FORWARD PRICES

An actual forward price for any asset is a price that be agreed now for a future transaction in that asset. The date of that future transaction defines the period of the forward price. Thus the one year actual forward price for gold is the price that can be agreed today for a purchase / sale transaction of gold taking place in one year's time.

The 'fair', 'implied', 'theoretical' or 'arbitrage free forward price' for an asset is that which is determined with reference to the currently existing 'spot' price, interest rates, yield on the asset and other cost items incurred in holding that asset to the forward date. Such an arbitrage free forward price is precisely the forward price that allows no arbitrage between spot price and forward price. To see this examine the following set of data :

At t₀ Gold trades at £100 per ounce.
The interest rate on money for one year is 10%
Storing one ounce of gold risk-free for one year costs £1.
The yield derived from holding gold for one year is £ 0.

The arbitrage free forward price for gold is then determined by setting up a cash flow representation in which the arbitrageur buys one ounce of gold today using borrowed money, stores the gold for one year and resells it in one year's time at a price agreed at time 0 making no profit or loss in so doing. In other words the entire transaction must make no profit for the arbitrageur if the forward price agreed at time 0 is to be called an arbitrage free forward price.

	t0	t1
Borrow £101 at 10%	+101
Repay loan		-111.1
Buy 1 ounce of gold	-100
Sell gold at forward price		+f
Pay storage costs	-1
Total cash-flow	0	0

The forward seller of one ounce of gold at a forward price of £f per ounce of gold agreed at time 0 must set his price f such that no arbitrage is possible. In other words the totals of column t₁ must sum to zero as shown. (We can see that the totals in column t₀ sum to zero since the arbitrageur borrows exactly what is needed to finance the purchase and storage of gold for one period). The only value for f that can satisfy the requirements of column t₁ is obviously 111.10 and in agreeing a forward sale of one ounce of gold at £111.10 for one year delivery, no arbitrage profit is possible.

The process of purchasing and holding an asset until delivery is made into a forward contract is sometimes termed 'cash and carry'. Since the cash and carry procedure here yields no profit, no arbitrageur should engage this arbitrage. Of course the existence of transaction costs would make the process yield an actual loss.

The difference between actual forward prices and arbitrage free forward prices is often very small or non-existent in financial markets. However many traders specifically watch for arbitrages of this kind between the forward market price and the cash market price in many products, for instance foreign exchange. Where actual and implied forward prices differ by a sufficient amount, a profit can be made through cash and carry or 'reverse cash and carry'. This latter arbitrage is simply the act of selling the asset now and agreeing a forward purchase simultaneously. In our above example, if the actual forward price bid in the market at t₀ was £112, a cash and carry trade would yield a £0.90 profit. If the actual forward price offered in the market was £110, reverse cash and carry would yield a profit of £1.10.

There is one assumption here, namely that the trader could borrow the gold to sell at t₀ without associated costs. Selling an asset one does not own ('shorting') in a cash or spot market, usually involves some kind of arrangement to borrow the asset for a given period (here one year) for which the asset's owner will usually demand a fee in one form or other. Without having access to the borrowing facility a reverse cash and carry would not be possible. Furthermore, the fee charged for the borrowing facility would have to be included in the calculation of the implied forward price.

Some assets such as bonds yield a return over a given holding period. Gold in contrast offers no return to the holder. Where an asset offers some kind of return, the arbitrageur will have extra cash-flows to take into account in his calculation of the implied forward price. Let us take the following example ;

Asset is a long bond, current spot price 100%
Bond has a running yield of 10% over one year.
Money market rates for a one year loan are 5%

We can now set up a cash-flow representation of the arbitrage procedure as before :

	t0	t1
Borrow £100 @ 5%	+100
Repay loan		-105
Buy bond	-100
Sell bond at forward price		+f
Receive interest on bond	+10
Total cash-flow	0	0

F_a = S_a (1 + r_f - r_a )^t

Where,
F_a = Forward price of asset a
S_a = Spot price of asset a
r_f = Financing rate on borrowed money
r_a = Yield on asset over holding period

For our above bond example, the one year implied forward price F_a is :
F_a = 100 ( 1 + 0.05 - 0.10)¹
= 100 (0.95)
= 95

 
FORWARD RATES

The fully arbitraged cash-flow method of calculating forward rates is an alternative to the more simple methods found in elementary treatments of the subject. The elementary method suffers from a problem which can best be highlighted by showing an exampe of it in action:

One year spot rate = 5% per annum
Two year spot rate = 10% per annum
One year, one year forward rate (F) is calculated as follows :
(1 + 0.05 ) (1+F) = 1.12
1 + F = 1.21 / 1.05
F = 0.1524 or 15.24%

If the forward rate of 15.24% is correct as an arbitrage free rate for the period, then there should be no arbitrage profits available between the two spot rates and the forward rate. However the following arbitrage is possible.

At time 0 ;
Borrow £100 for one year at 5% annual.
Enter one year forward contract to borrow £95 in one year's time at 15.24%
Lend £100 for two years at 10% annual.
Net Cash-flow at time 0 is zero.

At time 1 year ;
Receive £10 interest on two year loan.
Borrow £95 at 15.24% for one year.
Repay £105 on one year loan.
Net Cash-flow at time 1 is zero.

At time 2 years ;
Repay £109.478 on loan of £95 at 15.24%
Receive £110 (repayment of principal and final interest payment on two year loan).
Net Cash-flow is +£0.522
Remaining liabilities and assets are zero.

The arbitrage profit available from the forward rate of 15.24% is £0.522 on the sums transacted. Clearly then the forward rate that was calculated earlier is not arbitrage free.

It is a simple matter now to adjust the 15.24% to a rate that erodes the arbitrage profit of £0.522 to zero. Here, the true arbitrage free rate would have to be such that the repayment of a £95 loan after one year exactly cancelled the receipt of £110 from the two year loan repayment. In other words the net cash-flow at time 2 years would then be zero. The true arbitrage free forward rate must therefore be 15.789%, since £95 borrowed for one year at 15.789% leads to a repayment of £110. The above procedure is termed the 'fully arbitraged cash-flow method' of calculating forward rates. It compensates for the assumption implied in the simple forward rate method described earlier, namely that the first interest payment of 10% received at the end of year one can be reinvested at the same rate of 10% for the second year. Clearly even the simple forward rate calculation provides a reinvestment rate of 15.24% for the second year and to assume a reinvestment of rate of 10% in the very same calculation seems contradictory.

 
ZERO-COUPON DISCOUNT FACTORS

A bond that pays no interest during its life is said to be a 'zero-coupon' bond. This is due to the fact that interest payments on bonds are usually referred to as 'coupons'. In the early days of bond issuance, and in fact to this day, it was common for a bond investor to tear off a perforated piece of the bond paper (a coupon) and return it to the issuer in order to claim the interest due. Those unfamiliar with bonds may wonder why any individual would wish to purchase a bond that pays no interest. The answer is that a bond that promises to repay £100 at final maturity with no coupons in the meantime, will have a market price today of less than £100. If the current market price of the bond were £90, then the investor would know that by holding the bond to maturity he would gain the amount of £10, being the difference between redemption value and price paid for the bond. Now, if such a zero-coupon bond were to redeem in one year's time at £100 and have a current price of £90, one could say that the return to the holder would be :
10 / 90 .1111 i.e. 11.11%.

Thus we might say that the yield on a one year zero coupon bond is 11.11%.
By taking a two year zero coupon bond that is priced at 80% of face value (and which repays at 100%), one could say that the return to the holder is ;
20 / 80 = .25 = 25% over the two year period.
The implied yearly rate derived from the two year return of 25% is ;
(1.25)^0.5 -1 = 0.118034 i.e. 11.8034% per annum.

The rate of 11.8034% per annum assumes that reinvestment of interest if any were paid would occur at this same rate of 11.8034%. In fact since all the return to the holder of a zero-coupon bond comes in the form of capital gain, the reinvestment rate is said to be guaranteed since the capital gain received in place of interest is automatically reinvested at the zero-coupon rate. This is one major attraction of zero-coupon bonds to investors.

Now that we have seen how to calculate zero-coupon rates from zero-coupon bond prices we can develop a further theory of how to determine a fair forward yield. Notice first however that the price of a zero-coupon bond is determined mathematically as :

P = F (1 + r)^-t

P = present value of bond (here this is the theoretical current bond price)
F = Future value of bond at time t (here the redemption value of the bond, 100%)
r = interest rate used in present valuing (here the zero-coupon bond yield)
t = number of periods to the redemption date In our above example ;
P = 100 (1.118034)^-2
= 100 (0.8) = 80

The amount 0.8 representing the factor (1 + r)^-t in the above formula is of course the discount factor (the amount by which one must multiply the future value in order to arrive at the present value). In this case 0.8 is the two period discount factor. We might easily calculate the one, three, four etc. discount factors given the zero coupon bond prices for each maturity of bond. Conveniently, the discount factor for each maturity would be the percentage price of a zero-coupon bond which redeems at 100. A three year zero-coupon bond trading at 70 would imply a three period discount factor of 0.7. We can now progress to a discussion on forward rates.

A zero-coupon yield curve is derived from a given yield curve of interest rates. In our earlier example of a one year spot rate of 5% and a two year spot rate of 10%, we saw that the one year discount factor was (1.05)^-1 which equals approximately 0.952381. We could similarly calculate the two year discount factor as (1.10)^-2 which equals approximately 0.82644. But this two year discount factor is not the same as the two year zero-coupon discount factor. The problem here is once again that the two year discount factor of 0.82644 assumes that the first year's interest of 10% can be reinvested at a rate of 10% during the second year, whereas the forward rate for this second period is actually much higher.

The correct discount factor for the second year is the two year zero-coupon discount factor and it can be calculated as follows. Take the below series of cash-flows representing a two year money market loan of £100 where interest payments are agreed to be made annually at 10%.

t0	t1	t2
P	+10	+110

P, the present value of the loan, equals £100 in this example. P must also equal the present value of the £10 received by the lender at t₁ plus the present value of the £110 received at t₂. Now we know that P must equal the amount of £10 received at t₁ multiplied by the one year discount factor of 0.952381 plus the amount of £110 received at t₂ multiplied by the two year discount factor. In other words,
P = 10 × d₁ + 110 × d₂

Where,
d₁ = the one year zero-coupon discount factor
d₂ = the two year zero-coupon discount factor.
We know that d1 is 0.952381 since there is no reinvestment problem to take account of in its calculation, in other words that the discount factor and the zero-coupon discount factor for t₁ are the same.
Meanwhile, d2 is calculated as follows:
since,
P = 9.52381 + 110 d₂
and since,
P = 100,

then,
100 = 9.52381 + 110 d₂
d2 = 0.82251

So whilst the two year discount factor is 0.82644, the two year zero-coupon discount factor is 0.82251.

Zero-coupon discount factors help us greatly in calculating the arbitrage free forward rate for any given period within the yield curve. In our current example:

d₁ = 0.952381 and d₂ = 0.82251

The arbitrage free forward rate for the period can now be obtained by dividing the one year zero-coupon discount factor (d1) by the two year zero-coupon discount factor (d2) and subtracting one (0.952381 / 0.82251-1 = 15.79%). This result can also be obtained from the zero-coupon forward discount factor for the period d₁ to d₂, which equals d₂ / d₁ = 0.82251 / 0.952381 or 0.8636. The inverse of the zero-coupon forward discount factor, minus one, once more provides us with the arbitrage free forward rate for the period t₁ to t₂ (i.e. 1 / 0.8636 - 1 = .1579 or 15.79%). These are the same results (ignoring rounding errors) as the forward rate given by the FACF method used earlier. In general,

f_ab = d_a / d_b - 1

Where:
f_ab= forward rate for period a to b
d_a = zero-coupon discount rate for time 'a'
d_b = zero-coupon discount factor for time 'b'

THE FUTURES STRIP

Our previous examination of forward rates and spot rates showed that in order for there to be no arbitrage opportunities between any given set of spot rates and forward rates, there should be a distinct relationship between the two sets. In our discussion of yield curves we also saw that traders often compare yield curves of different assets to identify trading opportunities.

Many traders will spend their time deriving yield curves from a set of deposit rates or a set of forward rates or perhaps a set of short term interest rate future prices. Each set of rates should provide a yield curve of approximately the same shape in order for there to be no arbitrage opportunities. For instance, if a trader creates a yield curve from a set of deposit rates and finds that it is substantially below the yield curve derived from FRA rates, he may be advised to borrow in the deposit market and lend in the FRA market. The process of deriving a yield curve from a set of forward rates is described as 'stripping out' a yield curve from forward rates. Similarly the 'futures strip' is a yield curve derived from the interest rates implied by prices in the short term interest rate futures market. If any of these stripped rates are sufficiently different from deposit rates offered in other money markets for the same periods, the trader may wish to arbitrage the futures strip against that other money market instrument by simultaneously borrowing at the lower rate and lending at the other.