Pricing Mortgage Insurance with Asymmetric Jump Risk and Default Risk: Evidence in the U.S. Housing Market

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Pricing Mortgage Insurance with Asymmetric Jump

Risk and Default Risk: Evidence in the U.S.

Housing Market

Chia-Chien Chang&Wei-Yi Huang&So-De Shyu

# Springer Science+Business Media, LLC 2011

Abstract This study provides the valuation of mortgage insurance (MI) considering upward and downward jumps in housing prices, which display separate distributions and probabilities of occurrence, and the mortgage insurer’s default risk. The empirical results indicate that the asymmetric double exponential jump diffusion performs better than the log-normally distributed jump diffusion and the Black-Scholes model, generally used in previous literature, to fit the single-family mortgage national average of all home prices in the US. Finally, the sensitivity analysis shows that the MI premium is an increasing function of the normal volatility, the mean down-jump magnitudes, the shock frequency of the abnormal bad events, and the asset-liability structure of the mortgage insurer. In particular, the shock frequency of the abnormal bad events has the largest effect of all parameters on the MI premium. The asset-liability structure of the mortgage insurer and shock frequency of the abnormal bad events have a larger effect of all parameters on the default risk premium.

Keywords Mortgage insurance contract . Asymmetric double exponential jump diffusion process . Default risk

JEL Classification G1 . G2

C.-C. Chang

Department of Finance, National Kaohsiung University of Applied Science, 415 Chien Kung Road, Sanmin District, Kaohsiung 80778, Taiwan, R.O.C.

e-mail: cchiac@cc.kuas.edu.tw W.-Y. Huang

Fubon Financial, 3F., No.9, Xiangyang Rd., Zhongzheng Dist., Taipei 100, Taiwan, R.O.C. e-mail: vn750409@hotmail.com

S.-D. Shyu (*)

Department of Finance, National Sun Yat-sen University, No. 70, Lienhai Rd., Kaohsiung 80424, Taiwan, R.O.C.

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Introduction

Private mortgage insurance (MI) guarantees that if a borrower defaults on a loan, a mortgage insurer will pay the mortgage lender for any loss resulting from a property foreclosure up to 20–30% of the claim amount. Many previous studies indicate that the housing price change is a crucial factor in determining MI premiums (e.g., Kau et al.1992,1993, 1995; Chen et al.2010). Therefore, how to properly model the dynamic process of US housing price and price the MI contracts is an important issue. Chen et al. (2010) show empirical evidence of the jump magnitude phenomenon when using US monthly national average new home data. In contrast to the assumption of lognormality in the jump magnitude generally made in previous literature, we would like to investigate whether the jump risk is symmetric or asymmetric and how the asymmetric jump risk affects the value of MI premiums if the jump risk of US housing prices exists to a significant extent. Furthermore, in view of the rising foreclosure rates of the borrowers and the mortgage insurers’ huge losses, the default risk of the mortgage insurer has drawn more attention to the valuation of MI contracts, especially during the subprime mortgage crisis. Therefore, it is also vitally important to assess the impact of default risk of mortgage insurer on the MI premiums.

In addition to the interest rate change, the change in housing prices plays a crucial role in the pricing of MI contracts. In the previous literature, housing price change is assumed to follow a traditional Black-Scholes model (BSM), and this assumption is reasonable for relatively stable housing prices (e.g., Kau et al. 1992,1993,1995; Kau and Keenan1995,1999; Bardhan et al.2006). Kau and Keenan (1996) use a compound Poisson process to only consider the down-jump component of housing prices in the case of catastrophic events. Chen et al. (2010) assume that the housing price process follows the log-normally distributed jump diffusion (LJD) process, capturing important characteristics of abnormal shock events. This assumption is consistent with the empirical observation of the US monthly national average of new home returns from January 1986 to June 2008. However, the traditional lognormality assumption involves a “generic jump” whose magnitude fluctuates between minus one and infinity, thus allowing the generation of both downward and upward jumps. Although the lognormal distribution has many useful properties, one drawback of this approach is the constraining of upward jumps and downward jumps to both come from the same distribution as well as the lack of precise differentiation between the probabilities of the occurrence of each type of jump.

Figure1shows the US national average of all home price returns for single-family mortgages from January 1986 to October 2008. There were four occasions when the monthly housing price changed by more than two standard errors per month. Furthermore, it can be seen that there were nine occasions when the monthly housing price changed by less than two standard errors per month. Therefore, the US national average of all home price returns seems to have the properties of excess kurtosis, skewness and asymmetric jump phenomenon. The excess kurtosis and skewness of the housing price change can be partially explained through the modeling of jumps and also through the use of stochastic volatility (Heston 1993) or both as shown in Pan (2002) or

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Eraker et al. (2003).1 However, stochastic volatility is not considered here to allow a focus on the effects of jumps alone. This paper concentrates on extending the previous studies to relax the assumption of a common lognormal jump distribution spanning both upward and downward jumps by allowing each jump type to possess distinct distributional properties. Because the liquidity of the real estate market is lower than that of the financial market, the jumps are rare events, i.e., crushes and large drawdowns. This study uses the asymmetric DEJD process, a type of finite-activity Lévy process, to capture asymmetric jump characteristics related to good and bad shock events that influence housing price.2These good and bad shock events, as Kau and Keenan (1996) and Chen et al. (2010) define them, can be financial (e.g., sudden severe upward and downward jumps in housing prices due to favorable or unfavorable economic news for the local economy, such as announcements regarding expansionary or contractionary monetary policy). In general, downward jumps in the housing price are more sensitive than upward jumps in the housing price based on the valuation of MI premiums. Therefore, the asymmetric jump risk for housing prices plays an important role in pricing for MI contracts, and the frequency and magnitude of the downward jumps in the housing price are particularly important. Some studies (e. g., Sutton (2002), Borio and Mcguire (2004), and Tsatsaronis and Zhu (2004)) indicate a significant negative relationship between the real interest rates and housing prices. We also incorporate the interest rate process into the dynamic process of housing price change to capture the interest-rate sensitivity of the rate of change of housing prices.

In view of the subprime mortgage crisis in the US, Mortgage Insurance Companies of America (MICA) reports that large mortgage insurers reported $2.6 billion in losses in 2008, sparking concerns that rising foreclosure rates of the borrowers could compel the industry into a money crisis. For instance, shares of Radian Guaranty, Triad, and PMI Mortgage Insurance Group lost 90 percent of their value in 2007; Triad Guaranty Insurance Corporation failed to meet capital requirement on March 31, 2008 and is even going out of business. MICA reports that Triad’s risk-to-capital ratio, 27.7:1, exceeded the maximum (25:1) generally allowed by insurance regulations and Illinois insurance law. As we know, the default risk of the mortgage insurer is generally not considered by the previous pricing model of MI contracts.

There are three contributions to the pricing of MI contracts in this paper. First, we use US housing price data to find that the asymmetric DEJD process is the best fit by using the quasi-Newton algorithm, Bayesian information criterion (BIC) and

1

Stochastic volatility usually has a larger impact on long-term options, whereas the presence of jumps mostly benefits the pricing of short-term near-the-money options.

2_{In general, there are two types of Lévy processes: the finite-activity Lévy process and the infinite-activity}

Lévy process. The finite-activity Lévy process generates only a finite number of jumps during any finite time interval. Examples of such models are the Merton jump-diffusion model with Gaussian jumps and the Kou model with asymmetric double exponential jumps. On the other hand, the infinite-activity Lévy model can generate an infinite number of small jumps at any finite time interval. Because the liquidity of the real estate market is lower than that of the financial market, the number of jumps should be finite. Furthermore, in finite-activity Lévy processes, the dynamic structure of the process is easy to understand and describe because the distribution of jump sizes is known. Such processes are also easy to simulate, and it is possible to use efficient Monte Carlo methods for pricing path-dependent options. Hence, this paper uses a finite-activity Lévy model to describe asymmetric jumps in the housing price process.

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likelihood-ratio test (LR test). Next, to be consistent with the asymmetric jump behavior of US housing prices, the relationship between the interest rate and housing prices and mortgage insurers’ default risk, this paper develops a contingent-claim framework for valuing an MI contract. We adopt a structural approach to model the default probability of the mortgage insurer. The mortgage insurer’s total asset and liability value consists of two risk components: risk in interest rate and housing price. Finally, the sensitivity analysis examines how the asymmetric jump risk of housing prices and the default risk of the mortgage insurer impact the valuation of MI contracts and the default risk premium. We find that the shock frequency of the abnormal bad events has the most significant effect on the MI premium, and the asset-liability structure of the mortgage insurer and shock frequency of the abnormal bad events show the greatest effect of all parameters on the default risk premium. This implies that the insurer must carefully consider the impact of the shock frequency of the abnormal bad events when pricing the MI contracts.

The remainder of this paper is organized as follows. Section“Model” illustrates

the model. Section“Valuation of Mortgage Insurance Contract” derives the pricing

formulae for MI contracts under asymmetric DEJD. Empirical and numerical analyses are presented in Section “Empirical and Sensitivity Analysis”. Section

“Conclusions” summarizes the paper and gives conclusions.

Model

This study adopts a structural approach to model the default probability of the mortgage insurer. Because the interest rate, housing prices and the mortgage insurer’s asset–liability structure specifications are crucial factors in determining the value of MI contracts, we assume that the interest rate, housing prices and the mortgage insurer’s liability are related and that the interest rate and the mortgage insurer’s assets are related. This section outlines the dynamic processes of the interest rate process, the borrower’s housing price, the mortgage insurer’s assets, and the mortgage insurer’s liability under the risk-neutral measure Q.3

The Instantaneous Interest Rate Process

Following previous studies, (e.g., Kau et al.1992,1993,1995), the instantaneous interest rate is assumed to follow the square-root process of Cox et al. (1985). Therefore, the interest rate process under the physical probability measure P is as follows:

drðtÞ ¼ hrðq rðtÞÞdt þ v

ffiffiffiffiffiffiffiffi rðtÞ p

dW_rPðtÞ; ð1Þ

whereηris the mean-reverting force measurement,θ denotes the long-run mean of the interest rate, ν presents the volatility parameter for the interest rate, and WP

rðtÞ is a

Wiener process under the physical probability measure P. According to Girsanov’s

3_{As argued by Bardhan et al. (}₂₀₀₆_{), the valuation of MI can be also obtained if the assumption of the}

risk neutrality of agents is relaxed and insurance contracts are assumed to be traded. Therefore, we can price MI contracts without assuming the risk neutrality of the agents and instead assuming that MI contracts are traded.

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theorem, dW_rQðtÞ ¼ dW_rPðtÞ lr

ffiffiffiffiffiffiffiffi rðtÞ p

=n dt, the interest rate process under the risk-neutral measure Q can be described as

drðtÞ ¼ h»r q

» rðtÞ

dtþ vpffiffiffiffiffiffiffiffirðtÞdW_rQðtÞ; ð2Þ where the term1ris the market price of interest rate risk, and following Cox et al.’s (1985) assumption that1ris a constant and that WrQðtÞ is a Wiener process under the

risk-neutral measure Q,h»_r¼ hr lr,θ* = ηrθ/ηr–1r. The Housing Price Process

Kau and Keenan (1996) and Chen et al. (2010) use the LJD process to describe the change in housing prices. However, Fig.1 seems to show that the change in US housing prices is asymmetric. Therefore, this study uses asymmetric DEJD to describe up-jump and down-jump components of the change in housing prices. Furthermore, previous studies (e.g., Harris (1989), Abraham and Hendershott (1996), Englund and Ioannides (1997), Sutton (2002), Borio and Mcguire (2004), and Tsatsaronis and Zhu (2004)) indicate a significant negative relationship between the real interest rates and housing prices. Therefore, this paper assumes that the change rate of housing price dynamics under the physical probability measure P is governed by the following process:

dHðtÞ HðtÞ ¼ mHdtþ frHdWrPðtÞ þ sHdWHPðtÞ þ d X NP_ðtÞ n¼0 V_nP 1 ; ð3Þ where μH and sH¼esH ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 r2 rH p

are the drift and volatility terms of the rate of change rate housing prices,esHis the total volatility of the rate of change of housing

-30 -25 -20 -15 -10 -5 0 5 10 15 20 25 1986/2 1987/2 1988/2 1989/2 1990/2 1991/2 1992/2 1993/2 1994/2 1995/2 1996/2 1997/2 1998/2 1999/2 2000/2 2001/2 2002/2 2003/2 2004/2 2005/2 2006/2 2007/2 2008/2 Time(month)

Monthly national average all home price return(%)

Fig. 1 US national average all home price returns for single-family mortgage. Note that the solid (dashed) line represents the mean of the housing price return plus (minus) two standard errors

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prices andρrHis the correlation coefficient of interest rates and housing price,frH¼

esHrrHis negative and represents the instantaneous interest rate sensitivity of change

rate of house price and WP

HðtÞ is a Wiener processes under the physical probability

measure P. NP_{ðtÞ ¼ N}

uðtÞ þ NdðtÞ is an independent Poisson process with intensity

parameters 1=1u+1d, where 1u and 1d represent the intensity of up-jumps and down-jumps, respectively. V_nP is the jump magnitude and a sequence of independent identically distributed nonnegative random variables such that YP ¼ ln V _nP has an asymmetric double exponential distribution with the density function:

fYPðyÞ ¼ ph_uehuy1_f_y0_gþ 1 pð Þh_dehdy1_f_y<0_g; h_u> 1; h_d > 0: ð4Þ

Equation4 represents the distribution of the logarithm of the jump magnitudes under the asymmetric DEJD, which has a jump intensity 1, and YP has an independent identically distributed mixture distribution of exponential (ηu) and exponential (ηd) with probabilities p and 1-p, respectively. According to the Esscher transform, as the following martingale condition is satisfied,4

rðtÞ ¼ m_Hþ f_rHlr ffiffiffiffiffiffiffiffi rðtÞ p n s2Hhþ Z R ey 1 ð Þehy_{n dy}_{ð Þ;} _ð5Þ

wheren dyð Þ ¼ lfYPðyÞdy is a Levy measure of Y, and h is a real value, the change rate

of housing price dynamics under the risk-neutral measure Q can be written as follows: dHðtÞ HðtÞ ¼ rðtÞ lQk dtþ f_rHdW_rQðtÞ þ sHdWHQðtÞ þ d X NQ_ðtÞ n¼0 V_nQ 1 ; ð6Þ where lQ_{¼ l} phu huhþ 1p ð Þhd hdþh h i ; k ¼ ^p ^hu ^ hu1þ 1 ^p ð Þ^_h_d ^_h_dþ1 1; ^hu¼ hu h; ^_h d¼ hdþ h; ^p ¼ phuhd þh ð Þ phuðhdþhÞþ 1pð ÞhdðhuhÞ; and VQ

n is the jump magnitude under Q such that YQ¼ ln VnQ

has an asymmetric DEJD with the density function:

fYQðyÞ ¼ ^p^h_ue ^huy1_f_y0_gþ 1 ^pÞ^h_de^hdy1_f_y_<0_g; ^h_u> 1; ^h_d> 0:

ð7Þ Focusing on the jump specification of the housing price in Eq.7, four special cases can be delineated:

Case (1) Suppose that ^hu ¼ ^hd and ^lu¼ ^ld (i.e.,^p¼ 0:5); then the distribution of

jumps will be symmetrical with a higher peak and a positive kurtosis relative to normal.

Case (2) Suppose that ^hu¼ ^hd and ^lu6¼ ^ld (i.e.,^p6¼ 0:5); then, relative to the

geometric Brownian motion, the distribution of the return of the housing price will be skewed and have excess kurtosis, and the relative size of ^lu

and ^ld will lead to negative or positive skewness.

4

The detailed description of the Esscher transform and similar detailed proof of the martingale condition can see Carr and Madan (1999).

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Case (3) Suppose that ^hu6¼ ^hd and ^lu¼ ^ld; again, the resulting return of the

housing price will be skewed and show excess kurtosis. However, the relative size of ^hu and ^hd will determine whether the distribution is

negatively or positively skewed.

Case (4) Suppose that ^lu¼ ^ld ¼ 0; again, the resulting return of the housing price

will be reduced to the geometric Brownian motion. The Mortgage Insurer’s Liability Process

In the previous literature, the liability process follows a lognormal diffusion process, such as Cummins (1988). However, this modeling fails to particularly consider the impact of stochastic interest rates and housing prices. This shortcoming leads to the need to pay attention to modeling the liability value of the mortgage insurer, as falling house prices and rising interest rates are the accelerating factors for the catastrophic nature of MI. Therefore, extending Duan et al. (1995) and modeling the mortgage insurer’s total liability value as consisting of two risk components, i.e., interest rate and house price risks, the change rate of the liability of the mortgage insurer under the physical probability measure P can be described as

dLðtÞ LðtÞ ¼ mLdtþ frLdWrPðtÞ þ fHLdWHPðtÞ þ sLdWLPðtÞ; ð8Þ whereμLandsL¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi es2 L f2rL f2HL p

are the drift and volatility terms of the rate of change of the liability value, and frL¼esLrrL is the instantaneous interest rate

sensitivity of the rate of change of the liability value, whereesLis the total volatility

of the rate of change of the liability value, andρrLis a correlation coefficient of the interest rate and the mortgage insurer’s liability. fHL ¼esLðrHL rrHrrLÞ=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 r2

rH

p is the instantaneous house price sensitivity of the rate of change of the liability, whereρHLis the correlation coefficient between the housing price and the mortgage insurer’s liability. WP

LðtÞ is a Wiener process. As the following martingale condition

is satisfied, rðtÞ ¼ mLþ frL lr ffiffiffiffiffiffiffiffi rðtÞ p v fHLsHhþ sLhL; ð9Þ

in a risk-neutral measure Q, the rate of change of the liability of the mortgage insurer is governed by the following process:

dLðtÞ LðtÞ ¼ rðtÞdt þ frLdWrQðtÞ þ fHLdW Q HðtÞ þ sLdWLQðtÞ; dW_LQðtÞ ¼ dWP LðtÞ hLdt; ð10Þ where the termηLis the market price of the liability value, and the term WLQðtÞ is a

Wiener process under the risk-neutral measure Q. The Mortgage Insurer’s Asset Process

In addition to the typical way of modeling the asset dynamics by assuming a lognormal diffusion process for the asset value, the model explicitly takes into

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account the impact of stochastic interest rates on the asset value. This is important for modeling the asset value of the mortgage insurer, as it is common for mortgage insurers to hold a large proportion of fixed-income assets in their portfolios. The change rate of the asset value of the mortgage insurer under the physical probability measure P can be written as follows:

dAðtÞ AðtÞ ¼ mAdtþ frAdWrPðtÞ þ sAdWAPðtÞ; ð11Þ where μA and sA¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 r2 rA p

esA are the drift and volatility terms of the rate of

change of the asset value, and frA ¼ rrAesA is the instantaneous interest-rate

sensitivity of the rate of change of the asset value, where ρrA is a correlation coefficient of the interest rate and the mortgage insurer’s asset, and esA is the total

volatility of the rate of change of asset values. W_APðtÞ is a Wiener process. As the following martingale condition is satisfied,

rðtÞ ¼ mAþ frA lr ffiffiffiffiffiffiffiffi rðtÞ p v þ hAsA; ð12Þ

in a risk-neutral measure Q, the rate of change of the assets of the mortgage insurer can be described as follows:

dAðtÞ

AðtÞ ¼ rðtÞdt þ frAdWrQðtÞ þ sAdWAQðtÞ; ð13Þ

dW_AQðtÞ ¼ dW_APðtÞ hAdt; ð14Þ

where the termηAis the market price of the asset value. W_AQðtÞ is a Wiener process under the risk-neutral measure Q.

Valuation of Mortgage Insurance Contract

According to section above, we can know the risk-neutral dynamic processes of the interest rate, the housing price, the liability value and the asset value. Use of the four dynamic processes can lead to the valuation of the MI via discounting of the expected payoffs in the risk-neutral measure Q. At origination, t=0, the lender issues a T-year loan mortgage for the amount of B(0)=LRH(0). Let LRbe the initial loan-to-value ratio and H(0) be the initial housing price. We assume that the mortgage loan has an adjusted interest rate y and that installments c are paid annually. Therefore, with no prepayment or default prior to time t, the owed loan balance B(t) at time 0≤t≤T is as follows: BðtÞ ¼c y 1 1 1þ y ð ÞTt ! : ð15Þ

This equation shows that the unpaid loan balance is equal to the value of an ordinary annuity with an annual payment equal to c and a discount rate equal to the contract

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rate y. In addition, at time t=0, the mortgage insurer writes an MI contract that agrees to indemnify the lender if the borrower defaults.

We assume that each time interval is a year and that the borrower has the opportunity to default at these times. Considering the mortgage insurer’s default risk, the losses of MI at time t, LOSSD(t), can be written as follows:

LOSSDðtÞ ¼ LCBðtÞ if HðtÞ < 1 Lð CÞBðtÞ and AðtÞ LðtÞ LCBðtÞAðtÞ LðtÞ if HðtÞ < 1 Lð CÞBðtÞ and AðtÞ < LðtÞ BðtÞ HðtÞ if 1 Lð CÞBðtÞ HðtÞ < BðtÞ and AðtÞ LðtÞ BðtÞHðtÞ ð ÞAðtÞ LðtÞ if 1ð LCÞBðtÞ HðtÞ < BðtÞ and AðtÞ < LðtÞ 0 otherwise 8 > > > > > < > > > > > : ð16Þ

where LCdenotes the coverage ratio. The first and third terms of Eq.16follow the setup of Bardhan et al. (2006) and are expressed as the loss of the mortgage insurer if the borrower defaults at time 0≤t≤T, whereas the mortgage insurer does not default during the remaining life of the MI. Thus, during the remaining life of the MI, the value of the mortgage insurer’s total assets A(t) is higher than the value of the mortgage insurer’s total liability L(t), and thus the mortgage insurer does not default. The second and fourth terms of this equation are expressed as the recovery loss of the MI if the mortgage insurer defaults at time 0≤t≤T, i.e., A(t) is less than L(t), and the borrower also defaults at time 0≤t≤T during the remaining life of the MI. The recovery loss is equal to the original loss of the MI multiplied by the recovery rate, A (t)/L(t). Therefore, Eq. 12 indicates that the MI contract embeds a portfolio of vulnerable American puts that may be exercised when the mortgage borrowers default and the contract is compulsory to be terminated in the case of the default of the mortgage insurers.

The present value of the loss, i.e., the expected loss to the insurer conditional on the borrower’s default happening at time t∈T and discounted back to the present time, DL(t), can be described as follows:

DLðtÞ ¼ EQ _eR_tTrðsÞds_LOSS iðtÞ

; i ¼ ND or D: ð17Þ

Some special cases in Eq.17can be delineated:

(a) If A(t)/L(t)→∞ (i.e., the mortgage insurer would not default) and there is a constant interest rate, the closed-form solution of Eq.17using put-call parity is given by the following expressions:5

DLðtÞ ¼ P t; Kð 1Þ P t; Kð 2Þ ¼eak1_2p

R1 1eiwk1

ertf~

HðtÞðwi aþ1ð ÞÞ

a2_þaw2_{þi 2aþ1}_ð _Þwdw

eak2 2p

R₁

1eiwk2

ertf_HðtÞ~ ðwi aþ1ð ÞÞ

a2_þaw2_{þi 2aþ1}_ð _Þwdw þ Kð 1 K2Þert:

ð18Þ where

f~

HðtÞðw i a þ 1ð ÞÞ

5

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¼ exp i w i a þ 1ð ð ÞÞ ln Hð0Þ þ r 1 2s 2 H lQk t 1 2ðw i a þ 1ð ÞÞ 2_s2 Ht n o exp lQt p ^^hu ^ hui wi aþ1ð ð ÞÞþ 1^p ð Þ^_h_d ^

hdþi wi aþ1ð ð ÞÞ1

; k1 ln K1 ¼ ln BðtÞ; k2 ln K2¼ ln 1 Lð CÞBðtÞ; ~ HðtÞ ln HðtÞ: (b) If A(t)/L(t)→∞ and a constant interest rate and a lognormal jump component of

the housing price exist, the closed-form solution of Eq. 17 reduces to the closed-form formula of Chen et al. (2010).

(c) If A(t)/L(t)→∞ and a constant interest rate and no jump component of housing prices exist, the closed-form solution of Eq. 17 reduces to the closed-form formula of Bardhan et al. (2006).

Because the housing price is independent of the unconditional probability of the borrower’s defaulting, the MI premium (FPA) with an asymmetric jump risk is given by the following expression:

FPA¼ 1 þ qð ÞX

T t¼1

PðtÞDLðtÞ; ð19Þ

where q represents the gross profit margin, and PðtÞ ¼ 1 elbt_,1

bdenotes the default frequency of the borrower. Equation19implies that FPA is calculated by 1+q multiples of the fair price, i.e.,P

T

t¼1PðtÞDLðtÞ, which is the summation of a

series of the loss amounts of the insurer if the borrower defaults in each year from the beginning to expiration. Therefore, the insurer can decide for each year the probability that the borrower will default rather than at only maturity.

Empirical and Sensitivity Analysis Data and Empirical Results

Our data come from the Federal Housing Finance Agency and contain the term on conventional single-family mortgages and the monthly national average of all home prices in the US. We investigate the monthly average of the prices of all homes with adjustable-rate mortgages.6 Our sample period is from January 1986 to October 2008, leading to 274 observations for each variable. We use the asymmetric DEJD process (see Eq.3) to compare the model’s fitness for the national average of

single-6_{In addition to ARM loans, FRM loans are also available for the FHFA. US national average all home}

price returns for single-family FRM loans also seem to feature the asymmetric jump phenomenon. It shows that there were four occasions when the monthly housing price changed by more than two standard errors per month. And then it can be seen that there were six occasions when the monthly housing price changed by less than two standard errors per month. Hence, the asymmetric jump phenomenon in ARM loans seems to be higher than one in FRM loans. Because that this paper focuses on the asymmetric jump phenomenon of US national average all home price returns, for simplification, in the empirical study, only US national average all home price returns for single-family ARM loans were used. In further research, it could compare the asymmetric jump phenomenon of ARM loans and FRM loans, and then investigate the impacts of their asymmetric jump behaviors on MI premiums.

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