## Tuesday, May 21, 2019

### Delayed Interest Rate Concessions on 20-Year Student Loans

There has been a substantial increase in the number of elderly people with a student loan.  This post uses Excel finance functions to evaluate a student loan, where interest rates are cut to 0 after 12 years of payments.

Question:  Consider a student with a 20-year \$30,000 student loan at 4.0% interest.

What is the monthly payment on the student loan?

What are lifetime interest payments on this loan?

The government decides to lower the interest rate to 0 percent after 12 years.  The loan contract requires the borrow to continue with initial payments on the loan after the interest rate reduction.

What are the lifetime interest payments on the modified loan?

How many months does it take for the borrower to completely repay the loan under the new loan contract?

Analysis:

Discussion of traditional 20-year student loan:   The payment on the 20-year loan is obtained from the payment function in Excel.

\$181.79 = PMT(0.04/12,240,-30000)

The cumulative interest payments are the sum of all monthly payments minus the initial loan balance of \$30,000.   This figure is \$13,360.58.

This figure can be obtained directly from the CUMIPMT function.

-13630.58=CUMIPMT(0.04/12,240,30000,1,240,0)

Discussion of the modified student loan contract:

The amount of interest on the student loan when the interest rate is set to 0 percent after twelve years can be calculated directly from the CUMIPMT function by recognizing that all interest payments cease after 144 moths.

-11092.62=CUMIPMT(0.04/12,240,30000,1,144,0)

This is the total interest on the loan since no interest is charged after year 12.

The balance on the student loan at the end of 144 months can be calculated through the FV function.

\$14,914.27 =FV(0.04/12,144,181.79,-30000)

At a 0 percent interest rate, the entire payment goes towards principal.   The number of additional months after 12 years where payment ends is 82.04 (14,917.62/181.79).

The loan with the interest concession after 12 years is repaid in 226 payments.   (144+82).   I added \$7.15 to the final payment to account for the fraction over 82 months.

The annual rate of return can be obtained by taking the IRR of 227 payments (the initial loan balance of -30,000 followed by the 226 (144+82.04) monthly payments.) The monthly IRR from these cash flow streams is 0.294%

Multiply by 12 to get the annual interest rate on the loan with the interest concession after 12 year to get 3.53 %

Summary of Results:   Reducing interest rates on this 20 percent student loan to 0 after 12 years will reduce the average annual interest rate received by investors from 4.0 percent to 3.53 percent.   The interest reduction will also reduce the maturity of the loan from 20 years to 18.84 years.

Policy Discussions:  The figures presented in this post assume all payments are made as scheduled.   It is possible that some people will stop payments once interest rates are set to 0 percent because they would prefer to pay higher interest rate loans.   This could be deterred by reporting these people to credit bureaus.   It is also possible that some people who are having trouble making payments will simply give up if they do not receive some debt relief.

Lenders tend to oppose any form of loan forgiveness.   However, I believe that lenders would prefer this type of interest concession to Income Based Replacement loan programs where debt is actually discharged.

Authors Notes:   Please consider my book

Innovative Solutions to the College Debt Problem:
https://www.amazon.com/Innovative-Solutions-College-Debt-Problem/dp/1982999446