交易對手違約風險與抵押保險契約之評價---理論與實證

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交易對手違約風險與抵押保險契約之評價：理論與實證

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計 畫 類 別 ： 個別型 計 畫 編 號 ： NSC 98-2410-H-151-001- 執 行 期 間 ： 98 年 04 月 01 日至 99 年 07 月 31 日 執 行 單 位 ： 國立高雄應用科技大學金融系 計 畫 主 持 人 ： 張嘉倩 計畫參與人員： 碩士班研究生-兼任助理人員：黃惟怡 碩士班研究生-兼任助理人員：吳苑慈 碩士班研究生-兼任助理人員：王維安 大專生-兼任助理人員：吳曉玫 大專生-兼任助理人員：黃韋融 處 理 方 式 ： 本計畫可公開查詢

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交易對手違約風險與抵押保險契約之評價：理論與實證

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中 華 民 國 99 年 9 月 9 日

中文摘要 美國次級房貸風暴的發生導致美國及其他國家之銀行與保險公司面臨倒閉 或購併之窘境，此現象造成銀行與保險公司之違約風險不僅受到總體風險因子之 違約相關影響，並且會受到借款人違約所引發之對手違約風險影響。因此，本文 將提出同時考量違約相關與對手違約風險之縮減式模型，並且進一步推導出三種 不同類型的抵押保險契約之保費。在實證應用模型方面，本文將利用美國實證資 料進行縮減式模型之参數估計。在數值分析方面，本文將運用美國資料與推導出 的抵押保險契約公式解，分析在不同程度的違約相關、對手違約風險下，是否會 對抵押保險契約之保費有顯著的影響。 關鍵詞：違約相關, 對手違約風險, 縮減式模型, 抵押保險契約結構 Abstract

In view of the recent subprime mortgage crisis across the United States and other countries where banks and insurance companies go bankrupt or are forced into acquisition, the lender and insurer have not only correlated defaults exposing to common risk factors but also counterparty default risk triggered by mortgage defaults. Considering the correlated default and the counterparty default risk, we use the reduced-form approach to derive the closed-form formulas of mortgage insurance contracts and mortgage life insurance contracts with premium refund, annual premium and upfront premium cases. From the numerical analysis with parameter calibration, regardless of what the premium structures are, the numerical results convincingly demonstrate that both the correlated defaults and the counterparty default risk have significant impacts on mortgage insurance premiums, particularly in long-term mortgage loan.

Keywords: correlated defaults, counterparty default risk, reduced-form model,

報告內容

1. Introduction (前言)

The crisis began with the bursting of the United States housing bubble and high default rates on subprime. RealtyTrac Inc., the leading online marketplace for foreclosure properties, announced that nearly 1.3 million U.S. housing properties during 2007 were subject to foreclosure activity, up 79% from 2006. During 2007, at least 100 mortgage companies closed, suspended operations or were sold. Some of the well-known financial institutions listed in Table 1 have declared bankruptcy, and some of the main private mortgage insurers have been downgraded. For example, Triad Guaranty Insurance Corporation is going out of business on August 2008. On September 7, 2008, the Federal Housing Finance Agency (FHFA) announced its decision to place Fannie Mae and Freddie Mac, the two largest mortgage backing entities and main providers of mortgage insurance contracts in the United States, into conservatorship run by FHFA. Furthermore, on September 16, 2008, The Federal Reserve was forced to bail out American Insurance Group (AIG), the largest insurance company in the world, by providing an emergency loan of $85 billion for AIG to sell off assets to repay losses due to mortgage insurance. In sum, the mortgage defaults will spill over into the default probabilities of the lenders and insurers, a phenomenon which is termed ‘counterparty default risk’. Therefore, it is vitally important to assess the impact of counterparty default risk on the mortgage insurance premiums.

Table 1: Businesses in bankruptcy process due to subprime mortgage crisis

Business Type Date

American Freedom Mortgage, Inc. Subprime lender January 30, 2007 New Century Financial Largest U.S. subprime lender April 2, 2007 American Home Mortgage Mortgage lender August 6, 2007 Sentinel Management Group Investment fund August 17, 2007 Ameriquest Subprime lender August 31, 2007 NetBank Internet banking pioneer September 30, 2007 IndyMac Bancorp, Inc. Mortgage lender July 11, 2008 Lehman Brothers Investment banker September 15, 2008

Previous studies of pricing mortgage insurance contracts, such as Kau et al. (1992, 1993, 1995) and Kau and Keenan (1995, 1999), use a structural approach with two state variables, interest rate and housing price, to endogenously model prepayments as an American call option and defaults as an American put option. Other studies, such as Schwartz and Torous (1992), Dennis et al. (1997) and Bardhan et al. (2006), model the unconditional probability of default exogenously. Dennis et al. (1997) propose an actuarial pricing method in which the actuarially fair premiums of different mortgage insurance structures are determined by the present value of the expected losses (plus a gross margin) equal to that of the expected premium revenue; the expected losses are simply a constant fraction of the loan balance when a borrower is in default during the life of a mortgage. According to the assumptions of risk neutral agents, constant interest rate and housing price process following the geometric Brownian motion, Bardhan et al. (2006) develop a new option pricing model to explicitly derive the closed-form formulas of upfront mortgage insurance contracts. The expected losses for an insurer given mortgage defaults can be represented as a portfolio of put options on the collateral of borrowers.

2. The purpose of this research (研究目的)

It is important to incorporate the counterparty default risk, generally not considered by the previous studies, into the pricing model of mortgage insurance－ particularly in case of a mortgage crisis. Therefore, under stochastic interest rate, the first goal of this article is to fill this gap in the literature by considering the possibilities that lenders, insurers and borrowers may default prior to the maturities of mortgage insurance contracts and that mortgage defaults can trigger off a jump in the default rates of the lenders and insurers, a jump that lead to the increased default probabilities of lenders and insurers.

Mortgage insurance differs from other types of insurance in several respects. A noticeable difference, as pointed out by Dennis et al. (1997), is that default and prepayment rates of mortgages are highly dependent on macroeconomic variables, such as interest rates, housing prices and employment rates. When the processes of default and prepayment are modeled to incorporate the macroeconomic variables under a structural approach, it is hard to derive the closed-form formulas for mortgage insurance premiums; however, this problem can be solved by an alternative－the reduced-form approach. A relative merit of this approach is its flexibility and ability to derive an analytic solution with two correlated effects: correlated default and correlated prepayment. The processes of default and prepayment are modeled as

functions of exogenous variables, such as interest rates, housing price indexes and employment rates, all of which lead to the changes in probabilities of default and prepayment.

Different premium structures, such as upfront premiums, annual premiums and premium refunds, are used by various insurance/guarantee agencies, such as Federal Housing Administration, Fannie Mae, Freddie Mae and private mortgage insurance companies. However, most of the mortgage insurance pricing articles (e. g., Kau and Keenan (1995, 1999), Bardhan et al. (2006)) focus on the valuation of mortgage insurance as a lump sum (upfront premium) but not the annual premiums and premium refunds. Hence, the second goal of this article is to propose a risk-neutral pricing model to derive the fair premiums of different mortgage insurance premium structures with correlated defaults, correlated prepayments and counterparty default risk.

3. The model (研究方法)

Assuming that there are three types of agents in the economy－a lender, an insurer and a borrower － we first model the hazard processes of default and prepayment of the borrower. Then, considering the counterparty default risk, we design the hazard processes of the lender and the insurer, an abnormal jump of which is triggered by the borrower defaulting during the life of mortgage. Finally, we present the dynamics of common risk factors affecting the hazard processes of the lender, the insurer and the borrower.

Hazard processes of default and prepayment for the borrower

Let uncertainty in the economy be described by the filtered probability space *

0

(Ω, ,F Q F, ( _{t t})T₌ ). An arbitrage-free restriction guarantees the existence of a unique probability measure Q under which the mortgage insurance premium is determined when the present value of the expected losses (plus a gross margin) is equal to that of the expected premium revenues. There have been numerous empirical studies on this topic, such as Campbell and Dietrich (1983), Schwartz and Torous (1989, 1993), Quigley and Van Order (1990, 1995), Deng, Quigley and Van Order (2000), Deng and Quigley (2002), Lambrecht, Perraudin and Satchell (2003), Caselli, Gatti and Querci (2008)). They indicate that the patterns of default and prepayment are significantly explained by the following risk factors: interest rate, housing price return, loan-to-value ratio and unemployment rate. On the one hand, prepayment probability is positively related to housing price return but negatively related to unemployment rate, but on the other hand, default probability is an increasing function of

unemployment rate but a decreasing function of housing price return. Both probabilities are positively related to the interest rate. To incorporate the empirical results into our pricing model, the enlarged filtration F under which the default and prepayment are interrelated is defined as follows:

,

r M d p

t t t t t

F =F ∨F ∨H ∨H

where F_tr =σ

(

r s s t( ), ≤

)

, F_tM =σ

(

M s s t i_i( ), ≤ , =1,...,n

)

and H_ti =σ

(

1_{{ }τ}i_≤s ,s t≤

)

, for i d or p= ; 1_{{ }⋅} is an indicator function; r(t) is the spot rate at time t; M t , an ( )

n

R -valued stochastic process, denotes the time-t common risk factors, such as housing

price and unemployment rate. As a result, * *

r M

T T

F ∨F contains complete information on the interest rate and all common risk factors. Let τ and d τ stand for the default p time and the prepayment time of a borrower respectively and satisfy

inf : T ( ( ), ( )) i i i t t r s M s ds E τ = ⎧_⎨ λ ≥ ⎫_⎬ ⎩

∫

⎭, i d or p= ,

where the default time and prepayment time are thought as the first jump time of a doubly stochastic Poisson processes (also called a Cox process) with hazard rate process λi( ( ),r t M t( )); E is a unit exponential random variable independent of i state variables and λ . i

(

)

*

0

( ), ( ) T_t

r t M t ₌ , representing state variables underlying the evolution of the economy, is a right continuous with left limits Rn+1-valued process for predicting the likelihood of default or prepayment. Consequently, the conditional survival probability of

τ

i, i d or p= , is of the form:

(

* *

)

0 exp t [ ( ), ( )] i r M i T T P τ >t F ∨F = ⎛_⎜− λ r s M s ds⎞_⎟ ⎝

∫

⎠, [0, ] * T t∈ ,

Being a function of the interest rate and common risk factors, the hazard rates of default and prepayment can be rewritten as follows:

[

1

]

( ( ), ( )) ( ), ( ),..., ( ) i n r u M u f r u Z u Z u λ = , i d or p u= , ∈[0,T*],

where Z u denotes a geometric excess return of common risk factor j over time _j( ) interval [0, ]u and satisfies

*

( ) _{( )} ( ) (0)

( ) log log log log , 1, 2,..., , [0, ],

(0) (0) ( ) (0) j j j j j M u B u M u M Z u j n u T M B B u B ⎡ ⎛ ⎞ ⎛ ⎞⎤ ⎡ ⎛ ⎞ ⎛ ⎞⎤ ≡⎢ _⎜⎜ ⎟_⎟− ⎜ ⎟⎥=⎢ ⎜ ⎟− ⎜ ⎟⎥ = ∈ ⎢ ⎥ ⎢ ⎝ ⎠ ⎝ ⎠⎥ ⎣ ⎝ ⎠ ⎝ ⎠⎦ ⎣ ⎦ (1)

where log

(

M u M_j( ) _j(0)

)

represents a geometric cumulative return of common risk factor j up to time u;

0

( ) exp u ( )

B u = ⎛_⎜ r s ds⎞_⎟

the wealth accumulated by an initial one-dollar investment at spot rate r u( ) in each subsequent period, which means that log

(

B u B( ) (0)

)

is a cumulatively risk-free return up to time u.

Most studies use the Cox proportional hazard model to specify a hazard function as the product of a baseline hazard rate and an exponential function of covariates. However, under the pricing framework of martingale measure, Miltersen, Sandmann, and Sondermann (1997) indicate that a double exponential expression causes an infinite expectation of cumulative factors when influential variables are lognormally distributed. Therefore, Jarrow and Turnbull (1995), Duffee (1999), Jarrow and Turnbull (2000), Jarrow and Yu (2001), Calem and LaCour-Little (2004) and Liao, Tsai and Chiang (2008) assume that the hazard rate function is a linear function of the spot rate and influential variables. One problem of this assumption is that the hazard rate function may be negative. However, Jarrow and Turnbull (1997) indicate that this difficulty can be avoided by the use of non-linear transformations in lattice-based models. Furthermore, Duffee (1999) also argues that this problem can be ignored if the model accurately prices the relevant instruments.

Defining the hazard rates of default and prepayment as linear functions of the spot rate and the excess returns of influential variables, in view of Equation (1), we have the following representation:

0 1 1 ( ) ( ) ( ) ... ( ) i i i i i r n n u r u Z u Z u λ ≡ψ +λ +λ + +λ 0 1 1 (0) ( ) log ( ) log (0) ( ) n n j j i i i i j r j j j M M u r u B B u ψ λ λ λ = = ⎡ ⎛ ⎞⎤ ⎛ ⎞ =_⎢ − _{⎜ ⎟⎥}+ + _{⎜ ⎟} ⎝ ⎠ ⎝ ⎠ ⎣

∑

⎦

∑

0 1 ( ) ( ) log ( ) n j i i i r x j M u r u B u λ λ λ = ⎛ ⎞ = + + _{⎜ ⎟} ⎝ ⎠

∑

, for i d or p= , (2) where λ λ₀d( ₀p) denotes the baseline hazard rate of default (prepayment) at time u;

( )

d p

r r

λ λ measures the magnitudes of default (prepayment) to the level of spot rate; ( )

d p

j j

λ λ represents the magnitudes of default (prepayment) to the excess return of common risk factor j=1, 2,...,n.

Hazard processes of the lender and insurer

For a lender, in addition to a risk exposure to the interest rate and common risk factors, mortgage defaults will affect the default rate of the lender, particularly during a mortgage crisis. Moreover, when a borrower enters a mortgage insurance contract, an insurer abiding by the default risk of the borrower has a correlated default due to not only an exposure to common risk factors but also counterparty default risk. Therefore, we define the default times of the lender and the insurer as follows:

inf : T ( ( ), ( ), ) j j d j t t r s M s ds E τ = ⎧_⎨ λ τ ≥ ⎫_⎬ ⎩

∫

⎭, j I or L= , (3) where the superscript I and L denote the insurer and the lender, respectively. By virtue of Equation (3), their default times are connected with not only the interest rate and common risk factors but also the default time of the borrower. From this definition,

the conditional survival probability of τj is given by * * * 0 ( j | _Tr _TM _Td ) exp t j[ ( ), ( ), d] P τ >t F ∨F ∨H = _⎜⎛− λ r s M s τ ds⎞_⎟ ⎝

∫

⎠, [0, ] * T t∈ , j I or L= . Using the law of iterated expectation, we obtain

0 ( j ) Q exp t j[ ( ), ( ), d] P τ >t =E _⎢⎡ ⎛_⎜− λ r s M s τ ds⎞_⎟⎤_⎥ ⎝ ⎠ ⎣

∫

⎦, [0, ] * T t∈ , j I or L= .

Extending the model of Jarrow and Yu (2001), we assume that linear hazard rates of the lender and the insurer, incorporating n common risk factors and the spillover effect of the borrower’s default, have the following expression:

{ } 0 0 1 ( ) ( ) ( ) log 1 ( ) d n j j j j x j r x u x M u u r u B u τ λ λ λ λ _≤ α = ⎛ ⎞ = + + _{⎜ ⎟}+ ⎝ ⎠

∑

, j I or L= . (4) where λ λ₀I ( ₀L) denotes the baseline default rate of the insurer (lender); λ λI_r( L_r) measures the magnitudes of default to the level of spot rate of the insurer (lender); λ j_x has the anagogic definition; α α₀I( ₀L) represents the jump size of the default intensity of the insurer (lender) when a borrower’s default occurs. By virtue of Equation (4), when the borrower is in default, the default rates of the lender and the insurer increase as α₀j >0 or decrease as α₀j <0. Therefore, the changes in default rates depend on not only the unanticipated “normal” change caused by the interest rate and the excess returns on common risk factors but also the “abnormal” shocks caused by the transmission of default from the borrower.

nterest rate and common risk factor processes

We use a one-factor model－the extended Vasicek model (see Hull and White 1990)－to describe the evolution of the term structure, that is,1

r t rdW dt t r t t t r d ( )=[θ()−α( ) ( )] +σ , (5) where θ( ) / ( )t α t denotes the long-term equilibrium value of the process; α(t) is a nonnegative mean reversion speed; σ_r is the volatility of spot rate; Wtr is a

Brownian motion with respect to F . t

We assume that common risk factors, such as the housing price index and the unemployment rate, follow a geometric Brownian motion, namely,

( ) ( ) ( ) j j j t j dM t r t dt dW M t = +σ , j=1, 2,..., ,n (6) where σ_j is constant volatility of common risk factor j ; W_tj, a standard Brownian motion under Q , is correlated with W_tr and W_ti and satisfies E dW dW

(

_tr _tj

)

=ρ_rjdt

1

and E dW dW

(

_ti _tj

)

=ρ_ijdt, where ρ ρ_ij( _rj) is the correlation coefficient between the common risk factor i (spot rate) and the common risk factor j and satisfies

1.

jj

ρ =

4. 研究結果與討論

In this section, considering the counterparty default risk, we present a framework under which feasible mortgage insurance premium structures can be constructed. In a mortgage insurance contract, a borrower pays insurance premiums to an insurer. The premium structures can be upfront premiums, annual premiums or premium refunds. A feasible premium structure is to design a fair game in which the present value of the expected losses (plus a gross margin) is equivalent to the present value of the expected revenue.

Suitable premiums for different mortgage insurance structures

Thereinafter, we discuss three premium structures satisfying the insurer’s equilibrium conditions.

Refund case : Level annual premium rate, no upfront premium.

In the Refund case, the premium collected at time t is proportional to the time

t balance R_t, being refunded when the borrower prepays. Thus, the condition of no

arbitrage is ER= +(1 q EL) subject to c_s =cRF, which implies

3 2 * 1 (1 ) ( , ) ( , ) ( , ) RF q G t T G t T c G t T + + = , if τd>t τ, p>t τ, I >t, τL> (10) t, where * 1 0 , , ' ( , ) exp ( ) ' 2 T _{K t s} s X t s s t G t T = R ⎛_⎜−λ s t− + Γ μ +Γ Σ Γ⎞_⎟ds ⎝ ⎠

∫

.

In view of Equation (10), the annual premium rate is equal to the ratio of the sum of the discount values of the expected losses (plus a gross margin) and refund premiums to that of the remaining mortgage balance at each time conditional on no default and prepayment prior to evaluation time t .

Annual case: Level annual premium rate, no upfront premium, no refund.

In the Annual case, the annual premium collected at time t is proportional to the time- t balance R without premium refunded. The equilibrium condition is also _t

(1 )

ER= +q EL subject to AN

s

c =c and g_s =0,which implies 3 * 1 (1 ) ( , ) ( , ) AN q G t T c G t T + = , if τd >t τ, p>t τ, I>t τ, L> t. (11)

It is obvious that the Refund case can be reduced to the Annual case when 0

s

g = (G t T₂( , )=0). Hence, by virtue of Equation (11), an annual premium rate, given no default or prepayment prior to time t , is equivalent to the ratio of the present value of the expected losses (plus a gross margin) to the sum of present values of the remaining mortgage balance at each time.

Upfront case: Upfront premium only, no annual premium, no refund.

Collected in the Upfront case is neither annual premium nor premium refund; however, only an upfront fee is paid. In this case, the upfront premium in equilibrium is given by

3

(1+q EL) = +(1 q G t T) ( , ). (12)

Present value of the expected accumulated losses for the insurer

If the death time of a borrower is prior to the prepayment time of the borrower and the default times of the lender, the insurer and the borrower, the insurer will pay off the proportional to the outstanding mortgage immediately. Hence, the present value of the expected losses from t to T , denoted by EL , is given by _life

{ } { } { } { } { } ( ) 1 1 1 1 1 ( ) x x d x p x I x L x T Q life _t R x t s t B t EL E L R ds F B τ _τ < ≤τ τ τ> τ >τ τ >τ τ >τ ⎡ ⎤ ≡ _{⎢ ⎥} ⎣

∫

⎦ 3 ( ) exp ( ) ( , ) T s _{x v} v _{x u L} R v t L t R a b c t a b c du G t v dv ds + ⎛ + ⎞ = + _⎜− + _⎟× ⎝ ⎠

∫ ∫

∫

, if ,τx>t τd >t τ, p>t τ, I >t τ, L> t. (16) where ₃ , , 0 ' ( , ) exp ( ) ' . 2 L K t v X v v t G t v = ⎛_⎜−λ v t− + Γ μ +Γ Σ Γ⎞_⎟

⎝ ⎠ Appendix D provides the proof of Equation (16).

Using the equilibrium condition that the present value of the expected losses (plus a gross margin) is equal to that of the expected revenue, we have

(1 )

life life

ER = +q EL .

Sensitivity analysis shows that the mortgage insurance premium is an increasing function of loan maturity. Corresponding to a given maturity, the correlated defaults of the lender and the insurer are negatively related to the mortgage insurance premiums. The counterparty default risk, triggered by the mortgage defaults, increases the possibility of joint defaults of the insurer and the borrower and decreases the effectiveness of the insurer’s protection against mortgage defaults. The premium is lower and the decline in premium for a mortgage with longer maturity is larger when the counterparty default risk is considered. Moreover, in contrast to the mortgage insurance contract, the mortgage life insurance contract will be additionally early terminated if the borrower dies prior to loan maturity; accordingly, its premium with other things the same is higher.

In conclusion, correlated effects, time to loan maturity and counterparty default risk are crucial in pricing mortgage insurance contracts. Without considering either the correlated effects or the counterparty default risk, the premiums would be overpriced

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23. Kau, J., Keenan, D., 1999. Catastrophic default and credit risk for lending institutions. Journal of Financial Services Research 15(2), 87-102.

24. Kau, J., et al., 1992. A generalized valuation model for fixed-rate residential mortgages. Journal of Money, Credit and Banking 24, 280-299.

25. Kau, J., et al., 1993. An option-based pricing model of private mortgage Insurance. The Journal of Risk and Insurance 60(2), 288-299.

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國科會補助專題研究計畫成果報告自評表

請就研究內容與原計畫相符程度、達成預期目標情況、研究成果之學術或應用價

值（簡要敘述成果所代表之意義、價值、影響或進一步發展之可能性）

、是否適

合在學術期刊發表或申請專利、主要發現或其他有關價值等，作一綜合評估。

1. 請就研究內容與原計畫相符程度、達成預期目標情況作一綜合評估

█

達成目標

□ 未達成目標（請說明，以 100 字為限）

□ 實驗失敗

□ 因故實驗中斷

□ 其他原因

說明：

2. 研究成果在學術期刊發表或申請專利等情形：

論文：□已發表

█

未發表之文稿 □撰寫中 □無

專利：□已獲得 □申請中

█

無

技轉：□已技轉 □洽談中

█

無

其他：（以 100 字為限）

此研究成果已投稿於 Real Estate Economics 期刊，並已一審回覆中。

3. 請依學術成就、技術創新、社會影響等方面，評估研究成果之學術或應用價

值（簡要敘述成果所代表之意義、價值、影響或進一步發展之可能性）（以

500 字為限）

本計劃主要在研究考量交易對手違約風險下，抵押保險之最適保費率應如何訂定。 相較於先前文獻，本計劃研究主題在於除了 borrowers 之外，額外考慮 lenders 與 insurers 在 mortgage insurance 中的違約風險。研究成果發現不論抵押保險種類為 何，違約相關與交易對手違約風險對於抵押保險保費影響很大，尤其是長期的貸款 更甚。因此相當具有重要性或創新性的學術文獻，因為這些環節間的 correlated effects，特別是 defaults and prepayments 間的相關性，雖然文獻皆指出這些問題的 重要性，但是迄今似乎仍未出現公認的模型圭臬，因此本計劃研究主題的創新性、 價值或影響十分重大。

98 年度專題研究計畫研究成果彙整表

計畫主持人：張嘉倩 計畫編號：98-2410-H-151-001- 計畫名稱：交易對手違約風險與抵押保險契約之評價：理論與實證 量化 成果項目 實際已達成 數（被接受 或已發表） 預期總達成 數(含實際已 達成數) 本計畫實 際貢獻百 分比 單位 備 註 （ 質 化 說 明：如 數 個 計 畫 共 同 成 果、成 果 列 為 該 期 刊 之 封 面 故 事 ... 等） 期刊論文 0 0 100% 研究報告/技術報告 0 0 100% 研討會論文 0 2 100% 篇 論文著作 專書 0 0 100% 申請中件數 0 0 100% 專利 已獲得件數 0 0 100% 件 件數 0 0 100% 件 技術移轉 權利金 0 0 100% 千元 碩士生 3 3 100% 博士生 0 0 100% 博士後研究員 0 0 100% 國內 參與計畫人力 （本國籍） 專任助理 0 0 100% 人次 期刊論文 0 1 100% 研究報告/技術報告 0 0 100% 研討會論文 0 0 100% 篇 論文著作 專書 0 0 100% 章/本 申請中件數 0 0 100% 專利 已獲得件數 0 0 100% 件 件數 0 0 100% 件 技術移轉 權利金 0 0 100% 千元 碩士生 0 0 100% 博士生 0 0 100% 博士後研究員 0 0 100% 國外 參與計畫人力 （外國籍） 專任助理 0 0 100% 人次

其他成果

(

無法以量化表達之成 果如辦理學術活動、獲 得獎項、重要國際合 作、研究成果國際影響 力及其他協助產業技 術發展之具體效益事 項等，請以文字敘述填 列。) 無 成果項目 量化 名稱或內容性質簡述 測驗工具(含質性與量性) 0 課程/模組 0 電腦及網路系統或工具 0 教材 0 舉辦之活動/競賽 0 研討會/工作坊 0 電子報、網站 0 科 教 處 計 畫 加 填 項 目 計畫成果推廣之參與（閱聽）人數 0

國科會補助專題研究計畫成果報告自評表

請就研究內容與原計畫相符程度、達成預期目標情況、研究成果之學術或應用價

值（簡要敘述成果所代表之意義、價值、影響或進一步發展之可能性）

、是否適

合在學術期刊發表或申請專利、主要發現或其他有關價值等，作一綜合評估。

1. 請就研究內容與原計畫相符程度、達成預期目標情況作一綜合評估

■達成目標

□未達成目標（請說明，以 100 字為限）

□實驗失敗

□因故實驗中斷

□其他原因

說明：

2. 研究成果在學術期刊發表或申請專利等情形：

論文：□已發表 □未發表之文稿 ■撰寫中 □無

專利：□已獲得 □申請中 ■無

技轉：□已技轉 □洽談中 ■無

其他：（以 100 字為限）

此研究成果已投稿於 Real Estate Economics 期刊，並已一審回覆中。

3. 請依學術成就、技術創新、社會影響等方面，評估研究成果之學術或應用價

值（簡要敘述成果所代表之意義、價值、影響或進一步發展之可能性）（以

500 字為限）

本計劃主要在研究考量交易對手違約風險下，抵押保險之最適保費率應如何訂定。相較於 先前文獻，本計劃研究主題在於除了 borrowers 之外，額外考慮 lenders 與 insurers 在 mortgage insurance 中的違約風險。研究成果發現不論抵押保險種類為何，違約相關與交 易對手違約風險對於抵押保險保費影響很大，尤其是長期的貸款更甚。因此相當具有重要 性或創新性的學術文獻，因為這些環節間的 correlated effects，特別是 defaults and prepayments 間的相關性，雖然文獻皆指出這些問題的重要性，但是迄今似乎仍未出現公 認的模型圭臬，因此本計劃研究主題的創新性、價值或影響十分重大。