# Individual bond futures

The principle that underlies the fair value price of a bond future is the CCM as discussed. However, the calculation is more elaborate because of the existence of coupon payments, clean and dirty (all-in) prices, ex and cum interest and so on. The fair value price (FVP) of an individual bond future is made up of:

Bond spot price (i.e. all-in price) + carry cost (i.e. rfr) - income.

An example is required: LCC15729 bond future:

Bond = LCC157

Maturity date = 15 September 2015

Coupon (c) =13.5% pa

Coupon payment dates (cd1 & cd2) =15 March and 15 September

Yield to maturity (ytm) = 8.2%

Carry cost (rfr) = 7.5% pa

Purchase (valuation) date of future (fvd) = 20 June

Termination date of future (ftd) = 31 August30

Books (register) closes = one month before coupon dates31.

As noted, the FVP of a bond future is made up of three parts:

FVP = A + B - C (i.e. bond spot price + carry cost (excl income) - income 32)

where

A = dirty (all-in) price of underlying bond at market (current) rate on bond futures valuation date (fvd) 33 = 105.71077 (note: this price is assumed so that it does not date)

B = A x {(rfr / 100) x [(ftd - fvd) / 365]} = 105.71077 x [0.075 x (72 / 365)] = 105.71077 x (0.075 x 0.19726) = 105.71077 x 0.014795 = 1.56394

C = (c / 2) x (1 + {(rfr / 100) x [(ftd - cd2) / 365)]})

[if the futures termination date crosses a books-closed date and its associated coupon date (i.e. is not ex-interest)]

or

= (c / 2) / (1 + {(rfr / 100) x [(cd2 - ftd) / 365)])

[if the futures termination date crosses a books closed date but not the associated coupon

date (i.e. is in ex-interest period, which is the case here)]

= (13.5 / 2) / (1 + {0.075 x [(cd2 - ftd) / 365]})

= 6.75 / {1 + [0.075 x (15 / 365)]}

= 6.75 / [1 + (0.075 x 0.04110)]

= 6.75 / 1.00308

= 6.72927.

Thus:

FVP = A + B - C

= 105.71077 + 1.56394 - 6.72927 = 100.5454.

**Figure 10: example of individual bond future**

# Equity / share index futures

We covered the case of equity / share index futures in our first example where the simple interest * net carry cost *calculation was introduced:

FVP = SP + CC

= SP + {SP x [(rfr - I) x t]} = SP x {1 + [(rfr - I) x t]}.

Here we provide another example (ALSI future):

SP (spot price, i.e. index value) = 10765

rfr = 11.5% pa

I (dividend yield, assumed) = 3.5% pa

t (number of days to expiry of contract / 365) 245 / 365

FVP = SP + CC)

= SP + {SP x [(rfr - I) x t]}

= SP x {1 + [(rfr - I) x t]}

= 10765 x {1 + [(0.115 - 0.035) x (245 / 365)]}

= 10765 x (1 + (0.08 x 0.6712329))

= 10765 x 1.05369863

= 11343.

# Individual equity / share futures

Individual equity / share futures are also called * single stock futures *(in short SSFs). Calculation of the FVP of SSFs is the same as above - i.e. as for equity / share index futures, except that the dividend yield will be easier to predict.

It is appropriate to mention a futures product which is closely allied with SSFs: the * dividend future *(DIVF). They are used to hedge against the dividend risk that accompanies a position in a SSF. As we have seen, dividend expectations (I) are part of the FVP calculation; therefore there is a need for such contracts.

# Commodity futures

With commodities, where insurance and storage is payable (such as maize), and the amount is not proportional to the spot price, it is simply added to the FVP. An example follows [we assume there are only storage costs (SC); note: there is no income (I)]:

Contract = WMAZ (white maize)

Contract size = 100 metric tons

Number of contracts = 1

Date of valuation = 31 March

Expiry of contract = 21 September

Days to expiry (dte) = 174 days (31 March to 21 September)

t = dte / 365 = 174 / 365

rfr = 7.5% pa

SP = LCC2 732.20 (per metric ton)

Storage costs (SC) = 36 cents per ton per day

FVP (per ton) = SP + CC

= SP + [SP x (rfr x t)] + (SC x dte)

= SP x [1 + (rfr x t)] + (SC x dte)

= 2732.20 x [1 + (0.075 x 174 / 365)] + (0.36 x 174)

= 2732.20 x 1.03575 + 62.64

= 2829.88 + 62.64

= LCC2 892.52

FVP (per contract) = 100 x 2892.52

= LCC289 252.00.

# Currency futures

Currency futures are similar to foreign exchange forward contracts, and the * covered interest parity formula *(a variation of the CCM) is therefore applicable:

where:

SR = spot rate

irvc = interest rate of variable currency for period to expiry

irbc = interest rate for base currency for period to expiry

t = number of days to expiry of contract / 365.

An example is called for [base currency (i.e. the 1 unit currency) = GBP; variable currency = USD]:

SR = GBP І USD 1.5 irvc = 5.5%

vc

irbc = S.5% pa t = 1S2 І 365

FVP = SR x {[1 + (irvc x t)] І [1 + (irbc x t)]}

= USD 1.5 x {[1 + (0.055 x 1S2 І 365)] І [1 + (0.0S5 x 1S2 І 365)]} = USD 1.5 x (1.027425 І 1.0423S4)

= USD 1.5 x 0.9S5649 = USD 1.47S47.

It will be evident here that the formula is similar to the CCM, with the difference being that there are two rates of interest taken into account: the foreign rate and the local rate.

# Futures on other derivatives

As in the case of forwards (forwards on swaps) there are futures on other derivatives, for example futures on FRAs and futures on swaps.

# Other futures

Another future listed on the JSE deserves mention: the * variance future *(VARF). Variance is a statistical measure of volatility (= risk). The generally accepted measure of risk in the Finance discipline is the standard deviation of an asset's return (= the extent of deviation from the mean return). Standard deviation is closely related to variance; it is the square root of variance.

The variances and standard deviations of returns on assets (like shares) change considerably from period to period. It is also a major input in the pricing of options. There is a need by some investors to hedge against this risk, and certain speculators seek exposure to this risk. These two parties make the trading of this instrument a possibility.

In short, a variance future is a futures contract on realised annualised variance of returns on assets / indices. This instrument is regarded by some as a new asset class.