Pricing and Valuation of Forward Commitments

2024 Curriculum CFA Program Level II Derivatives

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Introduction

Forward commitments include forwards, futures, and swaps. A forward contract is a promise to buy or sell an asset at a future date at a price agreed to at the contract’s initiation. The forward contract has a linear payoff function, with both upside and downside risk.

A swap is essentially a promise to undertake a transaction at a set price or rate at several dates in the future. The technique we use to price and value swaps is to identify and construct a portfolio with cash flows equivalent to those of the swap. Then, we can use tools, such as the law of one price, to determine swap values from simpler financial instruments, such as a pair of bonds with a cash flow pattern similar to those of our swap.

Look out for the big picture: value additivity, arbitrage, and the law of one price are important valuation concepts.

Forwards and swaps are widely used in practice to manage a broad range of market risks. As well, more complex derivative instruments can sometimes be understood in terms of their basic building blocks: forwards and option-based components. Here are just some of the many and varied uses for forwards, futures, and swaps that you might encounter in your investment career:

Use of equity index futures and swaps by a private wealth manager to hedge equity risk in a low tax basis, concentrated position in his high-net-worth client’s portfolio.
Use of interest rate swaps by a defined benefits plan manager to hedge interest rate risk and to manage the pension plan’s duration gap.
Use of derivatives (total return swaps, equity futures, bond futures, etc.) overlays by a university endowment for tactical asset allocation and portfolio rebalancing.
Use of interest rate swaps by a corporate borrower to synthetically convert floating-rate debt securities to fixed-rate debt securities (or vice versa).
Use of VIX futures and inflation swaps by a firm’s market strategist to infer expectations about market volatility and inflation rates, respectively.

Learning Outcomes

The member should be able to:

describe the carry arbitrage model without underlying cashflows and with underlying cashflows;
describe how equity forwards and futures are priced, and calculate and interpret their no-arbitrage value;
describe how interest rate forwards and futures are priced, and calculate and interpret their no-arbitrage value;
describe how fixed-income forwards and futures are priced, and calculate and interpret their no-arbitrage value;
describe how interest rate swaps are priced, and calculate and interpret their no-arbitrage value;
describe how currency swaps are priced, and calculate and interpret their no-arbitrage value;
describe how equity swaps are priced, and calculate and interpret their no-arbitrage value.

Summary

This reading on forward commitment pricing and valuation provides a foundation for understanding how forwards, futures, and swaps are both priced and valued.

Key points include the following:

The arbitrageur would rather have more money than less and abides by two fundamental rules: Do not use your own money, and do not take any price risk.
The no-arbitrage approach is used for the pricing and valuation of forward commitments and is built on the key concept of the law of one price, which states that if two investments have the same future cash flows, regardless of what happens in the future, these two investments should have the same current price.
Throughout this reading, the following key assumptions are made:
- Replicating and offsetting instruments are identifiable and investable.
- Market frictions are nil.
- Short selling is allowed with full use of proceeds.
- Borrowing and lending are available at a known risk-free rate.
Carry arbitrage models used for forward commitment pricing and valuation are based on the no-arbitrage approach.
With forward commitments, there is a distinct difference between pricing and valuation. Pricing involves the determination of the appropriate fixed price or rate, and valuation involves the determination of the contract’s current value expressed in currency units.
Forward commitment pricing results in determining a price or rate such that the forward contract value is equal to zero.
Using the carry arbitrage model, the forward contract price (F0) is:

F0 = FV(S0) = S0(1 + r)T (assuming annual compounding, r)

F 0 = FV ( S 0 ) = S 0 exp r c T (assuming continuous compounding, rc )
The key forward commitment pricing equations with carry costs (CC) and carry benefits (CB) are:

F0 = FV[S0 + CC0 – CB0] (with discrete compounding)

F 0 = S 0 exp ( r c + CC − CB ) T (with continuous compounding)

Futures contract pricing in this reading can essentially be treated the same as forward contract pricing.

The value of a forward commitment is a function of the price of the underlying instrument, financing costs, and other carry costs and benefits.
The key forward commitment valuation equations are:

Long Forward: V t = PV [ F t − F 0 ] = [ F t − F 0 ] ( 1 + r ) T − t

and

Short Forward: − V t = PV [ F 0 − F t ] = [ F 0 − F t ] ( 1 + r ) T − t ,

With the PV of the difference in forward prices adjusted for carry costs and benefits. Alternatively,

Long Forward: V t = S t − P V [ F 0 ] = S t − F 0 ( 1 + r ) T − t

and

Short Forward: − V t = PV [ F 0 ] − S t = F 0 ( 1 + r ) T − t − S t
With equities and fixed-income securities, the forward price is determined such that the initial forward value is zero.
A forward rate agreement (FRA) is a forward contract on interest rates. The FRA’s fixed interest rate is determined such that the initial value of the FRA is zero.
FRA settlements amounts at Time h are:

Pay-fixed (Long): NA × {[Lm – FRA0] tm}/[1 + Dmtm] and

Receive-fixed (Short): NA × {FRA0 – Lm] tm}/[1 + Dmtm].
The FRA’s fixed interest rate (annualized) at contract initiation is:

FRA0 = {[1 + LTtT]/[1 + Lhth] – 1}/tm.
The Time g value of an FRA initiated at Time 0 is:

Long FRA: Vg = NA × {[FRAg – FRA0] tm}/[1+ D(T–g) t(T–g)] and

Short FRA: –Vg = NA × {[FRA0 – FRAg] tm}/[1+ D(T–g) t(T–g)].
The fixed-income forward (or futures) price including conversion factor (i.e., adjusted price) is:

F0 = Q0 × CF = FV[S0 + CC0 – CB0] = FV[B0 + AI0 – PVCI],

and the conversion factor adjusted futures price (i.e., quoted futures price) is:

Q0 = [1/CF] {FV [B0 + AI0] – AIT – FVCI}.
The general approach to pricing and valuing swaps as covered here is using a replicating portfolio or offsetting portfolio of comparable instruments, typically bonds for interest rate and currency swaps and equities plus bonds for equity swaps.
The swap pricing equation, which sets rFIX for the implied fixed bond in an interest rate swap, is:

r F I X = 1 − PV n ( 1 ) ∑ i = 1 n PV i ( 1 ) .
The value of an interest rate swap at a point in Time t after initiation is the sum of the present values of the difference in fixed swap rates times the stated notional amount, or:

V S W A P , t = NA × ( FS 0 − FS t ) × ∑ i = 1 n PV i (Value of receive-fixed swap)

and

− V S W A P , t = NA × ( FS t − FS 0 ) × ∑ i = 1 n PV i (Value of pay-fixed swap).
With a basic understanding of pricing and valuing a simple interest rate swap, it is a straightforward extension to pricing and valuing currency swaps and equity swaps.
The solution for each of the three variables, one notional amount (NAa) and two fixed rates (one for each currency, a and b), needed to price a fixed-for-fixed currency swap are :

NA a = S 0 × NA b; r a = 1 − PV n , a ( 1 ) ∑ i = 1 n PV i , a ( 1 ) and r b = 1 − PV n , b ( 1 ) ∑ i = 1 n PV i , b ( 1 ) .
The currency swap valuation equation, for valuing the swap at time t (after initiation), can be expressed as:

V C S = NA a ( r F i x , a ∑ i = 1 n PV i ( 1 ) + PV n ( 1 ) ) − S t NA b ( r F i x , b ∑ i = 1 n PV i ( 1 ) + PV n ( 1 ) ) .
For a receive-fixed, pay equity swap, the fixed rate (rFIX) for the implied fixed bond that makes the swap’s value (VEQ) equal to “0” at initiation is:

r F I X = 1 − P V n ( 1 ) ∑ i = 1 n P V i ( 1 ) .
The value of an equity swap at Time t (VEQ,t), after initiation, is:

VEQ,t = VFIX(C0) – (St/St–1)NAE – PV(Par – NAE)

where VFIX (C0) is the Time t value of a fixed-rate bond initiated with coupon C0 at Time 0, St is the current equity price, St–1 is the equity price at the last reset date, and PV() is the PV function from the swap maturity date to Time t.

2.5 PL Credit

If you are a CFA Institute member don’t forget to record Professional Learning (PL) credit from reading this article.