Fixed Income Securities II: Securitization, Duration, Convexity, and Portfolio Immunization

The Sub-Prime Crisis and the Mechanics of Securitization

  • Rationale for Securitizing Loans:     * Repackaging Risks: Securitization allows for the restructuring of risk into more homogeneous categories, which makes them easier to trade and value.     * Efficient Risk Allocation: It facilitates a more efficient distribution of risk across different participants in the financial system.     * Increased Risk-Bearing Capacity: By spreading risk, the system as a whole can support and manage a larger volume of total risk.     * Transparency: The process is intended to provide greater transparency into the underlying assets and their performance.     * Economic Support: It supports overall economic growth by providing liquidity to lending markets.     * Sub-prime Market Benefits: It extends credit to borrowers who might otherwise be excluded from traditional lending markets.

  • Requirements for Successful Securitization:     * Diversification: The underlying pool of assets must be sufficiently diversified to mitigate idiosyncratic risks.     * Accurate Risk Measurement: The ability to precisely quantify the probability of default and loss given default is critical.     * "Normal" Market Conditions: The model relies on stable market environments to function correctly.     * Sophisticated Investors: Investors must have the analytical capability to understand complex structured products.

Illustrative Example of Simple Securitization (CDO Structure)

  • Initial Setup:     * Consider a portfolio consisting of two identical one-period loans.     * Each loan has a Face Value of $1,000\$1,000.     * These are risky loans with a probability of default of 10%10\%.     * The single loan price is calculated as: \text{Price} = 90\% \times \1,000 + 10\% \times \0=$9000 = \$900.

  • Portfolio Creation (Independent Defaults):     * Two loans are packed into a portfolio with a total potential value of $2,000\$2,000.     * Assuming defaults are independent, the possible outcomes are:         1. Both loans perform: Portfolo Value = $2,000\$2,000; Probability = 0.90×0.90=81%0.90 \times 0.90 = 81\%.         2. One loan defaults: Portfolio Value = $1,000\$1,000; Probability = (0.10×0.90)+(0.90×0.10)=18%(0.10 \times 0.90) + (0.90 \times 0.10) = 18\%.         3. Both loans default: Portfolio Value = $0\$0; Probability = 0.10×0.10=1%0.10 \times 0.10 = 1\%.

  • Tranching the Portfolio (S and J):     * Two new claims (tranches) are issued on this portfolio: a Senior Tranche (S) and a Junior Tranche (J).     * The Senior Tranche has a $1,000\$1,000 face value and higher priority (it is paid first).     * The Junior Tranche has a $1,000\$1,000 face value and lower priority (it takes the first losses).

  • Outcome and Pricing (Independent Defaults):     * Senior Tranche (S):         * Receives $1,000\$1,000 if the portfolio value is $2,000\$2,000 (81% chance).         * Receives $1,000\$1,000 if the portfolio value is $1,000\$1,000 (18% chance).         * Receives $0\$0 ONLY if the portfolio value is $0\$0 (1% chance).         * Bad State Probability: 1%1\%.         * Price for Senior Tranche: 99\% \times \1,000 + 1\% \times \0=$9900 = \$990.     * Junior Tranche (J):         * Receives $1,000\$1,000 if the portfolio value is $2,000\$2,000 (81% chance).         * Receives $0\$0 if the portfolio value is $1,000\$1,000 (18% chance, as the first $1,000\$1,000 goes to the Senior tranche).         * Receives $0\$0 if the portfolio value is $0\$0 (1% chance).         * Bad State Probability: 19%19\%.         * Price for Junior Tranche: 81\% \times \1,000 + 19\% \times \0=$8100 = \$810.

  • Scenario: Perfectly Correlated Defaults:     * If defaults are perfectly correlated, both loans either perform together or default together.     * Portfolio Outcomes:         1. Value = $2,000\$2,000; Probability = 90%90\%.         2. Value = $0\$0; Probability = 10%10\%.     * Impact on Tranches:         * Senior Tranche Price: 90\% \times \1,000 + 10\% \times \0=$9000 = \$900 (Decreased from $990\$990).         * Junior Tranche Price: 90\% \times \1,000 + 10\% \times \0=$9000 = \$900 (Increased from $810\$810).

  • Market Implications:     * Initially, new securities experienced very low default rates and very low correlation of defaults.     * The Senior Tranches were rated Aaa (almost riskless), attracting pension funds.     * Junior Tranches were sought after by hedge funds seeking higher yields.     * When the National Real-Estate Market declined: Default correlations spiked, the value of senior tranches fell, ratings were downgraded, leading to a cycle of unwinding and subsequent losses.

Interest Rate Sensitivity and Duration

  • Bond Price and Yield Relationship:     * The relationship is inverse; as Yield to Maturity (YTM) increases, bond prices fall.     * Longer maturity bonds exhibit greater percentage price changes for a given change in yield.     * Lower coupon bonds (especially Zero-Coupon Bonds) are more sensitive to interest rate changes than higher coupon bonds.

  • Duration Definition:     * Duration is a measure of the effective average maturity of a bond's promised cash flows.     * Macaulay's Duration: A weighted average of the times until each payment is received.     * The weight applied to each time (tt) is the proportion of the total bond value accounted for by that specific payment (i.e., the Present Value of the payment divided by the bond price).     * Duration for Zero-Coupon Bonds: Duration equals the time to maturity (TT).     * Duration for Coupon Bonds: Duration is always less than the time to maturity.

  • Duration Calculation Formula:     * D=∑t=1Tt×wtD = \sum_{t=1}^{T} t \times w_t     * wt=CFt/(1+y)tPw_t = \frac{CF_t / (1+y)^t}{P}     * Where CFtCF_t is the cash flow at time tt, PP is the price of the bond, and yy is the yield to maturity.

  • Modified Duration (D∗D^*):     * Used as a measure of interest rate sensitivity.     * D∗=D1+yD^* = \frac{D}{1+y}     * Price Change Approximation: ΔPP=−D∗×Δy\frac{\Delta P}{P} = -D^* \times \Delta y     * Example Calculation: A 2-year, 8%8\% coupon bond selling at $964.54\$964.54 with a YTM of 10%10\% and a duration of 1.88521.8852. If YTM increases to 10.5%10.5\% (Δy=0.005\Delta y = 0.005):         * ΔPP=−1.88521.10×0.005\frac{\Delta P}{P} = -\frac{1.8852}{1.10} \times 0.005         * ΔP=−1.8852×0.0051.10×$964.54=−$8.27\Delta P = -\frac{1.8852 \times 0.005}{1.10} \times \$964.54 = -\$8.27

Five Fundamental Rules of Duration

  • Rule 1: The duration of a zero-coupon bond is equal to its time to maturity.
  • Rule 2: Holding maturity constant, a higher coupon rate results in a lower duration.
  • Rule 3: Holding the coupon rate constant, duration generally increases with time to maturity (though it is not a linear relationship).
  • Rule 4: Holding other factors constant, duration is higher when the bond's yield to maturity is lower.
  • Rule 5: The duration of a level perpetuity is calculated as: D=1+yyD = \frac{1+y}{y}.

Convexity in Bond Pricing

  • Concept of Convexity:     * The price-yield relationship for bonds is not linear but curved (convex).     * Duration is only a linear approximation and is accurate only for small changes in yield.     * Convexity measures the rate of change of the slope (duration) of the price-yield curve.

  • Adjusted Price Change Formula including Convexity:     * ΔPP=−D∗Δy+12×Convexity×(Δy)2\frac{\Delta P}{P} = -D^* \Delta y + \frac{1}{2} \times \text{Convexity} \times (\Delta y)^2

  • Investor Preference for Convexity:     * Investors prefer higher convexity because bonds with more curvature gain more in price when yields fall than they lose when yields rise (asymmetrical benefit).     * Bonds with greater convexity typically trade at higher prices and offer lower yields to maturity.

Callable Bonds and Mortgage-Backed Securities (MBS)

  • Callable Bonds:     * As interest rates fall, the price of a callable bond is capped near its call price (price compression).     * They exhibit Negative Convexity in the region where the price-yield curve is below its tangency line.     * Effective Duration Calculation: Effective Duration=−ΔP/PΔr\text{Effective Duration} = -\frac{\Delta P / P}{\Delta r}.     * Example: A bond with call price $1,050\$1,050 sells for $980\$980. If rates rise by 0.5%0.5\%, price falls to $930\$930. If rates fall by 0.5%0.5\%, price rises to $1,010\$1,010.         * Δr=0.5%−(−0.5%)=1%=0.01\Delta r = 0.5\% - (-0.5\%) = 1\% = 0.01         * ΔP=$930−$1,010=−$80\Delta P = \$930 - \$1,010 = -\$80         * Effective Duration=−−$80/$9800.01=8.16 years\text{Effective Duration} = -\frac{-\$80 / \$980}{0.01} = 8.16\, \text{years}.

  • Mortgage-Backed Securities (MBS):     * MBS are portfolios of callable amortizing loans (homeowners can prepay/refinance).     * They typically have negative convexity.     * Because homeowners do not always refinance immediately when rates drop, the call price is not a firm ceiling; MBS often sell for more than their principal balance.

Passive Bond Management: Immunization and Rebalancing

  • Immunization Strategy:     * A technique to shield financial status from interest rate risk by matching the duration of assets and liabilities.     * Used extensively by pension funds, insurance companies, and banks.     * It balances reinvestment rate risk (risk that future coupons are reinvested at lower rates) against price risk (risk that bond prices fall when rates rise).

  • Rebalancing Example (Insurance Company):     * Situation: Must pay $19,487\$19,487 in 7 years. Current rate is 10%10\%. PV of obligation is $10,000\$10,000. Asset choice: 3-year zero-coupon bonds and perpetuities.     * Step 1: Liability duration = 7 years7\, \text{years}.     * Step 2: Asset durations. Zero = 3 years3\, \text{years}. Perpetuity = 1.100.10=11 years\frac{1.10}{0.10} = 11\, \text{years}.     * Step 3: Asset mix. Solve for weight (ww) in zero: w×3+(1−w)×11=7w \times 3 + (1-w) \times 11 = 7. This gives w=0.5w = 0.5. Invest $5,000\$5,000 in each.     * Step 4 (Rebalancing): One year later, PV of obligation grows to $11,000\$11,000 (due in 6 years). Zero duration becomes 2 years2\, \text{years}. Perpetuity duration is still 11 years11\, \text{years}.     * New weights: w×2+(1−w)×11=6→9w=5→w=5/9w \times 2 + (1-w) \times 11 = 6 \rightarrow 9w = 5 \rightarrow w = 5/9.     * The manager must now invest \11,000 \times (5/9) = \6,111.116,111.11 in the zero-coupon bond, requiring a shift of funds.

  • Cash Flow Matching and Dedication:     * Cash Flow Matching: Automatically immunizes by matching the specific timing of asset cash flows with liability payments.     * Dedication Strategy: A multiperiod cash flow matching approach. It is a "once-and-for-all" method to eliminate interest rate risk but imposes significant constraints on bond selection.