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Lecture Notes 3, Spot & Forward Prices (Ch 5) and Rates(Ch 6, 7) K Foster, CCNY, Spring 2010 |
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Learning Outcomes (from CFA exam)
Students will be able to:
§ explain why the futures price must converge to the spot price at expiration;
§ determine the value of a futures contract;
§ explain how forward and futures prices differ;
§ describe the monetary and nonmonetary benefits and costs associated with holding the underlying asset, and explain how they affect the futures price;
§ describe backwardation and contango;
§ discuss whether futures prices equal expected spot prices;
§ compare and contrast interest rate options with forward rate agreements (FRAs);
§ define interest rate caps, floors, and collars;
§ describe the characteristics of swap contracts and explain how swaps are terminated;
§ define, calculate, and interpret the payment of currency swaps, plain vanilla interest rate swaps, and equity swaps
Chapter 5, Spot and Forward Prices
How can we develop a relationship between the current price of an asset (spot) and its future value? First we have to think about how these two prices are set. Clearly there is a relationship between them, but what?
Consider if the spot price were 100 and the forward price, for delivery in a year, were 110. Would you rather buy it now for 100 or spend a little more to lock in the price?
There are a couple things
that we immediately notice we're missing.
First, since we're comparing money in two different times, we need to
worry about the relative values of these dollars i.e. the interest rate, which gives the price
of next year's dollars. We also need to
know something about how/if the value of the underlying asset changes
if we're buying ripe tomatoes then they'll go
bad long before a year is up; if we're buying oil then we have to store it
somewhere; if we're buying stock shares they pay dividends.
Interest rate: assume the rate is given as "r" and that we're working in continuous time so the present value of each dollar, paid in a year's time, is e-rT, where T=1 so it is e-r.
In the example above, where spot is 100 and forward is 110, if the interest rate is low then we could borrow money today to buy at spot, sell it at the forward price, make $10 per transaction and if the $100 borrowed costs, say, $3 or $4, then that's a nice profit from the arbitrage. On the other hand, if the interest rate were very high then the opposite transaction would be more worthwhile. If I have $100 I could put it in the bank and get more than $110 after a year. Sell short at the spot rate (100) and buy forward at 110 to lock in the price at which I return the underlying asset. The difference (how much more I earn from interest over the 110 forward price) is arbitrage profit. In either case, the arbitrage trades work to change demand and supply to bring the prices back into line.
Examples of leverage:
suppose that the interest rate (for borrowing) is 6%. Then I want to borrow $100 from the bank, buy
the asset, sell it forward, and make $4 for each $100 that I can borrow today. Banks (alas!) don't just give away money,
they require some amount of money be put up.
Suppose they require $50; in this case for every $50 that I put up, I
borrow 50 more, pay back $3 on the borrowed funds, and make $7 on the $50 this is a 14% return. If I could put up a lower margin, say $25,
then I would borrow $75 and have to pay back $4.50; my $25 initial investment
returns $5.50, which is a 22% return!
The less of my own money that I must put up, the higher I can ratchet up
my own returns.
This is great, in these examples where the return can be locked in. But it's not generally easy to lock in easy money, everybody is looking to do that (just like you rarely find a $20 bill lying on the sidewalk). With risk then the higher returns go both ways: leverage generates the possibility of larger positive and negative risks.
Back to forward/spot
We might be confused
because we might think that the forward price is a predictor of the price that
will be set at that future date. But
it's not the spot price is a predictor. Why?
Again, we consider what actions might be taken by a smart financial
trader. Suppose that it is known that,
on Friday, the price of some asset will jump from 50 to 75. Clearly, someone who holds the asset on
Friday will get a huge return on their money.
So what is likely to be the demand for that asset on Thursday? Wednesday?
Tuesday? Today? The argument gets more complicated if the
asset is difficult to store or if it changes value when held. (Below we discuss the implications of the CAPM
model to show that, under some circumstances, the futures price can be an
unbiased estimate of the expected future spot price.) But the core arbitrage argument is
clear.
These examples assume that the value of the asset does not change much over the time period. So we differentiate between an investment asset and a consumption asset. This tells if large numbers of market participants will be able to arbitrage (as outlined above) or whether large numbers will be eating what they buy. (I wouldn't sell short a pint of Ben & Jerry's because I'd eat it and wouldn't have anything to deliver at the end of the contract!)
In some way we can think of
putting money in the bank as buying money forward: if I put $100 in the bank
and get (without risk) some return, r, so that after a time of T, I get 100e-rT. This is like buying forward 100e-rT
at a price of $100. Any other forward
contract can be thought of as delivering in some different units of measure but still, in the end, I should get the same
rate of return. Whether I buy forward 1
gallon of crude oil, or some equivalent number of liters, doesn't matter. Similarly it doesn't matter whether I buy
forward dollars or yen or euro or hog bellies or gasoline or S&P index
contracts….
All of these arbitrage
arguments get us to our first equation: F0
= S0erT. This
is strictly true for investment assets in markets where arbitrageurs can borrow
and lend at the same riskless rate, there are no transactions costs or other
taxes, and there are enough (potential) arbitrageurs. You can think of it as just offering one more
way to invest you could get the riskless rate on the money
or buy an asset that would (again, risklessly) provide some payment in a year.
Of course some stocks have a known income or known dividend yield, so we can modify the equation to take account of these complications. Other assets have storage costs (negative known income) or convenience yields. The convenience yield is defined as the amount that we observe that market participants are willing to forfeit in order to have the actual physical asset rather than a futures contract.
If we generalize about the "cost of carrying" some asset forward, whether that is the interest rate to finance it, or the interest rate less the income actually earned, or the interest rate less the foreign interest rate, or interest rate plus storage cost, denote the "cost of carry" as c so that for investment assets,
,
while for consumption assets, where y is the convenience yield,
.
Also there are many
contracts that offer interest held in different currencies again the same arbitrage arguments should
hold. If I can risklessly get some r
interest rate in US dollars then I should be able to lock in an equivalent rate
in euro or yen or any other major currency.
If I have a unit of foreign currency (FX) then I can either buy dollars
at S0 and invest in the
and buy forward at F0 to end up
with F0
. Set these two end possibilities equal, S0erT
=F0
or
.
Next we move to valuing
these forward contracts. The forward
price is F0 but the value
of the contract (agreeing to buy at that forward price) is f. This sounds confusing but
it is the simple result of the distinction between the value of a contract and
its notional price for example you could buy insurance that will
pay $25,000 if you die
but you don't pay $25,000 for it! Of course a forward contract is not
probabilistic
the whole point is that there is an ironclad
agreement to trade at F0.
Go back to the example at
the beginning, where the spot is 100 and forward price is 110. If I enter into a forward contract that sets
a strike price (denote it as K) of 110 then the value of this contract, f, is
exactly zero. Tomorrow is a new day so
the prices will change (but not K that's written into the contract) and f = (F0
K)e-rT.
You might be asking why
anyone would enter into a contract where the value of it is zero. This is what arbitrage means that although everyone is trying to make
money, on net the prices must give no arbitrage profit. (Like the firms in micro that get economic
profit of zero even though they work hard to maximize profits!) As we have discussed, a significant fraction
of the parties buying and selling are hedging: they're not looking for arbitrage
profit but rather to lock in some price.
Later on, as the forward
price changes away from the strike price, the value of the forward contract
will change that's the point.
Finally we discuss the relation between the current futures price (F0) and the expected future spot price (E(ST)). Again consider an investor who might have some expected future spot price that is different from the market, so she could put the necessary cash in the bank today (cost F0e-rT) and expect to get ST. However this expected rate in the future should be discounted by the investor's required rate of return given the risk (systematic or non-systematic) that she is taking on. But speculators would enter the market until F0 = E(ST)e(r-k)T. If the asset risk is uncorrelated with the stock market, then r=k and F0 = E(ST). When the futures price is below the expected future spot price (so k>r) this is called "normal backwardation"; when the futures price is above the expected future spot price (so k<r) then it is "contango". (There are a number of linguistic theories about where that word comes from.)
From
Interest Rate Futures
Day Count Conventions: Ugh, crazy details! Interest is only credited at some low frequency (every 6 months, for example) but it accrues continuously and is calculated as a fraction of the period. But what fraction? Some contracts use the actual number of days between dates divided by the days in the reference period; some use 30/360; some use actual/360. US Treasuries use actual/actual but corporate and municipal bonds use 30/360 and money market instruments (such as T bills) use actual/360.
So Hull's example that, on Feb. 28, holding a corporate bond overnight would get 3 days of interest accrual (under 30/360 rule) while holding a US government bond overnight would get just one day of interest accrual. It's just a reminder that the financial "system" is not really a "system" in the sense of something that was designed, or is even coherent. Rather it's a series of sometimes-random choices and institutions that have evolved many different ways of interacting with each other.
Treasury bond prices: dollars and 32nds of a dollar for a bond with face value of 100.
Quote "clean price" which is not the cash delivery price, the "dirty price" = clean price + accrued interest since last coupon.
Treasury bond Futures
Short party deliver any bond with maturity over 15 years and not callable within 15 years. Conversion factors (created by CBOT at benchmark 6%) defines the price. Which bonds are cheapest to deliver? Depends on yield curve and current interest rates.
Futures price: F0
= (S0 I)erT. Where "I" is PV of coupons over
life of contract.
Eurodollar futures are most common for interest earned on $1m traded on CME for any 3-month interval in next
10 years! Contracts designed so each bp
move is 1,000,000 * (1/10000) * (3/12) = 25.
Eurodollar futures can be
used to extend the LIBOR zero rates past 12 months to 2 5 years.
For short contracts, Eurodollar futures are equivalent to FRA but at longer terms the settlement periods matter (so convexity adjustment).
Duration hedging
Swaps
"plain vanilla" swap on LIBOR, fixed for floating, BFIX = BFL.
notional principal is not exchanged; only the cash flows (So measurement of transactions is complicated)
the tenor is the frequency of payment (6 months, 3 months, 1 month, etc)
swaps can transform assets or liabilities
Confirmations are the legal agreements underlying
Also currency-based swaps to borrow/lend at different rates and currencies, different rates (plus/minus bp) amortizing or step-up swaps, constant maturity swap, equity swaps, options, swaptions (options on swaps), commodity swaps, etc., etc., etc. Exotic instruments.
Credit Risk remains and is held by the intermediary
Why Swaps? Comparative Advantage low-rated companies who would get a high fixed rate can instead borrow at floating rate, betting that they can do better than their rating agency believes.
Longer horizon LIBOR rates only to 12 months but swaps go
even farther than Eurodollar futures
Valuation of Swap: originally zero, then either as bonds or as FRA portfolio
bonds: VSWAP = BFIX - BFL.
FRA: zero curve to get forward rates, then calculate the swap cashflows, then discount using the same zero curve.