Valuation and Portfolio Valuation and Portfolio Risk Management with Risk Management with

Mortgage-Backed SecurityMortgage-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 valuationFramework 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……

PricingPricing 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

HoweverHowever ，，

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 (OOption-Adjusted Premia (OAP)AP)

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 ConvexityDuration 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 convexityStep 0 ： use equation(2) to compute the oap(ρj)

Step 1 ： Shift the term structure by –50 basis points and recalibrate the stochastic process of interest ratesStep 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 ratesStep 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

100

jj

j

pp

2

0

50

2 jjj

j

PPP

Holding Period ReturnsHolding Period Returns

jo

sj

sjs

j P

VFR

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 Portfolio Risk Management Management TechniquesTechniques

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 Classification of Portfolio Management Portfolio Management

ModelsModels 1. Static Model 2. Single-Period, Stochastic 3. Multiperiod, Dynamic, and Stochastic Model

Static Model : Duration Static Model : Duration MatchingMatching

Assume : unlimited borrowing is allowed.

0

1

1

Minimize

1

1

1

1

0

j

T

tt

tj

Jj

T

tt

jt

Jjjj

x

x

r

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Jj

ljjx

Based on Mean-Variance Model Static models hedge against small chang

es 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 : Stochastic Model : Capturing CorrelationsCapturing 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) )()( RERERw

Model 1

Jjux

x

xRts

xRRERw

jj

Jjj

Jjjj

Jjjjj

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allfor 0

1

Minimize

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

Jjux

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xR

SsxRRy

SsxRRyts

yjS

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allfor 0

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allfor 0

allfor 0

1 Minimize

,

,

,..

Model 2

A Multiperiod, Dynamic Model : A Multiperiod, Dynamic Model : Stochastic OptimizationStochastic Optimization Based on MAD Model. More flexible than previous two mode

ls. Consider transaction cost and include

scenarios not only of interest rate but also of prepayments, spread, risk premia and the like.

ApplicationApplicationss

Immunization of an Insurance Liability StreamCost of portfolio by using Treasuries Cost of portfolio by using Treasuries onlyonly(saving)(saving)

$166,163,861.$166,163,861.00000.00%0.00%

Cost of portfolio by using up to 25% Cost of portfolio by using up to 25% MBSMBS(saving)(saving)

$152,933,690.$152,933,690.00007.92%7.92%

Cost of portfolio by using up to 50% Cost of portfolio by using up to 50% MBSMBS(saving)(saving)

$142,529,529.$142,529,529.000016.58%16.58%

Cost of portfolio by using up to 100% Cost of portfolio by using up to 100% MBSMBS(saving)(saving)

$137,489,656.$137,489,656.000021.07%21.07%

Cost of mixed U.S. Treasury-MBS Cost of mixed U.S. Treasury-MBS portfolioportfolio(saving)(saving)

$136,124,130.$136,124,130.000022.07%22.07%

Exhibit 3 : Performance of Immunize and MAD Portfolios

100% MBS portfolio

Mixed Portfolio

Model Exp. Exp. ReturnReturn

Std. Std.

Dev.Dev.Exp. Exp. ReturReturnn

Std. Std. Dev.Dev.

ImmunizedMAD(equal risk)MAD(equal return)

10.46910.78310.469

0.4060.4050.234

10.44810.69210.488

0.2930.2930.206

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.