Chapter 12: Interest Rate Risk Management
1. Master Formula Bank
1. Forward Rate Agreement (FRA) Discounted Cash Settlement Formula:
$$\text{Settlement Amount} = \frac{N \times (RR - FR) \times \left(\frac{dtm}{DY}\right)}{1 + RR \times \left(\frac{dtm}{DY}\right)}$$ Where $N$ = Notional, $RR$ = Reference Rate, $FR$ = Forward Rate, $dtm$ = FRA days, $DY$ = Day-count (360 or 365).
$$\text{Settlement Amount} = \frac{N \times (RR - FR) \times \left(\frac{dtm}{DY}\right)}{1 + RR \times \left(\frac{dtm}{DY}\right)}$$ Where $N$ = Notional, $RR$ = Reference Rate, $FR$ = Forward Rate, $dtm$ = FRA days, $DY$ = Day-count (360 or 365).
2. Bond Price Sensitivity & Duration:
$$D^* = \frac{D_{\text{Macaulay}}}{1 + \frac{y}{m}} \quad \implies \quad \frac{\Delta P}{P} \approx -D^* \times \Delta y + \frac{1}{2} C \times (\Delta y)^2$$
$$D^* = \frac{D_{\text{Macaulay}}}{1 + \frac{y}{m}} \quad \implies \quad \frac{\Delta P}{P} \approx -D^* \times \Delta y + \frac{1}{2} C \times (\Delta y)^2$$
3. Generic Swap Net Settlement:
$$\text{Fixed Leg} = N \times \text{AIC} \times \left(\frac{180}{360}\right)$$ $$\text{Floating Leg} = N \times \text{SOFR} \times \left(\frac{\text{Actual Days}}{360}\right)$$ $$\text{Net Amount Paid by Fixed Payer} = \text{Fixed Leg} - \text{Floating Leg}$$
$$\text{Fixed Leg} = N \times \text{AIC} \times \left(\frac{180}{360}\right)$$ $$\text{Floating Leg} = N \times \text{SOFR} \times \left(\frac{\text{Actual Days}}{360}\right)$$ $$\text{Net Amount Paid by Fixed Payer} = \text{Fixed Leg} - \text{Floating Leg}$$
4. Amortized Cap Option Premium (PVAF Method):
$$\text{Periodic Amortized Premium} = \frac{\text{Total Lump Sum Premium Paid}}{\text{PVAF}(i, t)}$$ Where $i$ = Periodic interest discount rate and $t$ = Number of reset periods.
$$\text{Periodic Amortized Premium} = \frac{\text{Total Lump Sum Premium Paid}}{\text{PVAF}(i, t)}$$ Where $i$ = Periodic interest discount rate and $t$ = Number of reset periods.
5. Overnight Index Swap (OIS) Floating Interest:
$$\text{Compounded Amount} = N \times \prod_{k=1}^n \left( 1 + \text{MIBOR}_k \times \frac{d_k}{365} \right) - N$$ Where $d_k$ = days spanned (1 day on weekdays, 2 days on Saturday covering Sunday).
$$\text{Compounded Amount} = N \times \prod_{k=1}^n \left( 1 + \text{MIBOR}_k \times \frac{d_k}{365} \right) - N$$ Where $d_k$ = days spanned (1 day on weekdays, 2 days on Saturday covering Sunday).
2. FRA Notation & Settlement Guide
| Notation ($m \times n$) | Fixing Date (Settlement) | Underlying Loan Tenor | Relevant Reference Rate |
|---|---|---|---|
| $3 \times 6$ FRA | 3 months from today | $6 - 3 = 3$ months | 3-month benchmark rate on fixing date |
| $6 \times 9$ FRA | 6 months from today | $9 - 6 = 3$ months | 3-month benchmark rate on fixing date |
| $6 \times 12$ FRA | 6 months from today | $12 - 6 = 6$ months | 6-month benchmark rate on fixing date |
Settlement Direction Rule:
• If $RR > FR$: Bank pays Buyer (Borrower hedged successfully against rate hike).
• If $RR < FR$: Buyer pays Bank (Borrower pays cost of insurance when market rates fall).
• If $RR > FR$: Bank pays Buyer (Borrower hedged successfully against rate hike).
• If $RR < FR$: Buyer pays Bank (Borrower pays cost of insurance when market rates fall).
3. Global Alternative Reference Rates (ARRs) Quick Matrix
| Currency | Rate Name | Administrator | Secured vs Unsecured |
|---|---|---|---|
| USD | SOFR (Secured Overnight Financing Rate) | Federal Reserve Bank of NY | Secured (US Treasury Repos) |
| GBP | SONIA (Sterling Overnight Index Average) | Bank of England | Unsecured |
| EUR | €STR (Euro Short-Term Rate) | European Central Bank | Unsecured |
| JPY | TONAR (Tokyo Overnight Average Rate) | Bank of Japan | Unsecured |
| INR | MIBOR (Mumbai Interbank Offered Rate) | FBIL | Unsecured Call Money |
4. Interest Rate Options Comparison Matrix
| Instrument | Option Type | Hedging User | Strike Condition for Exercise | Net Cost Position |
|---|---|---|---|---|
| Cap | Call Option on Interest Rate | Floating Rate Borrower | $R_{\text{market}} > R_{\text{strike}}$ | Pays upfront premium; cost capped at strike. |
| Floor | Put Option on Interest Rate | Floating Rate Investor/Lender | $R_{\text{market}} < R_{\text{strike}}$ | Pays upfront premium; return floored at strike. |
| Collar | Long Cap + Short Floor | Borrower seeking lower hedging cost | Band between Floor and Cap | Net premium reduced or zero ("Zero-Cost Collar"). |
5. Interest Rate Swaps (IRS) & Overnight Indexed Swaps (OIS)
Core Concept: Contract between two counterparties to exchange interest rate cash flows based on a specified notional principal without exchanging the underlying principal.
| Swap Parameter | Fixed Rate Payer (Borrower Hedge) | Floating Rate Payer (Asset Hedge) |
|---|---|---|
| Cash Flow Obligation | Pays fixed coupon rate ($R_{\text{swap}}$); receives floating (e.g. MIBOR / SOFR). | Pays floating benchmark rate; receives fixed coupon rate. |
| Market Expectation | Anticipates rising interest rates; locks in fixed borrowing cost. | Anticipates declining interest rates; benefits from falling floating costs. |
| Net Settlement Flow | $\text{Net Cash Flow} = (R_{\text{floating}} - R_{\text{fixed}}) \times N \times \frac{d}{360\text{ or }365}$ | $\text{Net Cash Flow} = (R_{\text{fixed}} - R_{\text{floating}}) \times N \times \frac{d}{360\text{ or }365}$ |
| OIS Structure | Floating leg is the geometric compound daily average of overnight rate (e.g. daily overnight MIBOR) settled against fixed rate at maturity. | |
6. Swaptions Architecture & Decision Rules
| Swaption Type | Holder Has Right To | Exercise Rule | Core Motivation |
|---|---|---|---|
| Payer Swaption | Pay Fixed & Receive Floating | Market Swap Rate $>$ Strike Rate | Hedge against rising fixed borrowing costs on anticipated future debt. |
| Receiver Swaption | Receive Fixed & Pay Floating | Market Swap Rate $<$ Strike Rate | Hedge against declining fixed yields on future portfolio investments. |
6. ICAI Exam Pitfalls & Traps
⚠️ Pitfall 1: Forgetting to Discount FRA Settlement
Never just calculate $N \times (RR - FR) \times \frac{dtm}{DY}$. That is the interest at maturity! Settlement happens at fixing date (start of loan period), so you must divide by $[1 + RR \times \frac{dtm}{DY}]$.
Never just calculate $N \times (RR - FR) \times \frac{dtm}{DY}$. That is the interest at maturity! Settlement happens at fixing date (start of loan period), so you must divide by $[1 + RR \times \frac{dtm}{DY}]$.
⚠️ Pitfall 2: Day-Count Bases in Swaps
Generic fixed swap legs use 30/360 ($\frac{180}{360} = 0.50$). Floating legs use actual days / 360 (e.g., $\frac{181}{360} = 0.502778$). Always watch for the differing denominators!
Generic fixed swap legs use 30/360 ($\frac{180}{360} = 0.50$). Floating legs use actual days / 360 (e.g., $\frac{181}{360} = 0.502778$). Always watch for the differing denominators!
⚠️ Pitfall 3: Weekend Handling in OIS Calculations
When compiling daily MIBOR tables, Saturday's rate applies for 2 days (Saturday + Sunday) because Sunday is a banking holiday. Multiply by $\frac{2}{365}$.
When compiling daily MIBOR tables, Saturday's rate applies for 2 days (Saturday + Sunday) because Sunday is a banking holiday. Multiply by $\frac{2}{365}$.