Explanation

To determine the notional amounts of the 5-year and 10-year swaps needed to hedge a USD 100 million notional amount of 7-year swaps, we need to set up a hedge that makes the portfolio insensitive to changes in both the 5-year and 10-year swap rates.

Step 1: Understanding the P&L Equation

The P&L of the position is given by:

$$-F^7 \times \left(\frac{DV01^7}{100}\right) \times \Delta y_t^7 - F^5 \times \left(\frac{DV01^5}{100}\right) \times \Delta y_t^5 - F^{10} \times \left(\frac{DV01^{10}}{100}\right) \times \Delta y_t^{10}$$

Where:

• $F^7 = 100$ million (notional of 7-year swap)
• $F^5$ = notional of 5-year swap (to be determined)
• $F^{10}$ = notional of 10-year swap (to be determined)
• $DV01^7 = 0.084$, $DV01^5 = 0.061$, $DV01^{10} = 0.115$

Step 2: Substituting the Regression Equation

From the regression model:

$$\Delta y_t^7 = \alpha + \beta^5 \Delta y_t^5 + \beta^{10} \Delta y_t^{10} + \varepsilon_t$$

Where $\beta^5 = 0.2471$ and $\beta^{10} = 0.6536$

Substituting into the P&L equation and ignoring the constant term (which doesn't affect the hedge ratios):

$$[-F^7 \times (DV01^7/100) \times \beta^5 - F^5 \times (DV01^5/100)] \times \Delta y_t^5 + [-F^7 \times (DV01^7/100) \times \beta^{10} - F^{10} \times (DV01^{10}/100)] \times \Delta y_t^{10}$$

Step 3: Setting Up the Hedge Conditions

For a perfect hedge, the coefficients of $\Delta y_t^5$ and $\Delta y_t^{10}$ must both be zero:

1. For $\Delta y_t^5$:

-F^7 \times (DV01^7/100) \times \beta^5 - F^5 \times (DV01^5/100) = 0 $`$2`. For $\Delta y_t^{10}$:

-F^7 \times (DV01^7/100) \times \beta^{10} - F^{10} \times (DV01^{10}/100) = 0

### Step 4: Solving for the Notional Amounts **For the 5-year swap:**

F^5 = -F^7 \times \frac{DV01^7}{DV01^5} \times \beta^5 = -100 \times \frac{0.084}{0.061} \times 0.2471 = -34.03 \text{ million}

$$**For the 10-year swap:**$$

F^{10} = -F^7 \times \frac{DV01^7}{DV01^{10}} \times \beta^{10} = -100 \times \frac{0.084}{0.115} \times 0.6536 = -47.74 \text{ million}

The negative signs indicate that the hedge positions should be in the opposite direction to the original 7-year swap position. Since the bank is the fixed-rate payer on the 7-year swap, they should be fixed-rate receivers on both the 5-year and 10-year swaps. ### Step 5: Final Answer The correct notional amounts are: - **USD 34.03 million of 5-year swaps** - **USD 47.74 million of 10-year swaps** This corresponds to option C.