In the previous section we covered the basic principle (formula) for valuing futures. However, there are a number of variations on the theme, because there are different types of futures contract traded.

The (valuation) mathematics pertaining to the different futures is illustrated with the following futures:

• Short-term interest rate futures.

• Individual bond futures.

• Equity / share index futures.

• Individual equity / share futures (aka single stock futures).

• Commodity futures.

• Currency futures.

• Futures on other derivatives.

• Other futures.

3.14.2. Short-term interest rate futures

In the case of short-term interest rate futures, the theoretical price or fair value price (FVP) is determined from the calculated forward-forward rate (which is also called the implied forward rate). An example is required here: the South African 3-month JIBAR27 future, the specifications of which are shown in Table 4.

UNDERLYING INSTRUMENT (CONTRACT BASE)

The 3-month Johannesburg Interbank Agreed Rate (JIBAR)

CONTRACT SIZE (NOTIONAL)

R100 000 nominal

QUOTATION STYLE

Effective interest rate

CONTRACT MONTHS

March, June, September and December

EXPIRY DATES & TIMES

11h00 on third Wednesday of the contract month (or previous business day)

MINIMUM TICK SIZE

0.001% (1/10 of a basis point)

BASIS POINT VALUE

ZAR 2.50 per basis point (rate change = 0.01% pa)

MARK-TO-MARKET (MTM)

Explicit daily fixing

SETTLEMENT

Cash

SETTLEMENT YIELD (DAILY MTM)

Closing MTM yield

SETTLEMENT YIELD (ON EXPIRY)

3-month JIBAR on expiry

INITIAL MARGIN

R100 per contract

Source: JSE (2010).

Table 4: Specifications of the 3-month jibar future

A note on how the basis point value (ZAR 2.50 per basis point) is arrived at is required. A basis point = 0.01% per annum. Because there are four 3-month periods in a year, 3 months is taken to be 91.25 days (365 / 4). Therefore, if the 3-month JIBAR rate changes from 7.81% pa to 7.80% pa (i.e. by 1 basis point), the profit on a 91.25-day asset = (0.01 / 100) x (91.25 / 365) x ZAR 100 000 = ZAR 2.50.

The theoretical price or fair value price (FVP) of a 3-month JIBAR future is arrived at by calculating the implied forward rate from the current spot rates. An example is required: shown in Figure 9 are the JIBAR rates quoted on the day a client wishes to buy a 3-month JIBAR futures contract (i.e. a 3-month rate in 3 months' time).

Figure 9: JIBAR spot rates and implied rate

The rate now (spot rate) for three months is 9.0% pa and the rate now (spot rate) for six months is 10.5% pa, and the period of the latter rate covers the period of the first rate. The rate of interest for the three-month period beyond the first three-month period can be calculated by knowing the two spot rates mentioned. This is called the forward rate of interest, or the implied forward rate, or the forward-forward rate. This is calculated as follows (assumption 3-month period = 91 days; 6-month period = 182 days):

where

IFR = implied forward rate

irL = spot interest rate for 6-month (i.e. long) period

irS = spot interest rate for 3-month (i.e. short) period

tL = 6-month (i.e. long) period, expressed as number of days / 365 (= 182 / 365)

tS = 3-month (i.e. short) period, expressed as number of days / 365 (= 91 / 365)

IFR = {[1 + (0.105 x 182/365)] / [1 + (0.09 x 91/365)] -1} x [365 / (182 - 91)] = [(1.05235616 / 1.02243836) -1] x (365 / 91) = 0.02926123 x 4.010989 = 0.11736647 = 11.736647% pa.

This derived interest rate may be tested as follows: if R1 million (present value, PV) is placed on deposit for 6 months (182 days) at the abovementioned 6-month rate of 10.5% pa, the future value (FV6m) amount would be:

FV6-m = PV x [1 + (0.105 x 182 / 365)] = R1 000 000 x 1.05235616 = R1 052 356.16.

Alternatively, if an investment were made for 91 days, the following would be the total:

FV3-m = PV x [1 + (0.09 x 91 / 365)] = R1 000 000 x 1.02243836 = R1 022 438.36.

If this amount (R1 022 438.36) is invested for another 91 days at the implied forward rate of 11.736647%,

the FV6-m:

6-m

FV6-m = PV x [1 + (0.11736647 x 91 / 365)] = R1 022 438.36 x 1.02926123 = R1 052 356.16.

As expected, this number is identical to the FV of the six-month investment calculated above.

As seen, the implied forward rate is 11.736647% pa. This is the fair value price / rate, i.e. the rate that should apply to the future.

Keep in mind that the fair value is not necessarily equal to the market value (= MTM value as determined by the exchange). It will also be apparent that the forward-forward pricing of futures is the same as the pricing of an FRA. An FRA can thus be seen as the OTC equivalent of the interest rate future. This calculation also applies to the forward-forward foreign exchange swap.

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