9.1 Pricing and Valuation of Forward Commitments: Equity, Fixed Income & FX

9.1 Pricing and Valuation of Forward Commitments: Equity, Fixed Income & FX

Core Insight: A forward commitment is a legally binding contract obligating the buyer (long) to purchase and the seller (short) to deliver an underlying asset at a predetermined price on a specified future date. At CFA Level II, the critical foundation is distinguishing between Pricing (determining the forward price $F_0(T)$ at inception such that the contract has zero initial market value, $V_0 = 0$) and Valuation (measuring the mark-to-market value $V_t(T)$ to the long or short counterparty at any point during the life of the contract, $0 < t \le T$). Mastery requires calculating no-arbitrage forward prices, carry benefits and costs across asset classes, bond futures conversion factors, Forward Rate Agreements (FRAs) with discounted cash settlement, and FX parity relationships.

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1. The Pricing vs. Valuation Framework

To avoid costly errors on the exam, candidates must rigorously maintain the conceptual divide between the contract price and the contract value:

                                  FORWARD CONTRACT TIMELINE
  Time t = 0 (Inception)              Time t (During Life)              Time T (Maturity)
  ┌───────────────────────┐           ┌──────────────────────┐          ┌───────────────────────┐
  │ Pricing: Set F_0(T)   │           │ Valuation:           │          │ Payoff to Long:       │
  │ No money changes hands│ ────────► │ V_t = PV of expected │ ───────► │ V_T = S_T - F_0(T)    │
  │ Contract Value:       │           │ difference from F_0  │          │ Payoff to Short:      │
  │ V_0(T) = 0            │           │ V_t != 0 (M-to-M)    │          │ -V_T = F_0(T) - S_T   │
  └───────────────────────┘           └──────────────────────┘          └───────────────────────┘

Core Principles

• Pricing ($F_0$): Forward price agreed upon at $t=0$. It represents the compounded future value of the spot price after accounting for the carrying costs (interest financing, storage) and carrying benefits (dividends, coupons, convenience yields) over period $T$.
• Valuation ($V_t$): The monetary worth of an existing forward contract. As the spot price $S_t$, interest rates $r$, or dividend yields fluctuate, $V_t$ shifts away from zero. For the long position, $V_t > 0$ when the spot price rises above expectations, while $V_t < 0$ for the short position.
• Forward vs. Futures Pricing: Forwards are private, bilateral, over-the-counter (OTC) agreements settled at maturity with credit risk. Futures are standardized, exchange-traded contracts that are marked-to-market daily with margin requirements and virtually zero counterparty default risk. When risk-free interest rates are deterministic (constant or predictable), forward prices and futures prices are mathematically identical. However, when interest rates are stochastic and positively correlated with underlying asset prices, futures contracts are more attractive to long investors (gains are received immediately and reinvested at higher interest rates, while losses occur when financing rates are lower), causing the futures price to exceed the forward price ($F_{futures} > F_{forward}$).

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2. Equity Forwards and Futures

Equity securities may generate cash inflows (discrete cash dividends) or continuous dividend yields over the holding period. These benefits reduce the net cost of carry.

Continuous Dividend Yield (Index Forwards)

When an equity index provides a continuous dividend yield $q$ and the annualized risk-free rate is $r$ compounded continuously:

$\mathbf{F_0(T) = S_0 e^{(r - q)T}}$

During the life of the contract at time $t$, with remaining maturity $\tau = T - t$, the mark-to-market value to the long position is:

$\mathbf{V_t(T) = S_t e^{-q\tau} - F_0(T) e^{-r\tau}}$

Discrete Cash Dividends (Single-Stock Forwards)

When an individual stock pays known discrete dividends $D_i$ at time $t_i < T$:

$\text{PV}(\text{Dividends}) = \sum_{i=1}^n D_i (1 + r)^{-t_i}$ $\mathbf{F_0(T) = [S_0 - \text{PV}(\text{Dividends})](1 + r)^T = S_0(1 + r)^T - \text{FV}(\text{Dividends})}$

The value to the long position at time $t$ with remaining dividends $\text{PV}_{t,\tau}(\text{Dividends})$ is:

$\mathbf{V_t(T) = [S_t - \text{PV}_{t,\tau}(\text{Dividends})] - F_0(T)(1 + r)^{-(T - t)}}$

Arbitrage Strategy: If market forward price $F_{market} > F_0(T)$, the forward is overpriced. Arbitrageur executes a Cash-and-Carry Arbitrage: borrow cash at rate $r$, buy spot stock $S_0$, and sell the forward contract short at $F_{market}$. At maturity, deliver the stock to receive $F_{market}$ and repay the loan balance $S_0(1+r)^T - \text{FV}(Div)$, locking in a riskless profit of $F_{market} - F_0(T) > 0$.

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3. Fixed Income Forwards and Bond Futures

Fixed-income securities introduce coupon payments and accrued interest conventions.

Forward Contracts on Coupon-Bearing Bonds

A bond's full price (dirty price) includes accrued interest: $S_0 = \text{Clean Price}_0 + AI_0$.

$\mathbf{F_0(T) = [S_0 - \text{PV}(\text{Coupons})](1 + r)^T = S_0(1 + r)^T - \text{FV}(\text{Coupons})}$

To quote the forward clean price at maturity $T$, subtract the accrued interest at maturity ($AI_T$):

$\mathbf{F_0^{clean}(T) = F_0(T) - AI_T}$

Bond Futures, Conversion Factors & Cheapest-to-Deliver (CTD)

Government bond futures contracts (such as US Treasury bond futures) allow the short seller to deliver any bond from a basket of eligible bonds with varying maturities and coupon rates. To standardize delivery, the exchange assigns each eligible bond a Conversion Factor ($CF$), representing the approximate clean price of $1 par value of that bond if it yielded 6.0% at the futures delivery date.

$\text{Delivery Invoice Price Paid by Long} = (\text{Quoted Futures Price} \times CF) + AI_T$

$\mathbf{\text{Quoted Futures Price (QFP)} = \frac{1}{CF} [S_0(1 + r)^T - \text{FV}(\text{Coupons}) - AI_T] = \frac{F_0^{clean}(T)}{CF}}$

                               CTD SELECTION MECHANISM
  ┌───────────────────────────┐         ┌──────────────────────────────────────┐
  │ Basket of Eligible Bonds  │ ──────► │ Calculate Net Basis for Each Bond:   │
  │ Bonds with various        │         │ Net Basis = Spot Clean Price         │
  │ coupons & maturities      │         │   - (Futures Price x CF)             │
  └───────────────────────────┘         └──────────────────┬───────────────────┘
                                                           │
                                                           ▼
                                        ┌──────────────────────────────────────┐
                                        │ Cheapest-to-Deliver (CTD) Bond:      │
                                        │ • Minimizes Net Basis (Delivery Cost)│
                                        │ • Maximizes Implied Repo Rate (IRR)  │
                                        └──────────────────────────────────────┘

• Cheapest-to-Deliver (CTD) Rule: The short position selects the eligible bond that minimizes the net basis (gross basis minus carry) or maximizes the implied repo rate (IRR).
• Yield Shift Heuristic:
  • If market yields are above 6%, the CTD is typically a long-maturity, low-coupon bond.
  • If market yields are below 6%, the CTD is typically a short-maturity, high-coupon bond.

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4. Forward Rate Agreements (FRAs)

A Forward Rate Agreement (FRA) is an over-the-counter forward contract on a short-term reference interest rate (such as SOFR term rates or Euribor). The long position in an FRA agrees to pay a fixed interest rate and receive a floating market reference rate on an agreed notional principal.

Advanced FRA Notation ($h \times m$)

• Notation: An $h \times m$ FRA indicates:
  • The contract expires in $h$ months (the forward starting date).
  • The underlying loan terminates in $m$ months from today.
  • The underlying rate covers a borrowing period of $m - h$ months ($d_{m-h} = [m - h] \times 30$ days in 30/360 convention).
  • Example: A $3 \times 9$ FRA expires in 3 months ($h=3$), covers a 6-month borrowing rate ($m-h = 6$), and settles based on the 180-day market rate prevailing at month 3.

                                  FRA $3 \times 9$ TIMELINE
  Today (t = 0)                   Month 3 (t = h)                        Month 9 (t = m)
  ┌──────────────────────┐        ┌────────────────────────────┐         ┌───────────────────────┐
  │ FRA is initiated     │        │ FRA Expires & Settles      │         │ Underlying loan period│
  │ Forward rate FRA_0   │ ─────► │ Reference rate L_h observed│ ──────► │ ends. (Payoff occurs  │
  │ locked in. V_0 = 0   │        │ Payoff discounted to t = h │         │ at h, not at m!)      │
  └──────────────────────┘        └────────────────────────────┘         └───────────────────────┘
                                  ◄───────── 6-Month Rate Span (180 Days) ────────►

FRA Pricing at Initiation ($t = 0$)

Using market spot zero-coupon rates $r_h$ (for $h$ days) and $r_m$ (for $m$ days) under the actual/360 money market convention:

$\mathbf{FRA_0(h, m - h) = \left[ \frac{1 + r_m \left(\frac{m}{360}\right)}{1 + r_h \left(\frac{h}{360}\right)} - 1 \right] \times \frac{360}{m - h}}$

FRA Valuation During Life ($t < h$)

At intermediate time $t$, value an existing FRA by comparing the original contract rate $FRA_0$ to the current market rate for an equivalent replacement FRA, $FRA_t$, discounted back to time $t$:

$\mathbf{V_t = \text{Notional} \times \left[ \frac{(FRA_t - FRA_0) \times \frac{m - h}{360}}{1 + r_{m - t} \left(\frac{m - t}{360}\right)} \right]}$

Cash Settlement Payoff at Expiration ($t = h$)

In the cash market, loan interest is paid in arrears at maturity ($m$). However, FRAs settle in cash at expiration ($h$). Therefore, the interest savings must be discounted back to month $h$ using the prevailing reference rate $L_h$:

$\mathbf{\text{Payoff to Long at Date } h = \text{Notional} \times \frac{(L_h - FRA_0) \times \left(\frac{m - h}{360}\right)}{1 + L_h \times \left(\frac{m - h}{360}\right)}}$

Exam Trap Alert: Forgetting to discount the cash settlement payoff by $[1 + L_h (d_{m-h}/360)]$ is the single most common student error in FRA questions. Because the payment occurs at the start of the loan period ($h$) rather than the end ($m$), this discounting factor is mandatory.

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5. Comprehensive Worked Numerical Model: FRA Pricing & Settlement

Scenario: Horizon Asset Management enters into a $3 \times 9$ FRA with a notional principal of $50,000,000. The current market term structure of spot zero-coupon money market rates (actual/360) is:

• 90-day spot rate ($r_{90}$): 4.00% (0.0400)
• 270-day spot rate ($r_{270}$): 4.80% (0.0480)

Step 1: Calculate the No-Arbitrage FRA Rate at Initiation ($FRA_0$)

$\text{Compounded 270-day factor} = 1 + 0.0480 \times \left(\frac{270}{360}\right) = 1 + 0.0360 = 1.036000$ $\text{Compounded 90-day factor} = 1 + 0.0400 \times \left(\frac{90}{360}\right) = 1 + 0.0100 = 1.010000$ $\text{Forward 180-day period return} = \frac{1.036000}{1.010000} - 1 = 1.0257426 - 1 = 0.0257426$ $\mathbf{FRA_0(3, 6)} = 0.0257426 \times \left(\frac{360}{180}\right) = 0.051485 = \mathbf{5.1485\%}$

Step 2: Calculate Payoff at Settlement (Day 90)

At day 90 ($t = h$), the 180-day reference rate ($L_{90}$) fixes at 5.80% (0.0580).

1. Calculate un-discounted interest differential: $\Delta \text{Interest} = \text{Notional} \times (L_{90} - FRA_0) \times \left(\frac{180}{360}\right)$ $\Delta \text{Interest} = \$50,000,000 \times (0.0580 - 0.051485) \times 0.50 = \$50,000,000 \times 0.006515 \times 0.50 = \$162,875.00$

2. Discount payoff to settlement date (Day 90) using $L_{90} = 5.80%$: $\text{Discount Factor} = 1 + 0.0580 \times \left(\frac{180}{360}\right) = 1 + 0.0290 = 1.029000$ $\mathbf{\text{Cash Settlement Paid to Long}} = \frac{\$162,875.00}{1.029000} = \mathbf{\$158,284.74}$

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6. Foreign Exchange (FX) Forwards

FX forward pricing is governed by Covered Interest Rate Parity (CIRP). Exchange rates are quoted as Price Currency / Base Currency ($P/B$), representing units of price currency per 1 unit of base currency.

Covered Interest Rate Parity Forward Pricing

$\mathbf{F_{P/B} = S_{P/B} \times \left[ \frac{1 + r_{price} \times \left(\frac{\text{days}}{360}\right)}{1 + r_{base} \times \left(\frac{\text{days}}{360}\right)} \right]}$

• Base Currency Rule: The currency with the higher interest rate must trade at a forward discount (forward rate < spot rate) to eliminate risk-free carry trade arbitrage.
• Forward Points: Forward quotes are often displayed as forward points ($F_{P/B} - S_{P/B}$): $\text{Forward Premium/Discount} \approx S_{P/B} \times (r_{price} - r_{base}) \times \left(\frac{\text{days}}{360}\right)$

Valuation of FX Forward During Life ($t > 0$)

To value an existing long position in the base currency (contracted at forward rate $F_0$) at time $t$ when the new forward rate is $F_t$ and remaining maturity is $\tau = T - t$:

$\mathbf{V_t = \text{Notional}_{base} \times \frac{F_t - F_0}{1 + r_{price} \times \left(\frac{\tau}{360}\right)} = \text{Notional}_{base} \times \left[ \frac{S_t}{1 + r_{base} \left(\frac{\tau}{360}\right)} - \frac{F_0}{1 + r_{price} \left(\frac{\tau}{360}\right)} \right]}$

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7. Forward Pricing & Valuation Formulas Summary Reference