Forward Rate Agreement (FRA) Pricing & Hedging Lab
Model institutional Forward Rate Agreements (FRAs), calculate no-arbitrage forward rates from spot yield curves, simulate in-advance discounted settlement cash flows, and evaluate borrower vs. lender hedging payoff profiles.
FRA Contract Specifications
+$12,352
Paid upfront at month 34.352%
Strike basis: +10.2 bps$12,500
Discount drag: -$148$247 / bp
Term: 90 days (0.25 yrs)Quantitative Cash Flow & Payoff Visualizer
FRA Cash Settlement & Hedging Decomposition
| Component | Analytical Equation | Value | Description |
|---|---|---|---|
| Contract Structure | A × B Notation |
3 × 6 | 90 Days (0.250 Years) |
| Agreed Strike Rate (K) | Fixed Rate Contracted |
4.250% | Annualized simple rate |
| Fixing Reference Rate (L) | Realized Spot at Month A |
4.750% | +50.0 bps above strike |
| Theoretical Fair Forward (F) | [(1 + r2·t2)/(1 + r1·t1) - 1] / Δt |
4.352% | +10.2 bps market premium |
| Undiscounted Arrears P&L | Notional × (L - K) × Δt |
$12,500 | Payoff if paid at maturity t2 |
| Discounted Settlement Payoff | [Notional × (L - K) × Δt] / (1 + L·Δt) |
$12,352 | Buyer Receives |
| Effective Net Borrowing Cost | Physical Loan Interest - FRA Payoff |
4.250% | Perfect hedge locks in strike K |
FRA Settlement Cash Payoff ($)
Fair Implied Forward Rate (%)
Institutional FRA Valuation & Hedging Foundations
1. FRA Market Conventions & Cash Settlement
A Forward Rate Agreement (FRA) is quoted in terms of months to fixing and months to maturity ($A \times B$). For example, a $3 \times 6$ FRA fixes at month 3 and covers interest accrual through month 6. Unlike physical loans where interest is paid in arrears at month 6 ($t_2$), standard market conventions settle the net cash difference in advance at month 3 ($t_1$).
Payoff_{\text{Seller}} = \frac{\text{Notional} \times (K - L) \times \Delta t}{1 + L \times \Delta t}
The denominator $1 + L \cdot \Delta t$ discounts the payment from loan maturity $t_2$ back to settlement date $t_1$, accounting for the time value of money.
2. No-Arbitrage Implied Forward Rate Derivation
Under money market no-arbitrage pricing, investing capital from $0$ to $t_2$ at spot rate $r_2$ must generate the exact same terminal wealth as investing from $0$ to $t_1$ at spot rate $r_1$ and reinvesting from $t_1$ to $t_2$ at the forward rate $F$:
F(t_1, t_2) = \left[ \frac{1 + r_2 \times t_2}{1 + r_1 \times t_1} - 1 \right] \times \frac{1}{\Delta t}
When market yield curves are upward-sloping ($r_2 > r_1$), the implied forward rate $F$ exceeds the spot rate ($F > r_2 > r_1$). When inverted, $F < r_2$.
3. Synthetic Borrowing & Lending Replication
A corporate treasury desk can synthetically manufacture an FRA using spot money market transactions:
- To replicate buying an FRA (lock in borrowing rate): Borrow for period $t_2$ at $r_2$, lend the proceeds for period $t_1$ at $r_1$. At $t_1$, retrieve the loan and deploy funds into business operations; at $t_2$, repay the long loan.
- To replicate selling an FRA (lock in lending rate): Borrow for period $t_1$ at $r_1$, lend for period $t_2$ at $r_2$.
The FRA derivative eliminates the need to expand balance sheet debt, achieving identical rate protection without balance sheet leverage.
4. Corporate Hedging & Net Effective Borrowing Cost
Suppose a company plans to borrow \$10M in 3 months for 3 months at floating SOFR. To eliminate interest rate risk, the CFO buys a $3 \times 6$ FRA at strike $K = 4.25\%$:
\text{FRA Payout Received} = +1.00\% \implies \text{Net Cost} = 4.25\%
\text{If } L = 3.25\%: \text{Physical Loan Cost} = 3.25\%
\text{FRA Payout Paid} = -1.00\% \implies \text{Net Cost} = 4.25\%
In all rate environments, the corporate borrower locks in an exact effective borrowing rate equal to strike $K$.