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2023/9/3 11:46 Corporate Finance 6E, Fall 1 2023-Yueheng Wang Student: Yueheng Wang Course: Corporate Finance 6E, Fall 1 2023 Instructor: Ko Wang Book: Berk/DeMarzo: Corporate Finance, 6e Date: 9/3/23 Time: 11:46 AM Your friend tells you he has a very simple trick for taking one-third of the time it takes to repay your mortgage: Use your holiday bonus to make an extra payment on January 1 of each year (that is, pay your monthly payment due on that day twice). Assume that the mortgage has an original balance of $100,000, has an original term of 30 years, and has an APR of 12.1%. a. If you take out your mortgage on January 1 (so that your first payment is due on February 1), and you make your first extra payment at the end of the first year, in what year will you finish repaying your mortgage? b. If you take out your mortgage on July 1 (so that the first payment is on August 1), and you make the extra payment each January, in how many months will you pay off your mortgage? c. How will the amount of time it takes to pay off the loan given this strategy vary with the interest rate on the loan? Note: Make sure to round all intermediate calculations to at least 6 decimal places. a. If you take out your mortgage on January 1 (so that your first payment is due on February 1), and you make your first extra payment at the end of the first year, in what year will you finish repaying your mortgage? Here is the cash flow timeline for this problem: Periods 0 1 2 360 Cash Flows $100,000 -C -C C For the original mortgage, we are paying interest every month at a monthly interest rate of 12.1%/12 = 1.0083%. Therefore, the monthly loan payment can be calculated using the following formula: PV 1 1 _— 1_ r [ <1+r>"] where C is the monthly payment, PV is the loan amount, r is the monthly discount rate, and n is the number of compounding periods per year. Therefore, $100,000 C= =$1,036.29 1 1 —_— 1 - 0.010083 [ (1+ 0.010083)360 ] The monthly loan payment is $1,036.29. Using a financial calculator or Excel: N Y PV PMT FV Given: 360 1.0083 100,000 0 Solve for: -1,036.29 Excel Formula: = PMT(RATE,NPER,PV,FV) = PV(0.010083,360,100000,0) Here is the cash flow timeline with the original monthly payments and extra payments: Periods 0 1 12 13 24 25 n Cash Flows$100,000 $1,036.29 -$1,036.29%$1,036.29 -$1,036.29%$1,036.29 -$1,036.29 -$1,036.29 -$1,036.29 Next, let's find the equivalent one time annual payment to these cash flows (as though we only made a single payment each January). The future value of the above cash flows is the future value of the monthly annuity plus the future value of the extra payment: (1+n7-1 FV=Cx f +C Therefore, (1+0.010083)12 - 1 0.010083 FVy yr = $1,036.20 [ ] +$1,036.29=$14,185.11 So, the new payment plan is equivalent to paying $14,185.11 at the end of every year. Then, to compute the number of years to pay off the loan, we set the present value of the loan payments equal to the original balance and solve for n using the following formula: https://xlitemprod.pearsoncmg.com/apifv1/print/en-us/highered 1/3
2023/9/3 11:46 Corporate Finance 6E, Fall 1 2023-Yueheng Wang lwa] PV=—x|1- r (140" where PV is the original loan amount, C is the annual payment, r is the annual interest rate, and n is the number of years. The equivalent annual interest rate is r= (1 + 0.010083)12 -1=0.127937 = 12.7937%. Therefore, $100.000 $14.185.11 [ 1 L ST o x - 0.127937 (1+0.127937)" 1 $100,000 x 0.127937 (1+0.127937)" $14,185.11 (1+0.127937)" =10.194774 log(10.194774) n= =19.29 log(1 + 0.127937) The approximate number of years to pay off the loan is 19.29 years. Using a financial calculator or Excel: N Y PV PMT FV Given: 12.7937 100,000 -14,185.11 0 Solve for: 19.29 Excel Formula: = NPER(RATE,PMT,PV,FV)=NPER(0.127937, - 14185.11,100000,0) It will take about 19 years to pay off the loan, which is close to 2/3 of its life of 30 years, and your friend is right. b. If you take out your mortgage on July 1 (so that the first payment is on August 1), and you make the extra payment each January, in how many months will you pay off your mortgage? Here is the cash flow timeline with the original monthly payments and extra payments: Periods 0 1 6 7 18 19 n Cash Flows$100,000 $1,036.29 -$1,036.29%$1,036.29 - $1,036.29%$1,036.29 -$1,036.29 -$1,036.29 -$1,036.29 Let's find the equivalent one time annual payment to these cash flows (as though we only made a single payment each July). The future value of the above cash flows is the future value of the monthly annuity plus the future value of the extra payment: (1+n"-1 FV=C><[ +Cx(1+nt where t is the number of periods the extra payment is compounded. Therefore, (1+0.010083)12 - 1 0.010083 FVy yr =$1,036.29 [ ] +$1,036.29 x (1 +0.010083)° = $14,249.41 So, the new payment plan is equivalent to paying $14,249.41 at the end of every year. Then, to compute the number of years to pay off the loan, we set the present value of the loan payments equal to the original balance and solve for n using the following formula: $100.000 $14,249.41 1 1 e —— x - 0.127937 (1+0.127937)" 1 __ $100,000x0.127937 (1+0.127937)" $14,249.41 (1+0.127937)" = 9.788632 log(9.788632) "= Tog(1+0.127937) _ The approximate number of years to pay off the loan is 18.95 years. Using a financial calculator or Excel: https://xlitemprod.pearsoncmg.com/apifv1/print/en-us/highered 2/3
2023/9/3 11:46 Corporate Finance 6E, Fall 1 2023-Yueheng Wang N Iy PV PMT FV Given: 12.7937 100,000 - 14,249.41 0 Solve for: 18.95 Excel Formula: = NPER(RATE,PMT,PV,FV)=NPER(0.127937, 14249.41,100000,0) It will take about 19 years to pay off the loan, which is close to 2/3 of its life of 30 years, and your friend is right. c. How will the amount of time it takes to pay off the loan given this strategy vary with the interest rate on the loan? This strategy will reduce the amount of time to pay off the loan by a greater amount as the interest rate increases. https://xlitemprod.pearsoncmg.com/apifv1/print/en-us/highered 3/3
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