Our Own Problems and Solutions to Accompany Topic 11

Transcription

1 Our Own Problems and Solutions to Accompany Topic. A home buyer wants to borrow $240,000, and to repay the loan with monthly payments over 30 years. A. Compute the unchanging monthly payments for a standard fixed-rate, fixed-payment mortgage (FRM) loan with a 7% annual percentage rate (APR) of interest. B. Compute the first and second years payments for an adjustable rate mortgage (ARM) loan with a first-year interest rate of 4%, a 200 basis point cap on -year rate increases, and a year-2 index level that is 320 basis points higher than the initial index level. C. Compute the first and second years payments for a price level adjusted mortgage (PLAM) loan if the real riskfree interest rate is 3% and there is 5% inflation during year. A. Recall that a fixed-rate, fixed-payment mortgage (FRM) loan is a present value of a level annuity application, and that we can always set up an annuity problem with the formula PMT x FAC = TOT With this formula in place, we can simply plug in the knowns and solve for the unknowns. Because the $240,000 lump sum TOT exists intact in the present, and because the payments are made at the end of each month and all payments are equal, we have a present value of a level ordinary annuity problem. With a 7% stated annual percentage rate (APR) of interest and monthly payments, our monthly periodic interest rate is.07 2 =, or.5833%. (Do you see that we could instead 2 have identified the annual interest rate as an EAR of 7.229%, such that =?) With 30 years of monthly payments, we have 30 x 2 = 360 payment periods. Therefore, with $240,000 borrowed we compute )360 ) = $240,000 PMT x = $240,000 So PMT = $240, = $, This monthly payment will remain the same for 30 years (or until the borrower terminates the loan by prepaying the remaining principal). Note that when computing loan payments we typically replace the present value of an annuity factor (here, ) with its reciprocal, the loan payment factor, so that we can multiply the amount borrowed by this payment factor to compute the periodic payment = , so PMT = $240,000 x = $, On a financial calculator (the steps described in this problem set are for a Texas Instruments BA II Plus financial calculator; other brands/models may follow slightly different key sequences), we would enter $240,000 PV, $0 FV, 360 N, 7 2 = I/Y (the periodic interest rate is 7% 2 =.5833%, or ); Compute PMT. It should show -$,596.73, equal in absolute value to what we computed manually above. (Financial calculator computations are based on the logic that an investment situation involves both positive and negative cash flows; if you are solving for a dollar value and do not enter a negative dollar amount the calculator will simply assign a negative value to the answer. Caution: if you are solving for I/Y or N and do not enter one of the dollar values as a negative you will get an error message. Trefzger/FIL 360

2 B. We compute the initial payment for an adjustable rate mortgage (ARM) loan the same way we compute the unchanging payment for a standard fixed-rate, fixed-payment mortgage loan. Here the first year s monthly periodic rate is.04 2 = , and we compute: PMT x FAC = TOT )360 ) = $240, PMT x = $240,000 So PMT = $240, = $,45.80 OR PMT = $240,000 x = $, With an initial stated annual interest rate (APR) of 4% instead of 7%, it should not be surprising that the ARM s year- payment is lower than the FRM s unchanging payment. But note that this lower ARM payment applies to the first year only; at the end of year a new year-2 payment is computed based on a new stated annual interest rate that reflects economic conditions, including expected inflation, that prevails at the end of the first year. Unless the year 2 APR is the same as the year APR, the year 2 payment will differ from that for year. On a financial calculator we enter -$240,000 PV (now we enter a negative dollar amount so the dollarbased answer shows as positive), $0 FV, 360 N, 4 2 = I/Y (the periodic interest rate is 4% 2 =.3333%, or ); Compute PMT. It should show $,45.80, just as we computed manually above. What about year 2 s monthly payment? First we compute the principal balance that remains owed at the end of year (after 2 months): $,45.80 ( ( ) ) OR $240,000 [ ( ( )2 ( ) 360 = $235, Then compute the payment for the new loan. This loan has a $235,773.5 original principal amount and a 29-year (348 month) life. With a 2% (200 basis points) annual cap on increases in the stated annual interest rate (a typical provision in real-life ARMs), the year-2 stated annual rate (APR) rises only to 4% + 2% = 6% (for a 6% 2 =.5% monthly periodic rate) even though the index on which the rate is, in theory, based has risen by 320 basis points (such that the rate would rise to 4% + 3.2% = 7.2% APR absent a cap). The year-2 payment is computed as:.005 )348 ) = $235, PMT x = $235,773.5 So PMT = $235, = $,43.6 OR PMT = $235,773.5 x = $,43.6. Note that if the new rate were 4%, then the new payment would be computed as Trefzger/FIL 360 2

3 )348 ) = $235, PMT x = $235,773.5 So PMT = $235, = $,45.80 OR PMT = $235,773.5 x = $,45.80 (if the interest rate does not change, then the monthly payment should not change from the year- level). On a financial calculator, let s assume that you have the original information already entered (or re-enter it): -$240,000 PV, $0 FV, 360 N, 4 2 = I/Y; Compute PMT and it shows $, Enter 2 N, Compute FV and it shows $235,773.5 in principal remaining owed at the end of month 2. Hit the +/- key to make this amount negative, and hit PV to enter this amount as the new loan balance to carry forward. Enter $0 FV, 348 N, 6 2 = I/Y; Compute PMT and it shows $,43.6, just as we computed manually above. C. We compute the initial payment for a price level-adjusted mortgage (PLAM) loan essentially the same way we compute the unchanging payment for a standard fixed-rate, fixed-payment mortgage loan. The only difference is that we base the PLAM payment on a very low interest rate, here an APR of 3% for a monthly periodic rate of.03 2 =.0025: PMT x FAC = TOT.0025 )360 ) = $240, PMT x = $240,000 So PMT = $240, = $,0.85 OR PMT = $240,000 x = $,0.85. With an initial stated annual interest rate (APR) of only 3% instead of 4% or 7%, it should be obvious that the PLAM s year- payment would be much lower than the FRM s unchanging payment or even the ARM s year- payment. But note that this lower PLAM payment would apply to the first year only; at the end of year a new year-2 payment would be computed based on a new principal amount that reflected inflation during the first year (so the year 2 payment would remain the same as year s only if the first year s measured inflation rate had been zero). On a financial calculator we would enter -$240,000 PV, $0 FV, 360 N, 3 2 = I/Y (the periodic interest rate is 3% 2 =.25%, or.0025); Compute PMT. It should show $,0.84, just as we computed manually above. What is the year-2 payment for the PLAM if there has been 5% measured inflation? Whereas with the ARM any expected inflation would have been reflected in the year-2 interest rate (limited by the cap), with the PLAM inflation directly increases the remaining principal balance (there is no limitation or cap). First compute the principal balance that remains owed at the end of year (after 2 months): Trefzger/FIL 360 3

4 $,0.85( (.0025 ) ) OR 240,000[ ( (.0025)2 (.0025) 360 = $234, Then compute the payment for the new loan. The original amortization plan shows a $234, principal amount remaining, but we multiply that amount by ( + inflation rate) to get a new inflationadjusted principal balance of $234, x.05 = $246, This loan is to be repaid over a 29-year (348 month) life at the same 3% stated annual interest rate (.25% monthly periodic rate):.0025 )348 ) = $02,807.8 $246, PMT x = $246, So PMT = $246, = $ OR PMT = $246, x = $, (vs. $,0.85 in year ). On a financial calculator, let s assume that you have the original information already entered (or re-enter it): -$240,000 PV, $0 FV, 360 N, 3 2 = I/Y; Compute PMT and it shows $,0.85. Enter 2 N, Compute FV and it shows $234, in principal remaining owed at the end of month 2. Enter x.05 = and then hit the +/- key to make this amount negative, and hit PV to enter this amount as the new loan balance to carry forward. Enter $0 FV, 348 N (the unchanging 3 2 = I/Y monthly periodic interest rate is already entered); Compute PMT and it shows the same $, computed manually above. 2. A home buyer wants to borrow $225,000 through an adjustable rate mortgage loan with payments made at the end of each month for 25 years. The contract calls for payments to be determined each year based on an index interest rate (perhaps the -year U.S. Treasury Bill rate) plus a margin of 250 basis points, with a 200 basis point cap on annual interest rate increases. The initial year s index interest rate is a 4.2% APR (such that index + margin = 6.7%), but the lender offers a first-year teaser rate of 4.8% APR. What would the monthly payment be during year 2 if the index rate in year 2 is 5.4%? Then what would the monthly payment be during year 3 if the index rate is 6.4%? [Note that loans with first-year teaser rates are unlikely to meet Qualified Mortgage, or QM, guidelines implemented by the Consumer Financial Protection Bureau in early 204. Under QM rules a borrower can sue a lender if the loan later is judged to have been affordable for the borrower only under the artificial conditions of the unrealistically low teaser rate.] To compute the year-2 payment on an ARM, we first have to compute the amount of principal still owed at the end of year. If we use our formula for the proportion of principal that has been repaid: ( (+r)short Period (+r) Long Period ) we don t have to bother computing the monthly payment during year (it would be $,289.24). For the balance still owed one year into the 25-year amortization period (with a =.004 monthly periodic rate in year ) simply multiply $225,000([ ( (.004)2 (.004) 300 )= $220, Trefzger/FIL 360 4

5 Then we simply compute a payment for the new loan, which has a 288-month life (with year gone, 24 years = 288 months remain). Here the index rate is 5.4%, indicating an index-plus-margin of 7.4%, but the 200 basis point cap limits the increase to 2% more than year s 4.8% rate, or 6.8%. With a new monthly periodic rate of 6.8% 2 =.5667%, the new monthly payment should be PMT x FAC = TOT )288 ) = $220, PMT x = $ So PMT = $220, = $,36.79 OR PMT = $220, x = $, (recall that the year payment would have been $,289.24, but we did not need to compute it since we used the formula for the proportion of principal repaid). On a financial calculator, let s assume that you have the original information already entered (or re-enter it): -$225,000 PV, $0 FV, 300 N, = I/Y; Compute PMT and it shows $, Enter 2 N, Compute FV and it shows $220, in principal remaining owed at the end of month 2. Hit the +/- key to make this amount negative, and hit PV to enter this amount as the new loan balance to carry forward. Enter $0 FV, 288 N, = I/Y; Compute PMT and it shows $,553.02, just as we computed manually above. Now for year 3: first, the amount of principal still owed at the end of year 2 is $220, ([ ( ( )2 ( ) 288 )= $26, The APR charged in year 3, based on market conditions, should be 8.4% (index rate of 6.4% plus margin of 2%). Because this rate does not exceed year 2 s 6.8% APR rate by more than the 200 basis point allowed cap, the lender can indeed charge an 8.4% APR during year 3. With a monthly periodic rate of 8.4%/2 =.7% or.007, and payments for 23 years x 2 = 276 months, the monthly payment is.007 )276 ) = $26, PMT x = $26, So PMT = $26, = $,36.79 OR PMT = $26, x = $, On a financial calculator, let s assume that you have the year 2 information already entered (or reenter it): -$220, PV, $0 FV, 288 N, = I/Y; Compute PMT and it shows $, Enter 2 N, Compute FV and it shows $26, in principal remaining owed at the end of month 2. Hit the +/- key to make this amount negative, and hit PV to enter this amount as the new loan balance to carry forward. Enter $0 FV, 276 N, = I/Y; Compute PMT and it shows $, just as we computed manually above. Trefzger/FIL 360 5

6 3. A 5/ hybrid adjustable rate mortgage loan has $260,000 in initial principal borrowed, a 5.4% APR initial interest rate, and a 30-year amortization period. Compute the monthly payment to be made during years through 5. Then compute the monthly payment to be made during year 6 if the interest rate after the adjustment is a 6.3% APR. Then compute the lender s effective annual rate of return (EAR) if the borrower repays the loan in full at the end of year 6. What would the EAR be if the lender had charged the borrower two discount points when the loan was closed? The only difference relative to the adjustable rate loan payments computed in part B of question above is that the remaining principal balance for the second phase is computed after 5 years rather than year. Here the monthly periodic rate for the first five years is =.0045, and we compute: PMT x FAC = TOT.0045 )360 ) = $260, PMT x = $260,000 So PMT = $260, = $, OR PMT = $260,000 x = $, On a financial calculator we enter -$260,000 PV, $0 FV, 360 N, = I/Y (the periodic interest rate is 5.4% 2 =.45%, or.0045); Compute PMT. It should show $,459.98, just as we computed manually above. What about year 6 s monthly payment? First we compute the principal balance that remains owed at the end of year 5 (after 60 months): $, ( (.0045 ) ) OR $260,000 [ ( (.0045)60 (.0045) 360 = $240, Then compute the payment for the new loan. This loan has a $240, original principal amount and a 25-year (300 month) life. With a new year-6 APR of 6.3% (for a 6.3% 2 =.525% monthly periodic rate), the year-6 payment is computed as: )300 ) = $240, PMT x = $240, So PMT = $240, = $,59.4 OR PMT = $240, x = $,59.4. On a financial calculator, let s assume that you have the original information already entered (or reenter it): -$260,000 PV, $0 FV, 360 N, = I/Y; Compute PMT and it shows $, Enter 60 N, Compute FV and it shows $240, in principal remaining owed at the end of month 60. Hit the +/- key to make this amount negative, and hit PV to enter this amount as the new loan balance to carry forward. Enter $0 FV, 300 N, = I/Y; Compute PMT and it shows $,59.4, just as we computed manually above. Trefzger/FIL 360 6

7 For the last part of the problem, the amount of principal that remains unpaid 72 months into the original 360 month amortization period, such that 288 months remain (or 2 months into the new loan s 300-month amortization period) is computed as: $,59.4 ( ( ) ) OR $240,077.00[ ( (.00525)2 (.00525) 300 = $235,99.5. Initially we assume that the lender gave the borrower $260,000 at the loan closing. In return the lender received $, at the end of every month for 5 years or 60 months, then $,59.4 at the end of every month during year 6, and then $235,99.5 along with the final $,59.4 payment at the end of month 72. We find the rate of return built into these values by solving for r in the equation: $260,000 = $,459.98( ( +r )60 ) + $,59.4 ( ( 2 +r ) 60 r r +r ) + $235,99.5( +r )72. [Or we could set up the equation showing the last monthly payment grouped with the balloon: $260,000 = $,459.98( ( +r )60 ) + $,59.4 ( ( +r ) 60 r r +r ) + $237, ( +r )72 ] If we use trial and error, the rate that solves the equation turns out to be a monthly periodic rate of about , for an APR of x 2 = % and EAR of ( ) 2 = %. Doublecheck: $,459.98( ( ) ) + $,59.4 ( ( ) )60 + $235,99.5( )72 = $76, $4, $$69, = $260,000. What if instead the lender had charged the borrower 3 points at the closing? Recall that points do not change the payments; they change only the amount that the lender ultimately extends/borrower ultimately receives in return for the agreed-upon payment stream. Here 3 points are charged, so the borrower receives only 97% of the $260,000 nominally being lent = $252,200. In return the lender receives $, at the end of every month for 5 years or 60 months, then $,59.4 at the end of every month during year 6, and then $235,99.5 along with the final $,59.4 payment at the end of month 72. We find the rate of return built into these values by solving for r in the equation: $252,200 = $,459.98( ( +r )60 ) + $,59.4 ( ( 2 +r ) 60 r r +r ) + $235,99.5( +r )72. Now with trial and error, the rate that solves the equation is a higher monthly periodic rate of about.00523, for an APR of x 2 = 6.47% and EAR of (.00523) 2 = %. It should make Trefzger/FIL 360 7

8 sense that the lender s rate of return is higher if the lender gives up fewer dollars initially to receive the same payment stream. Double-check: $,459.98( ( ) ) + $,59.4 ( ( ) )60 + $235,99.5( )72 = $75, $3, $ = $252,200. On a financial calculator you will use the automated IRR function. Type in CF 2 nd CLR WORK (this puts you in a new blank worksheet to handle cash flows). Then type $-252,200 ENTER, $, ENTER 60 ENTER (that tells calculator there are sixty $, cash flows), $,59.4 ENTER ENTER, $237, (which is $, $235,99.5) ENTER, then hit IRR and CPT (tells calculator to compute the IRR based on the cash flows specified). The screen will go blank as the calculator does its trial and error computations; then it should show You multiply that monthly rate by 2 to get the x 2 = 6.47% APR or compound it for 2 periods to get the ( ) 2 = % EAR. 4. Compute the payments for a $240,000 graduated payment mortgage (GPM) loan with payments growing by 7.5% per year for 5 years and then leveling off for years 6 30, if the interest rate is an APR of 7%. The graduated payment mortgage (GPM) loan is based on the deferred annuity idea. It is somewhat complicated, but we can handle it. First recall that, for any loan, the principal lent is the present value of the expected stream of payments. Thus the present value of all the payments to be received during years 30 on this GPM must be $240,000. Think back to the FRM in question above; with the $240,000 principal borrowed and a 7% APR, we computed each of the 360 equal payments to be $240,000 ( ( )360 ) = $, Turning it around, we can state that $, x ( ( )360 ) = $240,000. In other words, at the loan s origination date the stream of future payments, discounted to a present value, equals the original principal amount lent. This statement is true for any loan, including a graduated payment mortgage (GPM) loan. This particular GPM (with features popularized by an FHA program) calls for payments to start at a specified level and then to rise by 7.5% per year, leveling off in year 6 to remain the same through year 30. Here we want: Year payment stream, discounted back to a present value + Year 2 payment stream, discounted back to a present value + Year 3 payment stream, discounted back to a present value + Year 4 payment stream, discounted back to a present value + Trefzger/FIL 360 8

9 Year 5 payment stream, discounted back to a present value + Year 6 30 payment stream, discounted back to a present value = $240,000 principal originally lent. Let s think of the as-yet-unknown first-year payment amount as PMT. With a monthly periodic rate of.07 2 =, we can represent the present value of year s stream of 2 payments, for the moment, as PMT ( ( )2 ) = PMT x Now let s think of the as-yet-unknown second-year payment amount as PMT 2. But note that PMT 2 is simply PMT x (.075). Because we will receive 2 payments during year 2, but must wait until the 2 months of year have passed before we start collecting the 2 year 2 payments, we can represent the present value of year 2 s stream of 2 payments, for the moment, as PMT 2 ( ( )2 )2 = PMT 2 x x = PMT x (.075) x x = PMT x From this point we can just follow the pattern. We can think of the as-yet-unknown third-year payment amount as PMT 3. But PMT 3 is simply PMT x (.075) 2. Because we will receive 2 payments during year 3, but must wait until the 24 months of years and 2 have passed before we start collecting the 2 year-3 payments, we can represent the present value of year 3 s stream of 2 payments, for the moment, as PMT 3 ( ( )2 )24 = PMT 3 x x = PMT x (.075) 2 x x = PMT x The as-yet-unknown fourth-year payment amount, PMT 4, is simply PMT x (.075) 3. Because we will receive 2 payments during year 4, but must wait until the 36 months of years 3 have passed before we start collecting the 2 year 4 payments, we can represent the present value of year 4 s stream of 2 payments, for the moment, as PMT 4 ( ( )2 )36 = PMT 4 x x.8079 = PMT x (.075) 3 x x.8079 = PMT x Trefzger/FIL 360 9

10 The as-yet-unknown fifth-year payment amount, PMT 5, is simply PMT x (.075) 4. Because we will receive 2 payments during year 5, but must wait until the 48 months of years 4 have passed before we start collecting the 2 year 5 payments, we can represent the present value of year 5 s stream of 2 payments, for the moment, as PMT 5 ( ( )2 )48 = PMT 5 x x = PMT x (.075) 4 x x = PMT x Finally, the as-yet-unknown payment applying to years 6 30, PMT 6, is simply PMT x (.075) 5. Because we will receive 300 payments during years 6 30, but must wait until the 60 months of years 5 have passed before we start collecting the 300 year 6 30 payments, we can represent the present value of year 6 30 s stream of 300 payments, for the moment, as PMT 6 ( ( )300 )60 = PMT 6 x x = PMT x (.075) 5 x x = PMT x Look at what we have: PMT times a series of different factors. Using the distributive property, we can simply multiply PMT by the sum of ( ) = Because the present value of the payment stream must be $240,000, we can simply state that PMT x = $240,000 SO PMT = $240, = $,9.88. Now we know the amount of each of the first year s 2 payments. Then the payment stream will appear as follows: PMT = $,9.88 (months 2) PMT 2 = $,9.88 (.075) = $,28.27 (months 3 24) PMT 3 = $,4.25 (.075) 2 = $, (months 25 36) PMT 4 = $,4.25 (.075) 3 = $, (months 37 48) PMT 5 = $,4.25 (.075) 4 = $,59.72 (months 49 60) PMT 6 = $,4.25 (.075) 5 = $,7.0 (months 6 360) (vs. the FRM s $, payment for each of months 360). Let s double check: + $, ( ( )2 $,9.88( ( )2 ) + $,28.27( ( )2 )24 + $, ( ( )2 Trefzger/FIL )2 )36

11 + $,59.72 ( ( )2 )48 + $,7.0 ( ( )300 )60 = $,9.88 (.55720)+ $,28.27 ( ) + $, ( ) + $, ( ) + $,59.72 ( ) + $,7.0 ( ) = $3, $3, $3, $3, $3, $70, = $240, Financial calculator solution: beats the hell out of me. 5. A home buyer wants to borrow $292,000 through a price level adjusted mortgage (PLAM) loan, with payments computed based on a 20-year amortization. The current real risk-free interest rate is 3%, and inflation figures over the next three years are expected to be 5%, %, and 7%, respectively. What monthly payment does the borrower expect to make in each of years, 2, and 3? What has been the lender s effective annual rate of return (EAR) if the borrower is charged two discount points at the loan s closing and ends up paying off the loan in full at the end of year 3? Recall that with the PLAM we leave the interest rate unchanged and adjust the principal amount to reflect changing economic conditions (specifically, inflation). Let s start with the year payment. With a monthly periodic rate of 3% 2 =.0025, we compute the first year s monthly payment as PMT x FAC = TOT.0025 )240 ) = $292, PMT x = $292,000 So PMT = $292, = $,69.42 OR PMT = $292,000 x = $, Then the amount of principal that remains unpaid 2 months into the original 240-month amortization period (so there are = 228 month to go) is: $,69.42 ( (.0025 ) ) OR $292,000 [ ( (.0025)2 (.0025) 240 = $28, Then compute the payment for year 2. The original amortization plan shows a $28,78.92 principal amount remaining, but we multiply that amount by ( + inflation rate) to get a new inflation-adjusted principal balance of $28,78.92 x.05 = $295, This loan is to be repaid over a 9-year (228 month) life at the same 3% stated annual interest rate (.25% monthly periodic rate):.0025 )228 ) = $295, PMT x = $295, Trefzger/FIL 360

12 So PMT = $295, = $, OR PMT = $295, x = $, The principal remaining unpaid 2 months into the new 228-month amortization period (so there are = 26 month to go) is: $, ( (.0025 ) ) OR $295,237.86[ ( (.0025)2 (.0025) 228 = $283, The original amortization plan shows a $283,530.3 principal amount remaining, but we multiply that amount by.0 to get a new inflation-adjusted principal balance: $283,530.3 x.0 = $286, This loan is to be repaid over an 8-year (26 month) life at the same 3% stated annual interest rate (.25% monthly periodic rate), so the year 3 payment is.0025 )26 ) = $286, PMT x = $286, So PMT = $286, = $, OR PMT = $286, x = $,77.40 Principal remaining 2 months into the new 26-month amortization period (so there are 26 2 = 204 month to go) is: $,77.40 ( (.0025 ) ) OR $286, [ ( (.0025)2 (.0025) 26 = $274, The original amortization plan shows $274,80.96 in principal yet to be paid, but that amount must be multiplied by.07: the new inflation-adjusted principal balance is $274,80.96 x.07 = $293, This loan is to be repaid over a 7-year (204 month) life at the same 3% stated annual interest rate (.25% monthly periodic rate). (We need not compute the year-4 payment, but it would be $, ) Financial calculator solution: Enter $-292,000 PV, $0 FV, 240 N, 3 2 = I/Y; Compute PMT and it shows $, Enter 2 N, Compute FV and it shows $28,78.92 in principal remaining owed at the end of month 2. Enter x.05 = and then hit the +- key to make this amount negative, and hit PV to enter this amount as the new loan balance to carry forward. Enter $0 FV, 228 N (the unchanging 3 2 = I/Y monthly periodic interest rate is already entered); Compute PMT and it shows $, Then enter 2 N, Compute FV and it shows $283,530.3 in principal remaining owed at the end of month 24. Enter x.0 = and then hit the +- key to make this amount negative, and hit PV to enter this amount as the new loan balance to carry forward. Enter $0 FV, 26 N (the unchanging 3 2 = I/Y monthly periodic interest rate is already entered); Compute PMT and it shows $, Finally enter 2 N, Compute FV and it shows $274,80.96 in principal remaining owed at the end of month 36. Enter x.07 = and then hit the +- key to make this amount negative, and hit PV to enter this amount as the new loan balance to carry forward. Trefzger/FIL 360 2

13 Then we find the rate of return built into the relevant values by solving with trial and error for r in the following equation (note that if two points are charged the borrower gets the use of only.98 x $292,000 = $286,60): $286,60 = $,69.42 ( ( +r )2 ) + $,700.40( ( +r )2 r r +r )2 + $.77.40( ( +r )2 r +r )24 + $293,373.63( +r )36. [Or we could set up the equation showing the last monthly payment grouped with the balloon: $286,60 = $,69.42 ( ( +r )2 ) + $,700.40( ( +r )2 r r +r )2 + $.77.40( ( +r ) r +r )24 + $295,09.03( +r )36. The rate that solves the equation turns out to be a monthly periodic rate of about , for an APR of x 2 = % and an EAR of ( ) 2 = 8.253%. On a financial calculator you will use the automated IRR function. Type in CF 2 nd CLR WORK, then type -$286,60 ENTER, $,69.42 ENTER 2 ENTER, $, ENTER 2 ENTER, $,77.40 ENTER ENTER, $296, (which is $, $295,09.03) ENTER, then hit IRR and CPT. The screen will go blank as the calculator does its trial and error computations; then it should show.6629 a monthly rate since the computation was based on monthly payments and time periods. You multiply that monthly rate by 2 to get the APR or compound it for 2 periods to get the EAR. 6. An elderly person borrows money against the value of her home through a reverse annuity mortgage (RAM) loan. If she can borrow at a 5.7% stated annual interest rate, and if she wants the amount of principal she owes at the end of year 4 to be $220,000, how large a payment can she receive each year (based on monthly cash flows)? Here we might think of the bank as depositing an amount with the borrower each month that will grow over fourteen years (68 months) to total, with interest, $220,000. This situation involves a future value of an annuity application, not present value of annuity as most of our loan computations are. With a 5.7% 2 = monthly periodic interest rate, we compute PMT x FAC = TOT PMT ( (.00475)68 )= $220, PMT x = $220,000 So PMT = $220, = $ OR PMT = $220,000 x = $858.74, Trefzger/FIL 360 3

14 which gives the borrower an annual total of 2 x $ = $0,305 each year for 4 years. In this case, it is as though the bank deposits $ per month with the borrower. If the bank earns.475% per month on its growing balance, then the borrower should owe a total of $220,000 at the end of year 4. On a financial calculator, enter $-220,000 FV, $0 PV, 68 N, = I/Y; Compute PMT. It should show $ We show the FV as negative because that is the amount the home owner expects to pay in 4 years. 7. Five years ago a home buyer obtained a $20,000 fixed-rate mortgage (FRM) loan, which carried a 6.75% stated annual percentage rate (APR) of interest and was to be repaid with equal end-of-month payments over 30 years. Today he can obtain a new loan with a 5.73% stated annual percentage rate (APR) of interest. If he takes out a new loan he will incur a 4%-of-principal origination cost. If he has extra cash he can invest it to earn a 5.73% annual percentage rate (APR) of return. Should he refinance under each of the following scenarios? a. If he replaces the current loan he will get a new loan for the same remaining principal amount, and the new loan will have equal monthly payments for 25 years (thus he would make level payments over the next 25 years whether he continues with the current loan or gets a new one). He would pay the 4% origination costs on the new loan out-ofpocket, rather than rolling that cost into the new loan s principal. First step: compute the current monthly loan payment, based on a 360-month amortization period and a monthly periodic interest rate of = : PMT x FAC = TOT )360 ) = $20, PMT x = $20,000 So PMT = $20, = $, OR PMT = $20,000 x = $, Second step: compute the monthly loan payment that would be made on a new loan, based on a 300- month life and a monthly periodic interest rate of =. Principal that remains unpaid 60 months into the original loan s 360-month life (i.e., with = 300 months to go) is: $, ( ( ) ) OR $20,000 [ ( ( )60 ( ) 360 = $97, The monthly payment on a new $97,38.99, 25-year, 5.73% 2 = monthly periodic rate loan is: )300 ) = $97,38.99 PMT x = $97,38.99 So PMT = $97, = $, OR PMT = $97,38.99 x = $, Trefzger/FIL 360 4

15 Third step: So by refinancing after 5 years with a new 25-year loan, the borrower would realize a reduction in his monthly payment of $, $, = $ (Answers may seem to be off by in places, because my computations generally are based on many decimal places but in the solutions I show only two. Here the difference actually is $, $, = $ ) Thus he will save $24.22 every month for 25 years (300 months); with a monthly periodic reinvestment rate of 5.73% 2 =, the present value of this savings stream (the present value of an annuity) is PMT x FAC = TOT $24.22 ( ( )300 ) = TOT $24.22 x = $9, Fourth step: The cost of going through a refinancing, which he would pay from his savings account, would be the up-front cost of getting the new loan, or.04 x $97,38.99 = $7, Final step: So we can see that refinancing has a net present value of PV Savings Stream minus PV Costs of Refinancing = $9,784.0 $7, = $, NPV is positive, so refinance. Financial calculator: Enter -$20,000 PV, $0 FV, 360 N, = I/Y; Compute PMT. It should show $, Store that amount in memory register by hitting STO. Then enter 60 N; Compute FV. It should show $97, Hit the +- key and then PV, to make this amount the principal amount for a new 25-year, 5.73% loan. Then enter $0 FV, 300 N, = I/Y; Compute PMT. It should show $, Hit the +- key, then hit the + key, RCL (to retrieve the $, from memory register ), and the = key, to subtract the $, from the $, It should show $ Hit the PMT key, then enter $0 FV, 300 N, = I/Y; Compute PV. It should show $-9, Hit the +- key and STO 2 to store that value in the second memory register. Finally, enter.04 x $97,38.99 = ; it should show $7, Hit the +- key, then hit the + key, RCL 2 (to retrieve the $9,784.0 from memory register 2), and the = key, to subtract the $7, refinancing cost from the $9,784.0 refinancing benefit. It should show the $, NPV. Note that here we have treated the expected reinvestment rate as being equal to the new mortgage lending rate, a 5.73% APR. But an argument can be made that our assumed reinvestment rate should be closer to a risk-free rate, which is always less than the mortgage lending rate. After all, switching to a loan with a lower interest rate generates an assured monthly savings stream for the borrower, so we might want to find the present value of that assured stream based on an assured rate of return. And because discounting is a form of penalizing, when we discount at a lower rate (penalizing less) we generate a higher present value. The amount we would need in savings today to allow ourselves to withdraw an assured $24.22 per month for 300 months would be a much greater $24.22 ( (.003 )300 ) = TOT.003 $24.22 x = $24, Trefzger/FIL 360 5

16 if we used a low-risk monthly reinvestment rate of 3.6% 2 =.003. With that computed present value of the savings stream, we would compute the NPV of refinancing as a higher $24, $7, = $6, It should make sense that the benefit of refinancing is higher if we think of refinancing as generating a guaranteed stream of benefits. b. He plans to sell his house in 6 more years, and thus will repay the remaining principal balance at that time regardless of whether he keeps the current loan or refinances that amount of principal with a new 25-year, 5.73% APR loan. He would pay the new loan s origination costs out of pocket, rather than rolling them into the new loan principal. The cost to refinance is still $7,885.56, but with only 6 years of lower payments (not 25) the savings from refinancing will be less. Specifically, the $24.22 monthly payment savings will be realized not for 25 x 2 = 300 months, but for only 6 x 2 = 72 months, so the present value of this savings stream (the present value of an annuity) with a 5.73% reinvestment rate is $24.22 ( ( )72 ) = TOT $24.22 x = $7, Because the present value of the savings stream is slightly less than the cost of refinancing, it might seem that there must be a negative NPV from refinancing. But there will be another effect from refinancing, in terms of principal owed after 6 years, and this effect could be a benefit or an added detriment, depending on the circumstances. In this case, we see that years (5 years already past plus six added years) = 32 months into the original 6.75% APR loan (with = 228 months remaining), he would still owe $, ( ( ) ) OR $20,000 [ ( ( )32 ( ) 360 = $74, (Because the amount owed on a loan at any time is the present value of the stream of remaining payments discounted at the original contract interest rate, we could also compute the remaining balance six more years, or 72 more months, into the 300-month remaining life of the existing loan with $97,38.99 still owed: $97,38.99 [ ( ( )72 ( ) 300 = $74, ) But if he were to refinance with a new 5.73%, 300-month loan of $97,38.99 today, with a $, monthly payment, then in 6 years the amount he would owe on that new loan (72 months into its 300- month life, thus with 228 months remaining) would be: $, ( ( )228 ) OR $97,38.99 [ ( ()72 () 300 = $7, Trefzger/FIL 360 6

17 So in 6 years he would owe $7, on the new loan, and thus would be better off if he refinances by $74, $7, = $3,00.67 in six years. That amount s present value, based on an assumed monthly periodic reinvestment rate of =.4775%, is $3,00.67 ( )72 = $2, Thus refinancing has an NPV of $7, $7, $2,36.52 = $, With a positive NPV it still makes sense to refinance, though the benefit is far lower than if the new loan were to be held to its 300 month maturity. Note that one benefit of switching to the lower-rate loan is that the remaining principal balance will amortize faster. But if there were an unattractive feature connected to refinancing, such as a prepayment penalty on the initial loan, then the NPV of refinancing might well become negative. Financial calculator: Enter $-20,000 PV, $0 FV, 360 N, = I/Y; Compute PMT. It should show $, Store that amount into memory register (STO ). Then enter 60 N; Compute FV. It should show $97, Hit the +- key and then PV, to make this amount the principal amount for a new 25-year, 5.73% loan. Then enter $0 FV, 300 N, = I/Y; Compute PMT. It should show $,237.83; store this value in memory register 2 (STO 2). Hit the +- key, then the + key, RCL (to retrieve the value stored in memory register ), and the = key, to subtract the $, from the $, It should show $ Hit the PMT key, then enter $0 FV, 72 N, = I/Y; Compute PV. It should show $-7,553.54; hit the +- key and store this present value of the savings stream in memory register 3 (STO 3). Hit RCL to get the $, and hit PMT, $O FV, 228 N, = I/Y; Compute PV. It should show $-74, ; hit the +- key and STO 4 to store in memory register 4. Then RCL 2 to get $, and hit PMT, $O FV, 228 N, = I/Y; Compute PV. It should show $-7, ; hit the + key, RCL 4 (to get the $74, from memory register 4), and the = key; it should show $3,00.67 as the decrease in the year-6 principal owed if the new loan is obtained. Hit the FV key; then enter $0 PMT, 72 N, = I/Y; Compute PV. It should show -$2, Hit the +- key and the + key, RCL 3 (retrieving the $7, refinancing benefit stored in memory register 3), and the = key. It should show $9,690.06; subtract the $7, direct refinancing cost to get the $, NPV. What if we used a low-risk discount rate of 3.6% per year (.3% per month) instead of the new 5.73% mortgage lending rate? Then the NPV of refinancing would increase, because an assured stream of benefits (the savings generated by reduced payments) is worth more than a non-guaranteed stream. The $24.22 monthly payment savings for 72 months has a present value of $24.22 ( (.003 )72 ) = TOT.003 $24.22 x = $8, Then the reduction in principal owed in six years ($74, $7, = $3,00.67) has a present value, based on a.3% monthly discount rate, of $3,00.67 (.003 )72 = $2, Trefzger/FIL 360 7

18 Thus refinancing has a higher NPV, if we think of the borrower s benefits from refinancing as being assured and thus appropriate to discount at a low-risk monthly rate, of $8, $7, $2, = $2, c. If he refinances he will do a cash out by borrowing $260,000 under a new 25-year loan (here we assume that the house has increased substantially in value, to support a $260,000 loan), and will make the payments for the entire 25 remaining years. Costs for originating the new loan would be paid out-of-pocket. In this situation the payments actually go up instead of going down; the interest rate is lower but the principal is much greater. The monthly payment on a new $260,000, 25-year, 5.73% 2 = monthly periodic rate loan is: )300 ) = $260,000 PMT x = $260,000 So PMT = $260, = $, OR PMT = $260,000 x = $ So here the monthly savings in the payment stream is actually a negative amount: $, $, = $ per month for 25 years or 300 months, with a total present value (based on the 5.73% APR = =.4775% monthly periodic reinvestment or opportunity rate) of $ ( ( )300 ) = $43, He will also have immediate out-of-pocket refinancing costs, computed here as.04 x $260,000 = $0, So his total costs, in present value terms, are $43, $0, = $53, But he also walks away with $260,000 $97,38.99 = $62,86.0 in his pocket. The NPV of refinancing is PV of benefits $62,86.0 $43, $0, PV of costs = $9, It appears again that he should refinance; the benefit of the lower interest rate (and the receipt today of a large cash inflow) outweigh the cost of the higher payments. Financial calculator: Enter $-20,000 PV, $0 FV, 360 N, = I/Y; Compute PMT. It should show $,362.05; store that amount in memory register (STO ). Then enter 60 N; Compute FV; it should show $97, Store that amount in memory register 2 (STO 2). Then enter $-260,000 PV, $0 FV, 300 N, = I/Y; Compute PMT. It should show $, Hit the +- key and then the + key, RCL (to recall the $, from memory register ), and the = key; it should show the -$ expected change (a negative savings) in the monthly payment. Hit the +- key and the PMT key, then $0 FV, 300 N, = I/Y; Compute PV. It should show -$43,077.00; hit the +- key and store in memory register 3 (STO 3). Then compute.04 x $260,000 and hit the = key. It should show $0,400; hit + RCL 3 = to add the $0,400 to the $43,077 in memory register 3. Store the resulting $53, in memory register 4 by hitting STO 4. Finally, compute $260,000 RCL 2; it should show $62,86.0. Then hit the - key and RCL 4 key and the = key; it should show $9, Trefzger/FIL 360 8

19 But what if we were to use the.3% per month discount rate, rather than, on the logic that the higher payments to be made each month are an assured obligation, such that an account sufficient to provide for the higher payment stream would need to have a high initial balance: $ ( (.003 )300 ) = $53, With the $0,400 out-of-pocket refinancing costs his total refinancing costs, in present value terms, are $53, $0, = $63, He walks away with $62,86.0 in his pocket, but the NPV of refinancing is now a negative: PV of benefits $62,86.0 $53, $0, PV of costs = -$ So in this situation, with refinancing requiring higher monthly payments, the NPV of refinancing is negative if we use the low-risk discount rate. The benefit of the large inflow of cash at the closing is less than the new loan s closing costs plus the high negative present value of the obligation to make higher monthly payments for the subsequent 25 years. A few points of caution are in order. First, I selected a pretty high replacement loan principal to get the computed NPV to be negative (if the replacement loan is only $225,000 the NPV computed with a.3% monthly rate is $8,838.27; if the replacement loan is a larger $239,000 the computed NPV is $4,905.68; and even with $252,000 borrowed the NPV is slightly positive at $,253.99). To get the NPV negative a high principal is needed, because only a sizable difference in monthly payments will generate a big enough negative PV to overwhelm the benefit of the extra cash received at closing. Second, it probably is unrealistic to assume that the origination costs of the new loan would be 4% of principal regardless of the loan s size. While some costs would vary directly, if not fully proportionally, with the amount borrowed (application fee, which is often % of the amount borrowed; title insurance premium), many of the costs would be fixed or at least have fixed components (income and credit checks, appraisal, document recording fees). [The closing costs can vary considerably from state to state, based on state government taxes and fees.] Note that for a new-loan origination cost of less than $9, the NPV of refinancing is positive. d. If he refinances he will get a 30-year replacement loan for the amount of principal still owed, and will make the payments over 30 years. Closing costs of the new loan would be paid out-of-pocket. Recall that the principal remaining unpaid 60 months into the original loan s 360-month life is $97, Paying that principal balance over a new 30-year amortization period, with a 5.73% 2 =.4775% monthly periodic interest rate, would require a monthly payment of PMT x FAC = TOT )360 ) = $97,38.99 PMT x = $97,38.99 So PMT = $97, = $ OR PMT = $97,38.99 x = $,47.95 Trefzger/FIL 360 9

20 (less than the $, that would be paid each month if payments involving the same principal were spread over only 25 years). So the borrower will realize a savings of $, $,47.95 = $24. per month for the 300 months that remain on the original loan s amortization. The present value of that savings stream, if discounted based on the current monthly mortgage lending rate of.4775%, is $24. ( ( )300 ) = TOT $24. x = $34, However, he will then have to make 60 additional $,47.95 monthly payments (months ), whereas no payments during those months would be required if he kept the original loan in place. The (negative) present value of this 60-month deferred annuity obligation (deferred for 300 months) is $,47.95 ( ( )60 )300 = TOT $,47.95 x ( )( ) = TOT $59, ( ) = TOT = $4, (Note that $59, is the amount that will still be owed on the new loan at the start of month 30; the present value of that obligation today, 25 years before the fact, is $4,35.43.) Here we again assume that the origination costs on the replacement loan would be $97,38.99 x.04 = $7, Thus the NPV of refinancing is PV of benefits minus PV of costs = $34, $7, $4,35.43 = $, The NPV is positive, so once again the borrower should refinance. On a financial calculator, enter -$20,000 PV, $0 FV, 360 N, = I/Y; Compute PMT and it shows $, Hit STO to store this amount in memory register ; then enter 60 N, Compute FV and it shows $97, Then hit the +- key and PV, $0 FV, 360 N, = I/Y; Compute PMT to get $, Hit STO 2 to store in memory register 2. Then hit the +- key followed by the + key, RCL (to recall the $, stored in memory register ), and the = key; it should show $24. as the monthly payment savings. Hit the PMT key; then enter 300 N and Compute PV (the $0 FV and = I/Y should still be intact from when you entered them earlier); it shows -$34, Hit the +- key and STO 3 (to enter that value into memory register 3). Then enter RCL 2 (shows $,47.95) PMT, $0 FV, 60 N, = I/Y; Compute PV. It should show $-59, Hit the +- key and the FV key. Then enter $0 PMT, 300 N, = I/Y; Compute PV. It should show -$4, Hit the + key, RCL 3 (to recall the $34, in memory 3), and the = key; then enter - $7, followed by the = key and it should show the $, NPV of refinancing). If we used the low-risk monthly discount rate of 3.6% 2 =.3% or.003, the present value of the savings stream would be $24. ( (.003 )300 ) = TOT.003 $24. x = $42, Trefzger/FIL

21 But then also discounting back the obligation to make the added five years worth of payments (or just pay the lump sum still owed in 300 months) at the.3% low-risk discount rate yields $,47.95 ( ( ) )300 = TOT $,47.95 x ( )(.4078) = TOT $59, (.4078) = TOT = $24, (Note that $59, is the amount that will still be owed on the new loan at the start of month 30. Because the amount owed on a loan at any time is the PV of the remaining payment stream, the negative present value to the borrower today is the same regardless of whether the $59, will paid back through payments in months , or will be paid back all at once at the end of month 300. The present value of that obligation today, 25 years before the fact, based on the low-risk discount rate, is $24,33.60.) Here we again assume that the origination costs on the replacement loan would be $97,38.99 x.04 = $7, Thus the NPV of refinancing is PV of benefits minus PV of costs = $42,34.00 $7, $24,33.60 = $0, The NPV is positive, so once again the borrower should refinance. But it is less positive than the $, computed above with the higher discount rate. Using the lower discount rate makes both the present value of the savings stream from lower payments and the present value of the added payment obligation higher, but the PV of the savings stream increases by only 24% ($42,34.00 $34, =.2409), while the PV of the added payment stream increases by almost 70% ($24,33.60 $4,35.43 =.6997). e. If he refinances he will get a 30-year replacement loan that allows him to repay the principal still owed on the original loan, and will make the payments over 30 years, but he would roll the cost of refinancing into the principal borrowed under the new loan. Here again the remaining unpaid principal on the original loan is $97, But now the borrower wants to borrow, rather than pay out of pocket, the amount needed to pay closing costs on the new loan. It may not be intuitively clear initially, but the borrower must actually get a replacement loan that is bigger than.04 times the existing loan s remaining balance if he wants to incorporate the 4% origination fee into the new loan s principal. We do not multiply $97,38.99 by.04, but rather divide it by.96, for a slightly higher grossed up outcome: the new loan should be for $97, = $205,353.. (Then 4% of the grossed-up $205,353. is $8,24.2; the lender nominally lends $205,353., but then takes away $8,24.2 to cover closing costs, leaving $205,353. $8,24.2 = $97,38.99 for the borrower to retire the original loan.) Paying back $205,353. over a new 30-year amortization period, with a 5.73% 2 =.4775% monthly periodic interest rate, would require a monthly payment of PMT x FAC = TOT )360 ) = $205,353. PMT x = $205,353. Trefzger/FIL 360 2