Student loans: subsidized, unsubsidized, repayment plans.
Federal student loans amortize the same way as auto loans, with two twists: interest can accrue during school (unsubsidized) or not (subsidized), and there are multiple repayment plans you can pick at payback.
Two students take out the same $10,000 federal loan, freshman year. They graduate at the same time, take the same job, borrow at the same 6.5% rate. One student's monthly payment is $33 higher than the other's, every month, for ten years.
Why? One borrowed subsidized and one borrowed unsubsidized. The four years before graduation aren't free for the unsub borrower — interest is quietly accruing the whole time, then capitalizes at the start of repayment, rolling itself into a bigger principal. Same formula in L2; just a bigger P feeding into it. This lesson shows the math that produces the gap.
Same amortization formula. Different starting principal.
Same $10,000, same 6.5% rate, four years in school. The subsidized loan starts repayment at $10,000 (left bar). The unsubsidized loan starts at $12,600 (right bar) — the original $10,000 plus $2,600 of accrued simple interest. Same r and t in the amortization formula, but a bigger P → a bigger M. (Real federal loans actually accrue daily, which behaves like compound interest — see the “From the real world” sidebar.)
Subsidized loan: the federal government pays interest during in-school years. Unsubsidized loan: interest accrues and capitalizes (gets added to principal) at repayment. ALEKS uses the simple-interest formula I = Prt for the in-school accrual, where t is the in-school years; the new principal at repayment is P + I, and that becomes the P inside the amortization formula.
Vocabulary you'll see in word problems
- Subsidized loan Federal loan where the government pays interest during in-school years and deferment. Only available to students with demonstrated financial need.
- Unsubsidized loan Federal loan where interest accrues from disbursement. Available regardless of need.
- Capitalization Accrued in-school interest gets added to the principal at the start of repayment, increasing the amount you'll pay back. ALEKS uses simple interest (I = Prt) for the in-school accrual; real federal loans accrue daily.
- Deferment A pause in repayment (e.g. while in school). Subsidized loans don't accrue interest during deferment; unsubsidized loans do.
- Standard vs Income-Driven Repayment Standard: 10-year fixed term, computed by the L2 formula. IBR (income-driven): monthly payment is a percent of discretionary income, term up to 25 years, with possible forgiveness of remaining balance.
$10,000 loan, sub vs unsub, head-to-head.
"A student takes out a $10,000 federal loan freshman year at 6.5% APR. They're in school for 4 years, then begin a 10-year standard repayment plan. Compare the monthly payment, total paid, and total interest if the loan is subsidized versus unsubsidized."
Subsidized: in-school years are free.
The government covers interest while the student is enrolled. P at repayment = original loan amount. No T3-style compounding needed.
Unsubsidized: accrue simple interest.
Interest accrues for 4 years on the $10,000. ALEKS uses the simple-interest formula I = Prt; the result is added to the original principal to get P at repayment.
Apply the L2 amortization formula to each P.
r = 0.065, t = 10, so r/12 ≈ 0.005417 and n = 120. Same r, same t for both — only P differs.
Total paid + total interest, side by side.
Sum of 120 monthly payments. Subtract the original $10,000 to get total interest (both students borrowed the same amount; the extra is all interest).
The lifetime cost of "unsubsidized."
Subtract sub from unsub. This is what in-school interest accrual costs over the full life of the loan.
Three problems. Capitalization, monthly, comparison.
An unsubsidized $5,000 loan at 5% APR sits for 3 years in school accruing simple interest. What's the principal at the start of repayment?
Continuing problem 1: that $5,750 repayment principal goes into a 10-year standard repayment at 5% APR. What's the monthly payment?
From problem 2: the unsub student pays $60.99/mo; the sub student would pay $53.03/mo. Both for 120 months. How much more does unsubsidized cost over the full life of the loan?
Three fast questions before you move on.
Q1. For a subsidized federal loan, the principal P used in the amortization formula at repayment is...
Q2. For ALEKS-style problems, in-school accrual on an unsubsidized loan is computed using...
Q3. Two loans, same rate, same 10-year term. Loan A has P = $10,000; Loan B has P = $12,000. Which has the bigger M?
Subsidized always pays less.
One sharp insight worth internalizing: at any rate and term, a subsidized loan's monthly payment is lower than the equivalent unsubsidized loan's monthly payment, because the starting principal at repayment is smaller. The math is the L2 formula's monotonicity in P: same r, same t, smaller P → smaller M. T3 DQ2 asked you to explain this in writing; this lesson is the math behind that explanation.
Next: Mortgages — same amortization formula at scale, plus the principal/interest split per payment that explains why early mortgage payments feel like "all interest, no equity."
Continue to Lesson 06Different angle? Need another rep? These are optional — tap any that look helpful.
The Best Way to Apply for Student Aid!
Covers FAFSA, subsidized vs. unsubsidized loans, grants, and the federal-aid landscape. Note: 2020 release — use for the concept of sub/unsub + capitalization; the repayment-plan landscape has changed since (see next video for current detail).
Subsidized vs. Unsubsidized Student Loans: Understanding the Difference
2025 release. Outside the preferred-channel list but currently the cleanest concept-only treatment of the sub-vs-unsub distinction that survives recent policy changes. Pairs with studentaid.gov for the official documentation.