A comprehensive set of figures showing the sensitivities ofE[Vt] and S D[Vt] for various pa- rameter settings is provided in Appendix C.1. Tables 3.4 and 3.5 summarize the list of figures according to values ofµ,σ,randi. The range of values taken forrandiare the same from 1% to 20% with an increment of 1%. Plots only show the curves for the pairs (r,i) so thatdIt >0.
Table 3.4: List of figures for mean,E[Vt] and its derivatives.
µ Line of credit ratei Figure # Mortgage rater Figure #
0.05 0.03 Figure C.1 0.03 Figure C.4 0.05 Figure C.2 0.05 Figure C.5 0.08 Figure C.3 0.08 Figure C.6 0.10 0.03 Figure C.7 0.03 Figure C.10 0.05 Figure C.8 0.05 Figure C.11 0.08 Figure C.9 0.08 Figure C.12 0.15 0.03 Figure C.13 0.03 Figure C.16 0.05 Figure C.14 0.05 Figure C.17 0.08 Figure C.15 0.08 Figure C.18
3.3. Sensitivity ofInvestmentPortfolioValue 47
Table 3.5: List of figures for standard deviation,S D[Vt] and it derivatives.
µ σ Line of credit ratei Figure # Mortgage rater Figure # 0.05 0.10 0.03 Figure C.19 0.03 Figure C.22 0.05 Figure C.20 0.05 Figure C.23 0.08 Figure C.21 0.08 Figure C.24 0.20 0.03 Figure C.25 0.03 Figure C.28 0.05 Figure C.26 0.05 Figure C.29 0.08 Figure C.27 0.08 Figure C.30 0.30 0.03 Figure C.31 0.03 Figure C.34 0.05 Figure C.32 0.05 Figure C.35 0.08 Figure C.33 0.08 Figure C.36 0.10 0.10 0.03 Figure C.37 0.03 Figure C.40 0.05 Figure C.38 0.05 Figure C.41 0.08 Figure C.39 0.08 Figure C.42 0.20 0.03 Figure C.43 0.03 Figure C.46 0.05 Figure C.44 0.05 Figure C.47 0.08 Figure C.45 0.08 Figure C.48 0.30 0.03 Figure C.49 0.03 Figure C.52 0.05 Figure C.50 0.05 Figure C.53 0.08 Figure C.51 0.08 Figure C.54 0.15 0.10 0.03 Figure C.55 0.03 Figure C.58 0.05 Figure C.56 0.05 Figure C.59 0.08 Figure C.57 0.08 Figure C.60 0.20 0.03 Figure C.61 0.03 Figure C.64 0.05 Figure C.62 0.05 Figure C.65 0.08 Figure C.63 0.08 Figure C.66 0.30 0.03 Figure C.67 0.03 Figure C.70 0.05 Figure C.68 0.05 Figure C.71 0.08 Figure C.69 0.08 Figure C.72
48 Chapter3. SensitivityAnalysis
3.4
Conclusion
In this chapter we studied the behavior of the investment capital to different values of the mortgage rate, r and the line of credit rate i. The range of values for r andi is from 1% to 20% with an increment size of 1%. Considering all possible pairs of (r,i) we find that only if the difference, i−r, exceeds 5% willdIt become negative for somet > 0. A spread this large is unlikely to occur in practice. We collected mortgage and the line of credit rates from five different banks on February 23, 2012. The maximumr−ispread we get is 3.25%. Thus, we can safely say thatdIt will always be positive for realistic values ofrandi.
We investigate the sensitivity of the mean and standard deviation of the risky investment to different parameters such as time t, mortgage rate,r, the line of credit rate, i, growth rate,
µ, and volatility, σ. The mean, E[Vt], and the standard deviation, S D[Vt], grow with time. The derivatives of mean with respect totandµ, and the derivatives of standard deviation with respect to t, µandσ are increasing functions of time. The derivatives of mean and standard deviation with respect to i are decreasing functions of time. The derivative of mean with respect to r appears concave up. The function decreases between the time period 0 years and amortization term. After the amortization term it increases. The derivatives of standard deviations with respect torandiare decreasing functions of time.
After the amortization term the whole amount of mortgage payment is invested, therefore, if mortgage rate is higher the more money is available for new investment. It is for this reason that the mean, E[Vt] and its derivative with respect tot, are decreasing functions ofr before the amortization term and they are increasing functions ofr after the amortization term. The derivatives of mean with respect to r and i are increasing functions of r. The derivative of mean with respect to µis a decreasing function of r. The standard deviation, S D[Vt], and its derivatives with respect tot, σandµare decreasing functions ofr. The derivatives ofS D[Vt] with respect torandiare increasing functions ofr.
As the line of credit rate, i, increases less new money is available for investment, hence,
E[Vt] decreases. The derivatives of mean with respect totandµare also decreasing functions of i. The derivative of mean with respect to r is an increasing function of i. The standard deviation,S D[Vt], and its derivatives with respect tot, σ andµare decreasing functions of i. The derivatives ofS D[Vt] with respect torandiare increasing functions ofi.
The E[Vt] and its derivatives with respect to t and µ are increasing functions of µ. The derivatives ofE[Vt] with respect torandiare decreasing functions ofµ. The growth ofE[Vt] is not big enough ifµis small (between 1% to 10%). The higher theµthe higher the values of
E[Vt]. The S D[Vt] and its derivatives with respect to t, σandµincreases withµand σ. The derivatives ofS D[Vt] with respect torandidecrease withµandσ.
Chapter 4
Incorporating Variable Housing Prices
and Interest Rates
4.1
Introduction
In Chapter 1 we considered a fixed-rate mortgage and presented a mathematical model for the re-advanceable mortgage scheme. A homeowner has a mortgage of principal L with amorti- zation termmyears with fixed annual continuous rater. The homeowner repays the mortgage continuously at the annual ratePgiven by
P= Lre
rm
erm−1. (4.1)
The mortgage principal outstanding at timetis calculated by
Bt = L(erm−ert) erm−1 , if 0≤t ≤m, 0, if t >m. (4.2)
As the mortgage is paid the principal of the home equity line of credit increases by the principal portion of the mortgage payment. Thus, the time-tline of credit balance,Ct, is
Ct = L−Bt =
(
Leermrt−−11, if 0≤ t≤m,
L, if t>m. (4.3)
The rate of change ofCt is
dCt = −dBt =
( Lrert
erm−1dt, if 0≤t≤ m,
0, if t> m. (4.4)
The interest on the line of credit and the tax rebate are paid continuously at the annualized rates of iand c, respectively. If It is the time-t investment capital then the time-t amount of money available for new investment is
dIt =dCt −i(1−c)Ctdt+PdtIt>m = ( L erm−1 rert −i(1−c)(ert−1) dt, if 0≤t≤ m, L erm−1[re rm−i(1−c)(erm−1)]dt, if t> m, (4.5) 49
50 Chapter4. IncorporatingVariableHousingPrices andInterestRates
whereIA is an indicator function that is one ifAis true and is zero otherwise.
The homeowner invests the entire proceeds from the line of credit borrowing into a risky asset, a stock. The dynamics of the stock price,St, are given by geometric Brownian motion
dSt =µStdt+σSStdWt, (4.6) whereW is a Wiener process,µis the drift andσS is the volatility. The stochastic differential equation for the investment portfolio value,Vt, is given by
dVt = (µdt+σSdWt)Vt+dCt−i(1−c)Ctdt = ( (µVt+L re rt erm−1 −i(1−c)L ert−1 erm−1)dt+σSVtdWt, if 0≤t≤ m, (µVt+L re rm erm−1 −i(1−c)L)dt+σSVtdWt, if t> m. (4.7) A quantity of interest is the mortgage payofftime,τ, which we defined as the first time the value of the risky investment portfolio is at least the value of the original mortgage amount. The payofftime is defined as
τ= inf
t {t≥0|Vt ≥ L}. (4.8) Another quantity of interest is the total interest cost, Im∗, of the original mortgage which is defined as
Im∗ = Pm−L. (4.9)
The total cost of the re-advanceable mortgage can be different fromIm∗ depending on the payoff timeτand it is defined as
Iτ∗=
(
Pτ−Cτ, if τ <m,
Pτ−L, if τ≥m. (4.10)
Forτ <m,Pτis the total cost of repayingCτby timeτ.
The homeowner keeps on making mortgage payment after the amortization termmyears if
τ >m. This way there would be new money available for the investment as no more money can be re-borrowed in the line of credit onceCt isL. In other words, ifτ >mthen the homeowner needs to make up investment losses by making mortgage payments untilVt ≥ L. Whenτ= m,
Iτ∗is exactly equal toIm∗.
As mentioned earlier so far in this thesis a mathematical model for the re-advanceable mort- gage investment strategy was derived assuming that the mortgage rate remains fixed throughout the mortgage term. For simplicity we assumed that the mortgage term is same as the amorti- zation term. In reality this is not the case as the mortgage rate gets renewed on the expiry of a mortgage term (which is typically from six months to 10 years). In this chapter we incorporate the fact that the mortgage rate is not fixed by considering a variable rate mortgage. We still assume that the homeowner makes a fixed mortgage payment but the ratio between the prin- cipal and interest components of the payment fluctuates. When interest rates are low less of the mortgage payment goes to interest and more goes to principal. If rates are high more goes to interest and less to principal. This typically has the effect of introducing uncertainty in the payofftime for a traditional mortgage.
In this chapter we also incorporate the fact that housing prices change with time. If there is a rise in the house price then the home equity increases. This additional home equity allows the homeowner to increase the line of credit limit and invest more money in the stock.
4.2. InterestRateModel 51