Quantifying Negative Convexity in Mortgage-Backed Securities

This project develops a quantitative framework to model the cash flows and price sensitivity of Mortgage-Backed Securities (MBS). The primary focus is investigating the phenomenon of Negative Convexity—the asymmetric risk profile caused by the embedded homeowner prepayment option.

Key Features

• Custom Prepayment Model: Built a Python-based simulation engine for MBS valuation, featuring a path-dependent S-Curve model to quantify the impact of interest rate volatility and prepayment speeds on portfolio convexity.
• Comparative Benchmarking: Quantified the “Convexity Gap” by comparing MBS performance against a synthetic non-callable bond.
• Scenario Stress Testing: Analyzed portfolio Present Value (PV) sensitivity under ±25 and ±50 basis point interest rate shocks.

The Quantitative Challenge: Negative Convexity

Unlike standard fixed-income instruments, MBS investors are effectively Short Volatility. As rates drop, prepayments spike (prepayment risk), and as rates rise, prepayments dry up (extension risk).

Mathematical Foundation: The Geometry of Risk

To quantify how the Mortgage-Backed Security price ($P$) responds to interest rate fluctuations ($\Delta y$), we utilize a second-order Taylor expansion. This allows us to decompose price sensitivity into linear and non-linear components.

$$\frac{\Delta P}{P} \approx -D_{mod} \cdot \Delta y + \frac{1}{2} C \cdot (\Delta y)^2$$

1. First-Order Sensitivity: Modified Duration ($D_{mod}$)

Modified Duration represents the linear sensitivity of the bond price to a change in yields. In physics terms, this is the normalized first derivative—effectively the “velocity” of price change:

$$D_{mod} = -\frac{1}{P} \frac{\partial P}{\partial y}$$

2. Second-Order Sensitivity: Convexity ($C$)

Convexity captures the “curvature” of the price-yield relationship. It is the normalized second derivative—effectively the “acceleration” of price change:

$$C = \frac{1}{P} \frac{\partial^2 P}{\partial y^2}$$

3. The Negative Convexity Regime

For standard fixed-income instruments (e.g., non-callable Treasuries), $C > 0$. This Positive Convexity is a desirable trait, as it implies that prices rise faster than they fall for a given basis point move.

In the case of Mortgage-Backed Securities, the embedded homeowner call option introduces Negative Convexity ($C < 0$). As market yields decrease, the probability of prepayments spikes, causing principal to be returned prematurely. This results in a concave price-yield curve where:

$$\frac{\partial^2 P}{\partial y^2} < 0$$

This project benchmarks the MBS against a synthetic non-callable “Bullet” bond to visualize and quantify this Convexity Gap, measuring the cost of the “short volatility” position inherent in the mortgage market.

Results Summary

Our analysis demonstrates that the MBS exhibits significant “price capping” in falling-rate environments. At a -50bps shock, the MBS portfolio underperformed the benchmark by 3.83%, illustrating the substantial cost of the embedded call option.

Technical Skills Demonstrated

• Language: Python (NumPy, Pandas, Matplotlib)
• Finance: Fixed Income, Convexity/Gamma, Prepayment Modeling, Yield Curve Shifting.