5. Forwards, Eurodollars, and Futures 171
5.3 Forward Price of Assets That Pay Known Cash Flows
We use this market as an example through which we will explain the effect of the cash flows obtained from the asset during the life of the for- ward contract. The setting of the bond market is also as specified below by NarbitB. We use five approximating functions to estimate the term struc- ture and define the discount function asdis.
Table 5.1 A Simple Bond Market Specification.
Price/Time 1 2 3 Security
$94 $105 $0 $0 B1
$97 $10 $110 $0 B2
$85 $8 $8 $108 B3
> NarbitB([[105,0,0],[10,110,0],[8,8,108]],[94,97,85], 5,dis);
The no-arbitrage condition is satisfied.
The discount factor for time,1,is given by, 94
The interest rate spanning the time interval,[0,1],is given by,105 0.1170
The discount factor for time,2,is given by, 1849
The interest rate spanning the time interval,[0,2],is given by,2310 0.2493
The discount factor for time,3, is given by, 82507 124740
The interest rate spanning the time interval,[0,3],is given by, 0.5119 The function Vdis ([c1,c2,..]), values the cashflow [c1,c2,..]
The continuous discount factor is given by the function, ‘dis’, (.)
As we can see fromNarbitB, there are no arbitrage opportunities. As well, the continuous discount factor function, as seen in the previous figure, fits perfectly with the discrete discount factors. In this market there are three time periods and three different bonds. For simplicity one may think about the time period segments as years, that the current time, time zero, is a year prior to the first coupon payment and that the bonds make annual coupon payments.
Consider a forward contract written at time zero specifying delivery of bond three at time two, immediatelyafterthe second coupon is paid. Bond three’s cash flows are (8,8,108). According to the contract, the party with a long position will pay the forward price,F, at time two and will receive an instrument similar to a zero-coupon bond that will pay $108 at time three.
Hence, we can calculate the forward price based on the present value of the cash flow (0,−F,108) being equal to zero. Once we have solved forFwe can also graph the payoff, as of time two, from such a contract. We leave this derivation as an exercise for the reader.
Recall our discussion using the cost-of-carry model to calculate the for- ward price. The idea was to buy the asset now, hold it until maturity, and deliver it at that time. The purchase of the asset was financed by a loan and hence (at the delivery time) the loan had to be repaid. The amount of money borrowed was equal to the spot price of the asset, and hence the amount to be repaid was the future value of the spot price. Consequently, a forward price, to avoid arbitrage opportunities, must be equal to the future value of the spot price. The only difference is that for the purchase of an asset that pays known cash flows, buying the asset does not require taking a loan for the full amount of the spot price. The spot price of the bond is
$85, but the bond pays a coupon of $8 at time one and at time two.
When the coupons are received, they can be re-invested at the risk-free rate of interest until the maturity time of the forward contract, time two.
Hence, the cost of delivering the bond at time two will be less than the future value of the spot price. At time zero we can secure the rate that will be paid on the $8 invested from time one to time two. This will be exactly the forward rate r0(1,2), as we explained in Section 3.3. Similarly, the
$8 obtained at time two can be used at that time. Hence, the net cost of delivering the bond at time three will be reduced by the future value of the
$8 obtained at time one and at time two. Consequently, the cost will be
85
dis(2)−8(1+r0(1,2))−8.
(5.3) We have already mentioned that, in general,
1+r0(t1,t2) =dis(t1) dis(t2).
Thus, (5.3) can be rewritten in terms of the function disby applying this substitution. However, we wish to reinterpret this relationship here.
This will also be useful in understanding how to derive these types of rela- tionships.
One should always remember that a particular cash flow has two char- acteristics: its magnitude and its timing. The discount factors allow us to move cash flows through time, converting1cash of one time period to that of another. A useful methodology when converting cash in this manner is to “move” all the cash flows involved to a mutual point in time, and then to
“move” this amount to the required point in time.
Proceeding in this manner, to find the cost of delivering the bond at time two, we first calculate the cost as of time zero. The bond costs $85 at time zero but pays $8 at times one and two. The $8 paid at time one is worth
$8dã(1)at time zero. Similarly, the $8 paid at time two is worth $8dã(2) at time zero. Hence, the total cost of delivering the bond at time two
85−8ãdis(1)−8ãdis(2) (5.4) in terms of dollars at time zero. In our case this will be
> 85-8*dis(1)-8*dis(2);
82507 1155
Therefore, the cost of delivering the bond at time two, in terms of dollars of time two, will be the future value of this amount, i.e.,
1One can conceptualize two different points in time as two different markets, similar to a foreign market. For example, there is a market at timet1 and a market at timet2 with different currencies much the same as the U.S. and Canadian dollar. In the foreign market situation the exchange rate is the factor that allows us to convert from one currency to another. Similarly, the discount factor allows us to convert dollars of timet1to dollars of timet2.
85−8dis(1)−8dis(2)
d(2) . However this is exactly equation (5.3). In our example we calculate it as follows:
> (85-8*dis(1)-8*dis(2))/dis(2);
165014 1849 Or in a decimal form,
> evalf(%);
89.24499730
Therefore, we can make a general statement about the forward price of a forward contract on an asset that pays a known cash flow throughout the life of the contract. The forward price is the future value of the spot price, minus the future value, as of time zero, of the cash flow obtained during the life of the contract.
LetS(0)be the spot price of the asset andTthe maturity of the contract, and assume that the asset pays a cash flow ofc(t)at timet. Let the discount factor for timetbe denoted asd(t). The general result for the future price F is thus
F= S(0) d(T)−
T
∑
t=1
c(t)d(t)
d(T) , (5.5)
or in terms of the forward rates in equation (5.6), F=S(0) (1+r(T))−
T
∑
t=1
c(t) (1+r0(t,T)). (5.6) We can see that a positive cash flow obtained during the life of the contract reduces the forward price. It is best understood utilizing the cost- of-carry model. If the deliverable good produces positive cash flows, these cash flows can help with financing the purchase of the good in the spot market. Hence the amount of money borrowed to buy the good on the spot is reduced by the present value of the cash flow produced by the good to be delivered. In the foreign currency case, the positive cash flow was due to the interest earned on the currency in the foreign market.
In the case of a commodity, the cash flow produced (or its equivalent) can be positive or negative depending on the storage cost and the conve- nience yield. A similar argument applies to a forward contract on a stock that pays a fixed dividend yield. To cement the ideas explored in the last two sections we investigate in the next section the forward price, prior to maturity, of assets that pay a known cash flow — a coupon-paying bond.
5.3.1 Forward Contracts, Prior to Maturity, of Assets That Pay Known Cash Flows
Consider the same forward contract as in Section 5.3. Let us see what the forward price will be if the delivery time ist=1.5 instead of timet=2.
The maturity date of this contract does not coincide with any cash flow payment date of the asset. Indeed, there is no conceptual change. One needs only to be careful with the timing and the specification of the cash flows. In our example, we can apply equation (5.5) to obtain the forward price as
> (85-8*dis(1))/dis(15/10);
14292419328 157492693 or in decimal form
> (85-8*dis(1))/dis(1.5);
90.74972976
Suppose that a year has passed and we take another snapshot of the same market. The first bond has matured and we will assume that no new bonds have been issued. The second bond will mature in a year and will pay $110. The third bond will mature in two years and will pay $8 in one year, and $108 in two years. The prices of these two outstanding bonds are now assumed to be different than they were one year ago. They are $90 and $80, respectively. We run NarbitBbased on this specification to get an estimate of the term structure. The current time zero is of course time one of the last model. (We suppress the graph by adding a fifth parameter and assigning it a zero value.) If you would like to see the graph of the
discount factor function, rerun NarbitBas below but without the zero as the last input parameter.
> NarbitB([[110,0],[8,108]],[90,80],3,dis1,0);
The no-arbitrage condition is satisfied.
The discount factor for time, 1,is given by, 9
The interest rate spanning the time interval,[0,1],is given by,11 0.2222
The discount factor for time,2,is given by, 202
The interest rate spanning the time interval,[0,2],is given by,297 0.4703
The function Vdis ([c1,c2,..]), values the cashflow [c1,c2,..]
The continuous discount factor is given by the function, ‘dis1’, (.)
We name the current discount factor asdis1and we use three estimating functions for the continuous approximation of the discount function. As before, we see that there are no arbitrage opportunities and we obtain a perfect fit for the estimation.
Consider the forward contract discussed above. We will refer to this contract as the “old” forward contract. The holder of the short position of this contract needs to deliver the 8 percent bond in one year, immediately after the coupon payment. The value of the short position in the “old”
forward contract is no longer zero. The value of the long position in the same contract is not zero either: it is the negative of the value to the party with the short position. The short position in the forward contract obligates the investor to deliver the specified bond for $165014
1849 . We can use the functiondis1 to calculate the cash flows which are consequent from this contract, and so derive the value of this contract today.
In other words, the party with the long position in the bond will receive the bond. This is equivalent to obtaining $108 in two years and paying out
$165014
1849 =89.24499730 in one year. Hence the value of the cash flow [−89.24499730,108]can be calculated as
> dis1(1)*(-89.24499730)+dis1(2)*108;
0.43591130