Our Own Problem & Solution Set-Up to Accompany Topic 6. Consider the five $200,000, 30-year amortization period mortgage loans described below.

Transcription

1 Our Own Problem & Solution Set-Up to Accompany Topic 6 Notice the nature of the tradeoffs in this exercise: the borrower can buy down the interest rate, and thus make lower monthly payments, by giving the lender additional money up front (paying a greater number of points ). When a range of rate/point combinations is available, lenders refer to the higher annual contract interest rate that would be charged if no points were paid as the par rate. In real world situations a point tends to reduce the annual contract interest rate by12½ to 25 basis points. (If the lender absorbs the borrower s closing costs the loan can be said to carry negative points, with the borrower paying a contract interest rate higher than par.) When computing the answers, we are looking for the rate/point mix that leads to the lowest effective cost of borrowing (monthly IRR, annual APR, and annual EAR). While we must look at the specific numbers in each situation, what we tend to see is that it can make sense for the borrower to buy down the interest rate with points if he/she expects to keep the loan in force for a long number of years why pay considerable money up front to get the benefit of lower payments for only a short period of time? Consider the five $200,000, 30-year amortization period mortgage loans described below. Loan A: 7.200% Annual Coupon (Nominal) Interest Rate, 0 Discount Points Loan B : 7.000% Annual Coupon (Nominal) Interest Rate, 1 Discount Point Loan C : 6.875% Annual Coupon (Nominal) Interest Rate, 3 Discount Points Loan D : 6.750% Annual Coupon (Nominal) Interest Rate, 3.5 Discount Points Loan E: 6.0% Annual Coupon (Nominal) Interest Rate, 4 Discount Points For each of the five loans there are also five prepayment possibilities: Case 1: No early repayment (final payment made at the end of month ) Case 2: Final payment made at end of month (after 25 years of regular payments) Case 3: Final payment made at end of month (after 15 years of regular payments) Case 4: Final payment made at end of month (after 5 years of regular payments) Case 5: Final payment made at end of month 12 (after just 1 year of regular payments) Note that the key distinction within each of the five cases for loans A through E is whether there is early repayment of principal and, if so, when prepayment occurs. For each of the 25 possible scenarios (Loan A Case 1, Loan A Case 2, etc.) please: compute the payment that the borrower should make at the end of each month compute the amount of principal still owed when the final payment is made compute the monthly internal rate of return (IRR), annual percentage rate (APR) of return, and effective annual rate (EAR) of return earned by the lender Then identify the loan with the lowest periodic borrowing cost for each early repayment case. While you should be able to set up a problem of this nature in equation form (you are encouraged to try a few so that you can work with the equations on the exam), you should compute the answers requested above by setting up a spreadsheet template based on the instructions provided on the web site. Because you must compute answers for all 25 possible combinations, you should have 25 printouts of the results shown in the command area at the top of the spreadsheet. Your grader will be grateful if you also have a 26 th page containing a grid that shows the requested information for each of the 25 combinations and indicates which loan is the lowest cost alternative for each of the five early repayment cases. Trefzger/FIL 1

2 Computing the APR/EAR on a mortgage loan is a trial and error exercise, for which we would use the IRR function on a spreadsheet or the I/Y or IRR key on a financial calculator. For the exam you should be able either to set the problem up, or to solve it with a financial calculator. (Eventually we will construct our own spreadsheet template to solve all of these examples.) Loan A, Case 1: This loan carries no discount points, such that borrower receives the use of the full $200,000 nominal principal. With a monthly periodic interest rate of.072% 12 =.006% and 30 x 12 = monthly payment periods, we compute: PMT x FAC = TOT PMT.006 = $200,000 PMT x = $200,000 So PMT = $200, OR PMT = $200,000 x = $1, If there is no prepayment (final payment expected in month ), then borrower leaves the bank today with $200,000 and pays $1, back times. Because $1, x = $488,727.51, an amount far more than the $200,000 net amount lent, the transaction has to represent a positive rate of return for the lender. To find how big a return we use trial and error to solve for r in the following equation: r $1, r = $200,000. It turns out to be r =.006, such that the APR is.006 x 12 = 7.2% and the EAR is (1.006) 12 1 = %. Financial calculator: type $200,000 +/- PV, N, = I/Y, $0 FV, CPT PMT; should show the $ monthly payment. Now you have a payment and want to compute the rate of return represented when the lender receives that payment stream after investing $200,000, so type CPT I/Y; should show.6 (which is.6%, or.006). Loan B, Case 1: This loan carries 1 discount point, such that borrower must pay the lender 1% of the $200,000 stated principal as additional up-front interest. Thus borrower leaves lender s office with only 100% 1% = 99% of $200,000, or $198,000 net of what she had when she entered. But the payment is computed based on the entire $200,000 stated principal. With a monthly periodic interest rate of 7% 12 =.5833% and 30 x 12 = monthly payment periods, we compute: PMT x FAC = TOT PMT Trefzger/FIL 2 = $200,000

3 PMT x = $200,000 So PMT = $200, OR PMT = $200,000 x = $1,330.. If there is no prepayment (final payment expected in month ), then borrower leaves the bank today with $198,000 and pays $1,330. back times. Because $1,330. x = $479,017.80, an amount far more than the $198,000 net amount lent, the transaction has to represent a positive rate of return for the lender. To find how big a return we use trial and error to solve for r in the following equation: 1 1 $1, r r = $198,000. It turns out to be r = , such that the APR is x 12 = % and the EAR is ( ) 12 1 = %. Financial calculator: type $200,000 +/- PV, N, 7 12 = I/Y, $0 FV, CPT PMT; should show the $1,330. monthly payment. Now you have a payment and want to compute the rate of return represented when the lender receives that payment stream after investing only $198,000 ($200,000 minus 1 point); so type $198,000 +/- PV, CPT I/Y; should show Loan C, Cases 2, 4, and 5: This loan carries 3 discount points, such that borrower must pay the lender 3% of the $200,000 stated principal as additional up-front interest. Thus borrower leaves lender s office with only 100% 3% = 97% of $200,000, or $194,000 net of what she had when she entered. But the payment is computed based on the entire $200,000 stated principal. With a monthly periodic interest rate of 6.875% 12 =.5729% and 30 x 12 = monthly payment periods, we compute: PMT = $200,000 PMT x = $200,000 So PMT = $200, OR PMT = $200,000 x = $1, If prepayment occurs after months (25 years), when months (5 years) of payments have not yet been made, then the amount to be paid at the end of year 25, along with the th $1, payment, is $1, OR $200, = $66, To find the lender s return we use trial and error to solve for r in the following equation: Trefzger/FIL 3

4 1 1 1 r $1, r + $66, r = $194,000. The answer turns out to be a monthly r of.5984%, with an APR of x 12 = % and an EAR of ( ) 12 1 = %. Financial calculator: type $200,000 +/- PV, N, = I/Y, $0 FV, CPT PMT; should show the $1, monthly payment. Then type N, CPT PV; should show -$66, as remaining principal balance with months remaining. Then hit the key and FV to enter that value as the amount the lender will get back in a lump sum at the end of month. Then type in $194,000 +/- PV, N and CPT I/Y; should show If prepayment occurs after just months (5 years), when months (5 years) of payments have not yet been made, then the amount to be paid at the end of year 5, along with the th $1, payment, is $1, OR $200, = $188, To find the lender s return we use trial and error to solve for r in the following equation: r $1, r 1 + $188, r = $194,000. The answer here turns out to be monthly r =.6348%, with an APR of x 12 = % and an EAR of ( ) 12 1 = %. Financial calculator: type $200,000 +/- PV, N, = I/Y, $0 FV, CPT PMT; should show the $1, monthly payment. Then type N, CPT PV; should show -$188, as remaining principal balance with months remaining. Then hit the key and FV to enter that value as the amount the lender will get back in a lump sum at the end of month. Then type in $194,000 +/- PV, N and CPT I/Y; should show Finally, if prepayment occurs after just 12months (1 year), when 348 payments have not yet been made, then the amount that should be paid at the end of year 1, along with the 12 th $1, regular payment, is $1, OR $200, = $197, To find the lender s return we use trial and error to solve for r in the equation: Trefzger/FIL 4

5 1 1 $1, r.r 12 + $197, r 12 = $194,000. Here the monthly r solves to be.8380%, corresponding to an APR of x 12 = % and an EAR of ( ) 12 1 = %. Financial calculator: type $200,000 +/- PV, N, = I/Y, $0 FV, CPT PMT; should show the $1, monthly payment. Then type 348 N, CPT PV; should show -$197, as remaining principal balance with 12 months remaining. Then hit the key and FV to enter that value as the amount the lender will get back in a lump sum at the end of month 12. Then type in $194,000 +/- PV, 12 N and CPT I/Y; should show Loan D, Cases 1 and 3: This loan carries 3.5 discount point, such that borrower must pay the lender 3.5% of the $200,000 stated principal as additional up-front interest. Thus borrower leaves lender s office with only 100% 3.5% = 96.5% of $200,000, or $193,000 net of what she had when she entered. But again, the payment is computed based on the entire $200,000 stated principal. With a monthly periodic interest rate of 6.75% 12 =.5625% and 30 x 12 = monthly payment periods, we compute: PMT x FAC = TOT PMT = $200,000 PMT x = $200,000 So PMT = $200, OR PMT = $200,000 x = $1, If there is no prepayment (final payment expected in month ), then borrower leaves the bank today with $193,000 and pays $1, back times. Because $1, x = $466,990.63, an amount far more than the $193,000 net amount lent, the transaction has to represent a positive rate of return for the lender. To find how big a return we use trial and error to solve for r in the following equation: r $1, r = $193,000. It turns out to be r = , such that the APR is x 12 = % and the EAR is ( ) 12 1 = %. Financial calculator: type $200,000 +/- PV, N, = I/Y, $0 FV, CPT PMT; should show the $1, monthly payment. Now you have a payment and want to compute the rate of return represented when the lender receives that payment stream after investing only $193,000 ($200,000 minus 3.5 points); so type $193,000 +/- PV, CPT I/Y; should show Trefzger/FIL 5

6 If prepayment occurs after months (15 years), when months (15 years) of payments have not yet been made, then the amount to be paid at the end of year 15, along with the th $1, payment, is 1 1 $1, OR $200, = $146, (In this situation the exponents are deceptively easy, because mohtns have passed and remain. But be careful to tell yourself the correct story: in the left-hand representation above the exponent is because payments remain unpaid; in the right-hand approach the exponent in the numerator is the months that already have passed.) To find the lender s return we use trial and error to solve for r in the following equation: r $1, r + $146, r = $193,000. The answer turns out to be a monthly r of.5974%, with an APR of x 12 = % and an EAR of ( ) 12 1 = %. Financial calculator: type $200,000 +/- PV, N, = I/Y, $0 FV, CPT PMT; should show the $1, monthly payment. Then type N, CPT PV; should show -$146, as remaining principal balance with months remaining. Then hit the key and FV to enter that value as the amount the lender will get back in a lump sum at the end of month. Then type in $193,000 +/- PV, N and CPT I/Y; should show Trefzger/FIL 6