Inverse Floater Valuation

Step 1: Pricing the Constituent Components

The replication formula, $P_{\text{Inverse}}=(M+1)P_{\text{Fixed}}-M{P}_{\text{Floater}}$, requires us to find the prices of the fixed-rate bond and the floater. Let's tackle each one.

A. Valuing the Fixed-Rate Bond ($P_{\text{Fixed}}$)

The fixed-rate bond within the replication portfolio is a standard, "plain vanilla" bond. As you'll recall from Module 2, the most accurate way to price such a bond is by discounting each of its cash flows using the corresponding spot rate from the zero-coupon yield curve (ZCYC).

The price is the sum of the present values of all its cash flows:

where:

• ${\text{Coupon}}_i$ is the coupon payment at time $t_i$.
• $\text{Principal}$ is the face value returned at maturity.
• $z_i$ is the annualized spot rate for a cash flow at time $t_i$.
• $n$ is the total number of coupon payments.

B. Valuing the Floating-Rate Note ($P_{\text{Floater}}$)

Next, we need the price of the floating-rate note. In Module 6, we established a crucial property of standard floaters: on any coupon reset date, a floater is worth its par value.

This is because on a reset date, the coupon is adjusted to the current market rate, ensuring that the bond's value immediately after the coupon payment is precisely par. Therefore, for our valuation purposes, which are typically performed on these reset dates, we can confidently set:

This simplification is fundamental to making the replication strategy practical.

Step 2: Deriving the Inverse Floater Price - A Worked Example

With the valuation methods for our components established, we can now derive the price of an inverse floater. We will use a clear example that demonstrates the entire process.

The following reading walks through the valuation of an inverse floater created from an "even split" of a fixed-rate bond. In this common structure, the leverage factor $M$ is 1 (i.e., $N_{\text{Floater}}=N_{\text{Inverse}}$), so our general formula simplifies to:

Let's break down the calculation from the reading to ensure every step is clear.

1. The Setup:

• Inverse Floater: A 2-year inverse floater paying 11% minus the floating rate.
• Replication: This is equivalent to being long two units of a 2-year, 5.5% fixed-rate bond and short one unit of a floater. The fixed rate of 5.5% is half of the 11% cap in the inverse's coupon, confirming the structure $K=(M+1)c\to 11\%=(1+1)\times 5.5\%$.
• Zero Rates: The market provides the following semi-annually compounded spot rates:
  • $z_{0.5}$ = 5.54%
  • $z_{1.0}$ = 5.45%
  • $z_{1.5}$ = 5.47%
  • $z_{2.0}$ = 5.50%

2. Price the Fixed-Rate Component ($P_{\text{Fixed}}$):

• The bond has a $100 par value, a 5.5% annual coupon (paid semi-annually), and a 2-year maturity.
• The semi-annual coupon payment is $\$100\times 0.055/2=\$2.75$.
• We discount each cash flow with the appropriate spot rate (remembering to use semi-annual compounding for the rates): $$P_{\text{Fixed}}=\frac{2.75}{(1+0.0554/2{)}^1}+\frac{2.75}{(1+0.0545/2{)}^2}+\frac{2.75}{(1+0.0547/2{)}^3}+\frac{102.75}{(1+0.0550/2{)}^4}$$ $$P_{\text{Fixed}}=2.675+2.607+2.535+92.185=\$100.0019$$ As the reading states, the price of the 5.5% fixed-coupon bond is 100.0019.

3. Value the Floater Component ($P_{\text{Floater}}$):

• As established, a standard floater on a reset date is valued at its par value. Assuming a $100 par, $P_{\text{Floater}}=\text{100}$.

4. Calculate the Inverse Floater Price ($P_{\text{Inverse}}$):

• Now, we apply the replication formula for $M=1$: $$P_{\text{Inverse}}=2\times P_{\text{Fixed}}-P_{\text{Floater}}$$ $$P_{\text{Inverse}}=2\times 100.0019-100=200.0038-100=\text{\$100.0038}$$

This final value is the arbitrage-free price of the inverse floater. We arrived at it not by trying to forecast the floating rate, but by pricing its much simpler, static components.

Step 3: An Alternative Perspective - Replication with Swaps

To solidify your understanding, let's look at the same problem through a slightly different lens. An interest rate swap is an agreement to exchange fixed-rate payments for floating-rate payments. Its value can also be expressed through decomposition:

(This assumes you are receiving fixed and paying floating).

Given this, we can re-frame our inverse floater replication. If:

• Inverse Floater = $P_{\text{Fixed}}+(P_{\text{Fixed}}-P_{\text{Floater}})$
  Then it follows that:
• Inverse Floater = Fixed-Rate Bond + Interest Rate Swap

This shows that holding an inverse floater is economically equivalent to holding a standard bond and also entering into a swap to receive fixed payments. The following worked problem demonstrates this approach.

Let's trace the logic from Problem 18:

• Given:
  • Par value: $100,000
  • Price of 10-year, 6% fixed-rate bond: 97.99502% of par, or $97,995.02.
  • Value of 10-year floater: $100,000 (par).
• Part (a) - Value of Swap:
  • swap = fixed - floater
  • swap = 97,995.02 - 100,000 = -$2,004.98
  • The swap has a negative value because the fixed rate of 6% is below the prevailing market rates that priced the bond at a discount.
• Part (d) - Synthesizing the Inverse Floater:
  • The goal is to create an inverse floater paying 12% minus floating. This corresponds to the $M=1$ case, as $12\%=2\times 6\%$.
  • The replicating portfolio is: Long $100,000 par of the 6% fixed-rate bond plus Long $100,000 notional of the 6% swap.
• Part (e) - Market Value of Inverse Floater:
  • market value = Price_Fixed + Value_Swap
  • market value = 97,995.02 + (-2,004.98) = $95,990.04

This example perfectly validates our replication formula, arriving at the inverse floater's price by summing the values of its constituent bond and swap components.

Conclusion

In this lesson, we put the inverse floater replication strategy into action. You have learned how to derive the price of this complex instrument by breaking it down into a leveraged position in a fixed-rate bond and a short position in a floating-rate note.

Key Takeaways:

• Valuation by Replication: The price of an inverse floater is found by executing its replication formula: $P_{\text{Inverse}}=(M+1)P_{\text{Fixed}}-M{P}_{\text{Floater}}$.
• Pricing the Components: This process involves two familiar steps:
  1. Pricing the underlying fixed-rate bond by discounting its cash flows with the zero-coupon curve.
  2. Valuing the floating-rate note at its par value (on a reset date).
• Practical Application: We worked through two numerical examples, confirming that whether you frame the replication directly or by using an interest rate swap, the underlying principle of decomposition holds and yields the same arbitrage-free price.

You have now mastered the complete theoretical and mathematical framework for pricing inverse floaters. In our next lesson, we will translate this knowledge into code, implementing a Python function to perform this replication-based pricing automatically.