# Student Loans. The Math of Student Loans. Because of student loan debt 11/13/2014

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1 Student Loans The Math of Student Loans Alice Seneres Rutgers University 1 71% of students take out student loans for their undergraduate degree A typical student in the class of 2013 graduated with \$32,000 in student loan debt 2 Student loans are a financial commitment that span decades. Many students take out loans in their late teens and early twenties, unaware of the long-term impact of the loans. After realizing the extent of their debt, 39% said they would have done things differently. started saving earlier thoroughly researched financial aid looked for ways to save more spent less while at school 3 4 Because of student loan debt Young adults have decided: against pursing graduate school delayed getting married delayed having children to take a job they ordinarily would not have There is currently a trillion dollars out in student loan debt. Fetterman, M. & Hansen, B. (2006, November 20). Young people struggle to deal with kiss of debt. USA Today

2 Meet Hypothetical Student! Hypothetical Student Takes out a \$7,000 loan for the first year of college. Payments begin after graduation. How much is owed when repayment starts? 7 8 Two Types of Student Loans Subsidized Loans: The government pays interest that accrues until repayment begins Must demonstrate financial need Must fill out FAFSA form There are borrowing limits (\$3,500 to \$5,500) Unsubsidized Loans The loan is accruing interest while the student is in school. Assume it s 6.8% compounded annually. Every year the interest is added to the balance of the loan This new balance then accrues interest During the first year of the loan I PRT I: interest accrued (what we want to find) P: principal (\$7,000) R: rate (6.8%) T: time (1 year) During the first year of the loan I (7,000)(0.068)(1) I 476 After the first year, the balance is \$7,000 + \$476 \$7,476 Or in one step: Balance \$7000( ) \$7,476 (CCSS 7.RP.3) 11 But there are three more years to go! 2

3 During the second year of the loan What variables have a different value? During the second year of the loan What variable has a different value? I PRT I: interest P: principal R: rate T: time I PRT I: interest P: principal (\$7,476) R: rate (6.8%) T: time (1 year) During the second year of the loan I PRT I (\$7,476)(0.068)(1) I \$ More interest is being charged during the second year. The interest is accruing interest. Balance is \$7,476 + \$ \$7, Balance End of Year 1 \$476 \$7, (1.068) End of Year 2 \$ \$7, (1.068) End of Year 3 End of Year 4 Or in one step: (\$7476)( ) \$7, Balance End of Year 1 \$476 \$7, (1.068) End of Year 2 \$ \$7, (1.068) (7000)(1.068)(1.068) End of Year 3 End of Year 4 Balance End of Year 1 \$476 \$7, (1.068) End of Year 2 \$ \$7, (1.068) (7000)(1.068)(1.068) (7000)(1.068) 2 End of Year 3 End of Year

4 Balance End of Year 1 \$476 \$7, (1.068) End of Year 2 \$ \$7, (1.068) (7000)(1.068)(1.068) (7000)(1.068) 2 End of Year 3 (7000)(1.068) 3 End of Year 4 (7000)(1.068) 4 19 Balance End of Year 1 \$476 \$7, (1.068) End of Year 2 \$ \$7, (1.068) (7000)(1.068)(1.068) (7000)(1.068) 2 End of Year 3 \$ \$8, (7000)(1.068) 3 End of Year 4 \$ \$9, (7000)(1.068) 4 20 Compound interest FV PV(1 + n FV: Future Value PV: Present Value (\$7,000) i: interest rate per time period (0.068) n: number of time periods (4) FV 7000( ) 4 FV \$9, Geometric Sequence: Each term in the sequence is obtained by multiplying the preceding term by a constant (a common ratio). Common ratio is Sequence: 7000, 7000(1.068), 7000(1.068) 2, 7000(1.068) 3,, 7000(1.068) 4 (CCSS F.IF. 1, 2, 3) (CCSS F.LE.1 and F.IF.8b) Hypothetical Student s Loan Loan student took out: \$7,000 Amount due four years later: \$9, Grace Period Payments do not start until six months after graduation. Wait! There s more!

5 Grace Period Payments do not start until six months after graduation. But interest is accruing during this time! Tack on another half a year to that compound interest calculation. FV 7000( ) 4.5 \$ College is Four Years Assume a \$7,000 loan taken out every year Year 1: FV 7000( ) 4.5 \$9, Year 2: FV 7000( ) 3.5 \$8,8.47 Year 3: FV 7000( ) 2.5 \$8, Year 4: FV 7000( ) 1.5 \$7, Hypothetical Student s Loans Loan student took out: \$28,000 Amount due when repayment begins: \$34, (\$9, \$8, \$8, \$7,726) Now Repayment Begins Hypothetical Student will pay a set every month for 15 years. This is a present value annuity : money was provided up front and is paid back in equal payments made at regular time intervals. How much will the student pay every month? Now Repayment Begins The student will make 180 payments ( payments/year)(15 years) 180 The due is \$34, is Accruing The loan is accruing interest during repayment During repayment assume interest is 6.8% compounded monthly. Why isn t the monthly payment determined this way: \$34,201.55/

6 PV PMTa n i PV: Present Value (the balance of the loan when repayment begins) \$34, PMT: The of each payment a n i : The present value annuity factor a i n 1 a n i n PV PMTa n i 34, PMT( ) PMT So Hypothetical Student will pay \$ a month for fifteen years. (Derivation provided at the end of handout.) How Much Will be Paid? Loan student took out: \$28,000 How Much Will be Paid? Loan student took out: \$28,000 How much will the student pay in interest? Total student will repay: (\$303.60)()(15) 54,648 Student will repay \$54, How Much Will be Paid? Loan student took out: \$28,000 Total student will repay: (\$303.60)()(15) 54,648 Where Do Monthly Payments Go? The student is writing a check every month for \$ What is happening to that money? Amortization table Student will repay \$54,648. Student will pay \$26,648 in interest on her \$28,000 loan

7 Payment number Payment 1 \$ \$ Amortization table Principal Remaining balance Where Do Monthly Payments Go? is paid off as it accumulates It isn t being carried month to month for the first month is I PRT I ( )(0.068)(1/) I \$ \$ The rest of the \$ payment goes towards principal 38 Amortization table Amortization table Payment Principal 1 \$ \$ \$ \$ Remaining balance \$34, Payment Principal 1 \$ \$ \$ \$ \$ \$ Remaining balance \$34, \$33, for the second month is I PRT I ( )(0.068)(1/) I \$ Amortization table The payment is the same each time, but: going toward interest decreases going toward principal increases Amortization Table Feel like doing all 180 rows by hand? This is where a spreadsheet is handy!

8 What if Hypothetical Student paid an extra \$100 a month? What if Hypothetical Student paid an extra \$100 a month? What if Hypothetical Student paid an extra \$100 a month? Pay off the loan in 9 years 8 months Shaved more than 5 years off the payments Only repay \$46, rather than \$54,648 Save \$7, in interest Write apply extra to principal on check Thank you! Questions? Alice Seneres

9 Tips for a great conference! Rate this presentation on the conference app Download available presentation handouts from the Online Planner! Join the conversation! Tweet us using the hashtag #NCTMRichmond Links money.cnn.com/interactive/pf/college/studen t-debt-map/?iidel 50 Future value annuity Deriving the present value annuity formula Let s put the student s loan aside I deposit \$1 at the end of each year for four years into an account that earns 5% interest compounded annually. This is a future value annuity. How many years would the first dollar be accruing interest? Deposit from Year Deposit Years earning interest 1 \$1 3 (deposit at end of year) Future value \$1.16 1(1.05) 3 What are the mathematical expressions for Deposit from Year Deposit Years earning interest 1 \$1 3 (deposit at end of year) Future value \$1.16 1(1.05) 3 2 \$1 2 \$1.10 1(1.05) 2 3 \$1 1 \$1.05 1(1.05) 1 4 \$1 0 \$1 How do we find total? Add the future values. the other deposits?

10 Future Value Annuity Future Value Annuity FV (1)(1.05) 3 + (1)(1.05) 2 + (1)(1.05) FV \$4.31 is what we have saved after four years. Won t always be investing for 4 years Won t always be investing at 5% interest. We need a flexible formula Future value annuity The previous specific situation for how \$1 behaved: FV (1)(1.05) 3 + (1)(1.05) 2 + (1)(1.05) A flexible formula for how \$1 behaves: s n i (1)(1 + n-1 + (1)(1 + n-2 + +(1)( (1)( Original equation: s n i (1)(1 + n-1 + (1)(1 + n-2 + +(1)( (1)( Multiply both sides by (1 + : (1 + s n i (1)(1 + n + (1)(1 + n-1 + +(1)( (1)(1+ Now subtract original equation (1 + s n i (1)(1 + n + (1)(1 + n-1 + +(1)( (1)(1+ -s n i -(1)(1 + n-1 - (1)(1 + n (1)( (1)( s n i is the future value annuity factor i is interest rate, n is number of payments Need it in a more compact form Original equation: s n i (1)(1 + n-1 + (1)(1 + n-2 + +(1)( (1)( Multiply both sides by (1 + : (1 + s n i (1)(1 + n + (1)(1 + n-1 + +(1)( (1)( Now subtract original equation (1 + s n i (1)(1 + n + (1)(1 + n-1 + +(1)( (1)(1+ -s n i -(1)(1 + n-1 - (1)(1 + n (1)( (1)( is n i (1 + n - 1 s n i (1 + n - 1 i 59 Future value annuity Why is this so powerful? I can calculate the annuity factor no matter how many payments or the interest rate s n i (1 + n - 1 i Then I can use it to scale it up to the dollar I m investing 60 10

11 Future value annuity FV PMTs n i FV: Future Value (the of money saved at the end of the annuity) PMT: The of each payment s n i : Future value annuity factor But Hypothetical Student has the opposite situation. The student borrowed a certain of money that needs to be paid off. This is a present value annuity How can we calculate the student s monthly payment? Let s Pretend The student won t make recurring payments to the loan company. The student will pay off the entire loan in one lump sum at the end of 15 years. To save up for this, each month the student puts money into a bank account. At the end of 15 years, the balance of this bank account will pay off the loan. This would never happen, but if it did Calculating Monthly Payments How much student will owe in 15 years? Compound interest problem Calculate FV FV PV(1 + n Calculating Monthly Payments Will pay off this loan in one lump sum Student sets up a savings account Account pays 6.8% interest compounded monthly Student will make deposits every month for 15 years into the account What should the monthly deposit be? Calculating Monthly Payments FV PMTs n i FV: Future value, needed to pay off loan PMT: Monthly payment (need to find) s n i : future value annuity factor (can calculate)

12 Calculating Monthly Payments We know: FV PMTs n i FV PV(1 + n Let s set them equal to each other: PV(1 + n PMTs n i PV n PMTs PV PV PMT PMTs ni n s ni n ni where a ni sni 1 ( + n PV PMTa ni PV PMTa n i PV: Present Value (the balance of the loan when repayment begins) \$34, PMT: The of each payment a n i : The present value annuity factor Now we can calculate the monthly payment! First calculate s n i Then calculate a n i n s 1 ni sn i ani n s s i a a PV PMTa n i 34, PMT( ) PMT So Hypothetical Student will pay \$ a month for fifteen years. 71

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