Ann found an apartment that costs $500,000. She has saved $100,000 for a down pa
ID: 2810073 • Letter: A
Question
Ann found an apartment that costs $500,000. She has saved $100,000 for a down payment and will get a mortgage for $400,000. It will be a fully amortizing 30year fixed rate mortgage at 4% with monthly payments. The following 10 questions will use this information. a) What is Ann's Loan to Value ratio when she gets the loan? b) What will Ann's monthly mortgage payment be? c) What will be the balance on Ann's mortgage after the 7h monthly payment? d) How many dollars is Ann paying in interest in her 30th monthly mortgage payment? e) How much of Ann's 30h monthly mortgage payment will be principal? What is the total sum of all cash flows that Ann will pay the bank over the entire life of the loan? 9) Suppose the bank wants to sell Ann's mortgage after 10 years of payments and Ann has not prepaid. What is the market of the mortgage if rates stay the same h) Suppose the bank wants to sell Ann's mortgage after 10 years of payments and Ann has not prepaid. What is the market value of the mortgage if interest rates rise to 8%?Explanation / Answer
Ann is taking a loan of $400,000 which is a 30-year mortgage loan with monthly payments @4% pa.
a) Loan to Value ratio (LTV) is defined as the ratio of the Principal of the loan to the Market Value of the asset being purchased using the loan
LTV = 400,000/500,000 = 0.8
b) Monthly Payments (C) =?
Since payments are done monthly, No.of time periods (n) = 30*12 = 360 months & r = 4%/12 = 0.33%
Loan amount = $400,000
As a part of this mortgage, C dollars of monthly payments is done for 360 months @0.33% so that the present value of all these payments will be equal to the loan amount today which is $400,000
Therefore; 400,000 = PV of annuity of $ C paid every month for 360 months @0.33% per month
->400,000 = C*(1-(1/(1+r)n))/r
->400,000=C*(-(1/(1+0.0033)360))/0.0033
-> C = $1,909.66
Therefore Monthly Mortgage payment by Ann = $1,909.66
c) Total Payments are for 360 months.
After the 7th payment, there will be 360-7 = 353 payments left.
Therefore balance on Ann's Mortgage Loan after the 7th payment = Present Value of all the 353 future payments left after the 7th payment
Mortgage Loan balance = PV of the annuity paying $1,909.66 every month for 353 months from today @0.33%
= C*(1-(1/(1+r)n))/r
= 1909.66*(1-(1/(1.0033)353))/0.0033
=395,925.14
Therefore Mortgage Loan Balance after the 7th payment = $395,925.14
(d) Use the IPMT formula in Excel that retains the interest payment made in an nth period of a mortgage
IPMT(interest rate, period, no.of payments, PV)
Since we want the interest paid in the 30th monthly mortgage, period = 30, r=4%/12=0.33%,no.of payments(n)=360 & PV = $400,000
= IPMT(0.33%,30,360,400000)
= $1,274.94
Therefore the amount Ann paid as interest in her 30th monthly payment = $1,274.94
(e) Use the PPMT formula in Excel that retains the value of payment made towards principal in an nth period of a mortgage
PPMT(interest rate, period, no.of payments,PV)
Since we want the principal paid in the 30th monthly mortgage, period = 30, r=4%/12=0.33%,no.of payments(n)=360 & PV = $400,000
= PPMT(0.33%,30,360,400000)
= $634.72
Therefore the amount Ann paid as interest in her 30th monthly payment = $634.72
(Note: An ineteresting observation can be made that the total monthly payment is equal to the sum of the amounts paid as interest an principal of that particular period. Thus, $1,274.94+$634.72=$1,909.66)
(f) The sum of all payments done by Ann in the life of mortgage = 360 payments of $1909.66 each
= 360*1909.66 = $687,478.03
Thus, Sum of all payments done by Ann in the lifetime of mortgage = $687,478.03
(g) After 10 years of payments, Ann is left with 20 more years of payments.
Then, the market value of mortgage after 10 years of payments = Present value of all the payments of the future 20 more years left
Market Value of Mortgage = PV of an annuity of $1909.66 paid every month for 20*12=240months discounted @0.33% (since rates stay the same)
= C*(1-(1/(1+r)n))/r
= 1909.66*(1-(1/(1.0033)240))/0.0033
= $315,135.84
Therefore, Market Value of Mortgage after 10 years of payments = $315,135.84
(h) After 10 years of payments, Ann is left with 20 more years of payments.
Then, the market value of mortgage after 10 years of payments = Present value of all the payments of the future 20 more years left
Market Value of Mortgage = PV of an annuity of $1909.66 paid every month for 20*12=240months discounted @0.67%per month (8%/12) (since interest rater rose to 8%)
= C*(1-(1/(1+r)n))/r
= 1909.66*(1-(1/(1.0067)240))/0.0067
= $228,308.19
Therefore, Market Value of Mortgage after 10 years of payments = $228,308.19
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