Valuation and Portfolio Risk Management with Mortgage- Backed Security

1. Simulate term structure of interest rates 2. Prepayment model 3. Calculate cash flows 4. Calculate OAS 5. Total return 6. Holding period 7. Construct portfolio by MAD model

Framework of the valuation Phase 1 Generate arbitrage free interest rate scenario Phase 2 Generate cash flows for each interest rate scenario Phase 3 compute NPV 、 duration 、 convexity ……

Pricing Monte Carlo simulation of the term structure which is used to generate paths of risk free rates Generate security cash flows for each path Compute and average the present value of discounted cash flow

However ， Most fixed income securities cannot be priced using the riskless discount rates implied by the Treasury yield curve Price of the security has to reflect the credit ， liquidity ， default ， and prepayment risks

Option-Adjusted Premia (OAP) Multiplicative adjustment factor for the Treasury rates that will equate today ’ s (observed) market price with the fair price obtained by computing the expected present value of the cash flows

Option-adjusted price of the security but ， does not depend only on the state σ ， but also on the history of interest rates from t=0 to t= τ that pass through this state

Duration and Convexity The sensitivity of the computed prices to changes in the term structure How to use Monte Carlo simulations to calculate option-adjusted duration and convexity Step 0 ： use equation(2) to compute the oap(ρ j )

Step 1 ： Shift the term structure by – 50 basis points and recalibrate the stochastic process of interest rates Step 2 ： Sample interest rate paths { } from the stochastic process calibrated in step 1 ， and use the security cash flow projection model to compute option- adjusted prices ：

Step 3 ： Shift the term structure by +50 basis points and recalibrate the stochastic process of interest rates Step 4 ： Sample interest rate paths { } from the stochastic process calibrated in step 3 ， and use the security cash flow projection model to compute option- adjusted prices ：

Step 5 ： option-adjusted duration of the security option-adjusted convexity

Holding Period Returns For shorter time horizons the distribution is highly asymmetric Average price of the security converges to par ， as it should towards its maturity

價格較對稱 & 平均價格 接近面額

Portfolio Risk Management Techniques Indexation – passive portfolio managers the performance measure of such a portfolio is the difference in return between the portfolio and the index ， And this difference has to be very small

Liability payback — insurance and pension fund construct a portfolio of MBS that will pay the future stream of liabilities Debt issuance — government agencies ensure that the payments against the issued debt will be met from the available assets ， irrespectively of the timing of cash flows and fluctuations in interest rates

Classification of Portfolio Management Models 1. Static Model 2. Single-Period, Stochastic 3. Multiperiod, Dynamic, and Stochastic Model

Static Model : Duration Matching Assume : unlimited borrowing is allowed.

 Based on Mean-Variance Model  Static models hedge against small changes from the current state of the world.  Advantage : (1) simple (2) the least cost  Disadvantage : Too simplistic with the increased volatility of the term structure.

Stochastic Model : Capturing Correlations  The model recognizes the volatility of MBS price, and the correlation of prices in a portfolio, and develops the tradeoffs between return and volatility.  Based on Mean-absolute deviation (MAD)

Model 1  A MAD model is suitable for the fixed- income securities with embedded options since they exhibit highly asymmetric distributions of return.

Model 2

A Multiperiod, Dynamic Model : Stochastic Optimization Based on MAD Model. More flexible than previous two models. Consider transaction cost and include scenarios not only of interest rate but also of prepayments, spread, risk premia and the like.

Applications  Immunization of an Insurance Liability Stream Cost of portfolio by using Treasuries only (saving)$166,163, % Cost of portfolio by using up to 25% MBS (saving)$152,933, % Cost of portfolio by using up to 50% MBS (saving)$142,529, % Cost of portfolio by using up to 100% MBS (saving)$137,489, % Cost of mixed U.S. Treasury-MBS portfolio (saving)$136,124, %

Exhibit 3 : Performance of Immunize and MAD Portfolios 100% MBS portfolio Mixed Portfolio Model Exp. Return Std.Dev. Std. Dev. Immunized MAD(equal risk) MAD(equal return)

 Tracking a Mortgage Index  A Mad model was develop to track the Salomon Index of mortgage-backed securities.  The index consists of a representative of all traded fixed-rate, passthrough securities, issued by FNMA, GNMA and FHLMC.  Tracking cost is very high.