch
– failure to comply with margin calls results in bankruptcy
– which has a cascading impact across the whole financial sector
– when lenders find that their collateral becomes worthless
Buying on margin
• Buying on margin: investor pays a fraction of assets’ cost in advance (borrows remainder from
broker).
• Later, either (i) takes delivery and pays remainder; or (ii) sells the asset
collateral minus loan
collateral
collateral = market value of the asset
loan = amount borrowed from broker.
• Actual margin =
• Initial margin: actual margin required at the outset
• Maintenance margin: if the actual margin falls below this, a margin call is triggered
• Margin call: demand to deposit funds in the margin account
Example: buying on margin
An investor, A, instructs a broker, B, to purchase 100 shares of company XYZ when the market
price is $10 each. Suppose that A and B have an arrangement whereby A’s instructions are carried
out so long as B holds a margin of 40% of the transaction value. Hence, A makes an immediate
payment of $400 and B has effectively loaned A $600.
B holds the shares as collateral against the loan to A.
Sooner or later, A either (a) takes delivery of the shares (and pay B an additional $600 plus interest
and commission fees), or (b) instructs B to sell the shares (and repay the loan from B).
The margin agreement works smoothly so long as XYZ ’s share price increases above $10.
But suppose that the price falls, say, to $5. Now A owes B more than the value of the collateral,
$500. If the shares are sold, and if A does not pay B an additional $100 (plus transaction costs),
then B loses out. To guard against this potential loss, margin accounts may require replenishing. If
A does not provide additional funds when requested, then B might sell some or all of the shares to
avoid realizing a loss.
Suppose the share price falls to $5, A deposits an extra $300, thereby reducing the loan to $300, and
restoring the actual margin to its initial level of 40% = (500 − 300)/500.
Trigger prices for buying on margin: these are prices for which the investor (who purchased the
asset on margin) receives a margin call, i.e. the range of prices for which the actual margin falls
below the maintenance margin.
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To calculate the range of trigger prices, find the price at which the actual margin equals the
maintenance margin. Then all prices below this level are trigger prices because they would trigger a
margin call. In the above example, the actual margin equals
Actual margin =
Collateral − Loan
100p − 600
=
Collateral
100p
where p is the market price. Notice that at the transaction price, p = 10, so that the initial margin
equals the actual margin, 40%.
Suppose that the maintenance margin had been set at 25% (this would be part of the contract
agreed at the outset). Hence the market price, say pb, at which the actual margin equals the maintenance margin is given by:
Maintenance margin = 0.25 =
100b
p − 600
= Actual margin,
100b
p
which gives pb = 8. Thus the range of trigger prices is p < $8: any price below $8, triggers a margin
call for the investor to replenish funds in the margin account.
Short sales
• Short Sales: steps
1. borrow an asset, sell it.
2. Later, repurchase the asset and return to lender.
collateral minus loan
loan
collateral = funds (cash) in the margin account
loan = market value of asset short-sold.
• Actual margin =
• Initial and maintenance margins: as for buying on margin
• Market regulations often restrict which investors can short sell, and conditions under which
they can short sell
• Naked short sale: short seller does not borrow the asset, intending to repurchase before ‘settlement’ (usually a few days) – an illicit practice in many jurisdictions
Example: Margins with Short Sales
Investor A has an agreement with broker B which allows A to make short sales of company XYZ ’s
shares (the shares might be borrowed from B’s own portfolio or from the portfolio of one of B’s
other clients).
Suppose that A instructs B to short-sell 100 shares at a market price of $10 each.
B holds the proceeds, $1000 in A’s margin account and also demands an additional deposit of, say,
$400. Thus the initial margin is 40% in this example.
Sooner or later A will return the borrowed shares by instructing B to purchase 100 XYZ shares at
the ruling market price. If the price has fallen below $10, then A makes a profit (after allowing for
the deduction of B’s commission and other expenses, such as a fee for the loan of the shares).
However, if the share is purchased at a price above $10, then A makes a loss — a loss which might
be so large that an additional payment has to be made to B.
10
Suppose that the shares are re-purchased at a price of $16. Then A has to pay another $200 (plus
transaction costs) to B. If A defaults, then B makes a loss. To guard against possible loss margin
deposits are adjusted by margin calls in a similar way to when shares are purchased on margin.
If the share price rises to $16 and the short sale remains in place, a variation margin of $840 would
restore the actual margin to its initial value, 40% = (1400 + 840 − 1600)/1600. (Once again, the
rules for margins are prescribed by the relevant regulatory authorities. The detailed rules differ from
market to market.)
Trigger prices for short sales: these are prices for which the investor (who short-sold the asset)
receives a margin call, i.e. the range of prices for which the actual margin falls below the maintenance
margin.
To calculate the range of trigger prices, find the price at which the actual margin equals the
maintenance margin. Then all prices above this level are trigger prices because they would trigger a
margin call. In the above example, the actual margin equals
Actual margin =
Collateral − Loan
1000 + 400 − 100p
=
Loan
100p
where p is the market price. Notice that at the transaction price, p = 10, so that the initial margin
equals the actual margin, 40%.
Suppose that the maintenance margin had been set at 25% (this would be part of the contract
agreed at the outset). Hence the market price, say pb, at which the actual margin equals the maintenance margin is given by:
Maintenance margin = 0.25 =
1400 − 100b
p
= Actual margin,
100b
p
which gives pb = 14/1.25 = 11.20. Thus the range of trigger prices is p > $11.20: any price above
$11.20, triggers a margin call for the investor to replenish funds in the margin account.
5
Arbitrage
Arbitrage
• Law of one price (LoP) = identical assets have the same price Why? If not, there are unlimited
profit opportunities
• Arbitrage generalises LoP to link prices of non-identical assets
• Arbitrage strategies, definition:
– risk-free, and
– require zero initial outlay
• Prediction: zero payoff, in market equilibrium
• Prediction requires frictionless markets
– zero transaction costs, and
– no institutional restrictions on trades.
• Require only mild assumptions about investor behaviour,
– merely that more wealth is preferred to less.
• Arbitrage links asset prices (is only a partial theory).
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5.1
Arbitrage examples
Example 1: foreign exchange market
Suppose that:
£1
U110
$1
=
=
=
$1.30
£1
U100
Arbitrage opportunity:
• borrow £1 and exchange it for $1.30
• buy U130 with the $1.30;
• exchange U110 for £1
• Profit = U20, after repaying £1 loan
N OTE: The arbitrage principle cannot on its own predict an equilibrium pattern of exchange rates:
this can be done only with a more fully specified model. One possible equilibrium is £1 = $1.30,
U130 = £1, $1 = U100. But there are many other equilibria consistent with the arbitrage principle,
e.g., £1 = $1.10, U110 = £1, $1 = U100.
Remember: the arbitrage principle (absence of arbitrage opportunities) cannot on its own predict
the prices of all assets traded in a market. But it rules out many (an infinity!) of asset price patterns.
Example 2: bond market
• Bond payoff = $115.50, one period from today.
• Interest rate = 5%
• Bond rate of return =
115.5 − p
p
• Market equilibrium: interest rate = bond return
Why? Otherwise there would be an arbitrage opportunity
• Market equilibrium: 0.05 =
• Hence, p =
6
115.5 − p
p
115.50
= $110
(1 + 0.05)
Asset market efficiency
Asset market efficiency
Asset market efficiency has several meanings:
• Allocative efficiency (Pareto efficiency):
– does the capital market allocate resources efficiently?
12
• Operational efficiency:
– is the capital market organised efficiently?
• Informational efficiency:
– do asset prices reflect information? (too vague)
• Portfolio efficiency (mean-variance analysis):
– minimum variance of return for any level of expected return
Beware! Take care to define “asset market efficiency”
Summary
Summary
1. Financial markets are treated as markets for stocks. Equilibrium prices: asset prices such that
existing stocks are willingly held, given the decision rules adopted by investors.
2. Investors are assumed to make their choices consistently.
The implications of this analysis provide the decision rules for selecting portfolios.
3. In frictionless markets the absence of arbitrage opportunities enables definite predictions about
how asset prices are linked together.
4. Asset market efficiency is about how ‘well’ markets function but can be a highly ambiguous
concept.
Always be careful to explain precisely what you mean when you use ‘efficiency’ in financial economics.
*****
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