1. Student Loans Modeling Student Loans Conclusion
American Student Loans
Paolo Guasoni1,2
Yu-Jui Huang3
Dublin City University1
Università di Bologna2
University of Colorado at Boulder3
SIAM Conference on Financial Mathematics and Engineering
June 6th
, 2023

2. Student Loans Modeling Student Loans Conclusion
Outline
• Student Loans’ Features.
• Income-Driven Repayments and Forgiveness.
• Optimal Repayment and Valuation.
• Modeling Non-capitalized interest.

3. Student Loans Modeling Student Loans Conclusion
Largest Debt Type after Mortgages ($ Trillions)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022
Student Loans Car Loans Credit Cards

4. Student Loans Modeling Student Loans Conclusion
Federal Student Loans
• Borrow for tuition and living expenses while in college.
• Six months after graduation, repayments begin.
• By default, 10-year loans with fixed interest rate.
• Loans taken in different years may have different rates.
• Rates decided by Federal Government annually.
• Three types with different annual rates and borrowing limits.
Undergraduate 4.99% $57,500
Graduate or Professional 6.54% $138,500
PLUS 7.54% +∞
• Early repayments can be made without penalty, as for US mortgages.
• Pandemic pause with 0% rate.

5. Student Loans Modeling Student Loans Conclusion
Income-Driven Repayment and Forgiveness
• Enrolling in an income-driven repayment schemes triggers several
features.
• Monthly payments capped at 10% of disposable income above 150% of
poverty level. (No payments if income is lower.)
• Loans are forgiven after 20-25 years...
• ...but forgiven amount is taxable. Forgiven only in part.
(Forgiveness tax-free through 2025, but very few loans are affected.)
(Exception: PSLF forgives loans after 10 years, and no tax is due.)
• To enroll or not to enroll? When to enroll?
• High stakes in some sectors.
Median student loan balance for dental school graduates: $292,000

6. Student Loans Modeling Student Loans Conclusion
The Tradeoff
• Income-Driven Repayment means lower payments today.
• And delaying long enough may trigger forgiveness.
• But it also means more interest accruing to the loan.
• Without forgiveness, it would be cheapest to pay as soon as possible.
• Forgiveness creates a tension with accrued interest.
• How to minimize costs?

7. Student Loans Modeling Student Loans Conclusion
Literature
• Very little work in comparison to market size.
• Especially as student loans are so close to academia.
• More student loans:
(i) reduce home ownership (Mezza et al., 2019)
(ii) inhibit propensity to entrepreneurship (Krishnan and Wang, 2019)...
(iii) ... and public sector employment (Rothstein and Rouse, 2011)
(iv) delay marriage (Gicheva, 2016)
(v) postpone parenthood (Shao, 2015)...
(vi) ...enrollment in graduate degrees (Malcom and Dowd, 2012; Zhang, 2013)
(vii) increase parental cohabitation (Bleemer et al., 2014; Dettling and Hsu, 2018)
• Defaults counterintuitive:
(i) perhaps due to borrowers’ insufficient information (Delisle et al., 2018)
(ii) “majority of distressed student borrowers have their loans in disadvantageous
repayment plans even when eligible for more advantageous options”
(Cornaggia and Xia, 2020)
(iii) delinquency rate decreases as loan balances increase.
(iv) over 30% of loans less than $5,000 in default (Looney and Yannelis, 2019)
• $1 increase in maximum loan linked to increase of 60c in tuition price.
(Lucca et al., 2018). The “Bennett” hypothesis.

8. Student Loans Modeling Student Loans Conclusion
Model
• Borrower must repay balance x > 0.
• Discounts cash flows at rate r > 0.
• Borrowers’ opportunity cost, non necessarily risk-free rate.
• If borrower has a mortgage, mortgage rate is a useful benchmark for r.
• Loan carries higher interest rate r + β, β > 0.
Otherwise problem trivial: pay as late as you can.
• Balance evolution:
dbα
t = (r + β)bα
t dt − αt dt, b0 = x > 0
• αt : chosen repayment rate.
• Deterministic model. Goal is minimizing cost.
• Optimal solution already reduces risk through income-driven repayment.

9. Student Loans Modeling Student Loans Conclusion
Constraints and Forgiveness
• Repayment rate:
m(t) ≤ αt ≤ M(t)
• m(t): minimum payment required by repayment plan.
• M(t): maximum payment affordable to borrower.
• At time T > 0 the remaining balance bα
T is forgiven.
• Forgiven amount taxed at rate ω ∈ (0, 1).
• Total cost of loan:
J(x, α) :=
Z τ
0
e−rt
αt dt + e−rτ
ωbτ ,
• τ := inf{t ≥ 0 : bt = 0} ∧ T: time loan is paid off or forgiven.

10. Student Loans Modeling Student Loans Conclusion
The Critical Horizon
tc :=
T +
log ω
β
+
• Before this time, prepaying an extra dollar saves more than one dollar in
future payments.
• If forgiveness is near, there is no incentive to prepay.
• Critical horizon depends on time to forgiveness, on the interest spread,
and on the tax rate.
• It is zero if the spread is high or the tax rate is low.
• If there were no payment constraints m, M, it would be optimal to prepay
before this time and not prepay after.
• But there are constraints.

11. Student Loans Modeling Student Loans Conclusion
Optimal Repayment
Theorem
For any x 0, the strategy α∗
∈ A defined as
α∗
t :=
(
M(t)1[0,tc ](t) + m(t)1(tc ,T](t) t ∈ [0, T], if x x∗
, (max-min)
M(t) t ∈ [0, T], if x ≤ x∗
, (max)
attains the minimum loan value. Also, v(x) = v1(x) for x x∗
and
v(x) = v2(x) for x ≤ x∗
, where
v1(x) :=
tc
Z
0
e−rs
Msds +
T
Z
tc
e−rs
msds+ ωeβT

x−
tc
Z
0
e−(r+β)s
Msds−
T
Z
tc
e−(r+β)s
msds

 ,
v2(x) :=
tM
Z
0
e−rs
Msds, where tM 0 satisfies x =
Z tM
0
e−(r+β)s
Msds.
• What is x∗
?

12. Student Loans Modeling Student Loans Conclusion
The Critical Balance
• If the balance is low (x x∗
), max strategy. (Pay as soon as possible.)
• If the balance is high (x x∗
), max-min strategy.
Pay maximum before critical horizon, then minimum.
• If critical horizon is zero, always pay minimum.
• Critical balance is the balance for which two strategies are equivalent.
x∗
:=
Z t∗
0
e−(r+β)s
M(s)ds 0,
• t∗
∈ (tc, T) is the unique solution to
Z t∗
tc
e−rs
M(s)(1 − ωeβ(T−s)
)ds =
Z T
tc
e−rs
m(s)(1 − ωeβ(T−s)
)ds.

13. Student Loans Modeling Student Loans Conclusion
Critical Balance
100000
125000
150000
175000
200000
225000
250000
275000
300000
325000
1 2 3 4 5 6 7 8
0
2
4
6
8
10
• Horizontal: loan spread β. Vertical: discount rate r.
• Forgiveness horizon T = 25, annual growth of income and poverty level
g = 4%, tax rate ω = 40%, minimum and maximum payments are 10%
and 30% of income above subsistence of $32,000.

14. Student Loans Modeling Student Loans Conclusion
Critical Balance
• Highly nonlinear in model parameters.
• Higher with a lower spread.
• Higher for very high or very low discount rate.
• It suffices to compare the cost of the two candidate strategies.

15. Student Loans Modeling Student Loans Conclusion
To Enroll or not to Enroll? And When?
0 100000 200000 300000 400000
10
15
20
25
30
0 100000 200000 300000 400000
10
15
20
25
30
0 100000 200000 300000 400000
10
15
20
25
30
• Horizontal: loan balance. Vertical: forgiveness horizon.
• Vertical lines: maximum borrowing
• Light: immediate enrollment in income-driven repayment.
• Dark: later enrollment.

16. Student Loans Modeling Student Loans Conclusion
Loan Valuation
0 50000 100000 150000 200000 250000 300000
0.9
1.0
1.1
1.2
1.3
1.4
1.5
• Cost-to-balance ratio (vertical) against loan balance (horizontal) for PLUS
loans (7.54% rate), with discount rate of 3% (solid) and 6% (dashed).

17. Student Loans Modeling Student Loans Conclusion
Loan Valuation
• Unit cost increases for small balances.
• Decreases for large balances.
• Forgiveness benefits only borrowers with big loans.
• For very large loans, the cost of an extra dollar is just ωeβT
.

18. Student Loans Modeling Student Loans Conclusion
Simple Interest
• Previous analysis omits a peculiar feature.
• For many income-driven schemes, interest is not capitalized.
• If the monthly payment does not cover interest,
the balance increases only by the interest accrued on the principal, not on
outstanding interest.
• But any payment is applied towards interest before the principal.
• Mathematically, what does it mean?
• The loans splits in two.
• The principal, carrying the original rate.
• Unpaid interest, carrying zero rate.
• But you cannot repay the principal while outstanding interest is zero.
• Simple interest is rather complicated.
• Two-dimensional problem?

19. Student Loans Modeling Student Loans Conclusion
Modeling Simple Interest
• The budget equation is replaced by
dbα
t = ((r + β)pα
t − αt )dt, b0 = x 0;
pα
t = inf
0≤s≤t
bα
s .
• Only the remaining principal generates interest.
• The remaining principal is the running minimum of the total balance,
because accrued interest must be repaid first.
• Define the first time of principal repayment:
θ(α) := inf {t ∈ [0, T] : pα
t x}

20. Student Loans Modeling Student Loans Conclusion
Improving Repayments
Lemma
For any x 0,
(i) if θ(α) = T, then J(x, m) ≤ J(x, α).
(ii) if θ(α) T, then there exists a unique t0 ∈ [0, θ(α)] that satisfies
Z θ(α)
0
αsds =
Z t0
0
m(s)ds +
Z θ(α)
t0
M(s)ds.
Moreover, α defined by
αt := m(t)1[0,t0](t) + M(t)1(t0,θ(α)] + αt 1(θ(α),T](t), ∀0 ≤ t ≤ T,
satisfies θ(α) = θ(α) and J(x, α) ≤ J(x, α).
• Minimum: cheapest strategy among those that never repay the principal.
• Otherwise can improve a strategy by starting with a min-max segment.

21. Student Loans Modeling Student Loans Conclusion
Max-Min in Positive Amortization
Lemma
Fix any x 0 and α ∈ A with θ(α) T. Suppose that there exist
a, c ∈ [θ(α), T] with a c such that t 7→ pα
t is strictly decreasing on [a, c]. If
α ∈ A does not belong to the collection
B[a,c] := {α ∈ A : ∃s0 ∈ [a, c] s.t. αt = M(t)1[a,s0]+m(t)1(s0,c](t) for a.e. t ∈ [a, c]},
then there exists u ∈ (a, c) such that α(u) ∈ A defined by
(α(u))t := αt 1[0,a](t) + M(t)1(a,u](t) + m(t)1(u,c](t) + αt 1(c,T](t) ∀t ∈ [0, T] (1)
satisfies J(x, α(u)) J(x, α).
• When amortization positive, first max then min.
• Constant payment rates strategies cannot be optimal.
• But constant payments are the default! Inaction is costly.

22. Student Loans Modeling Student Loans Conclusion
Optimal Strategy with Simple Interest
• For a large enough balance, minimum payments are optimal.
• For a small enough balance, maximum payments are optimal.
• For large balances, simple interest makes huge difference.
• Marginal cost of extra dollar for large loan:
• With compound interest ωe−rT
e(r+β)T
= ωeβT
(insensitive to discount rate r).
• With simple interest ωe−rT
(1 + (r + β)T) (depends on r).
• From $1.09 to $0.52 with r = 3%, β = 4%, ω = 40%, T = 25.
• A priori, we cannot rule out optimality of min-max-min.
• But we have not found any concrete setting where this happens.
• General result?

23. Student Loans Modeling Student Loans Conclusion
The Road Ahead
• Student Loans are very peculiar debt contracts.
• Features seem to have accumulated partly by design, partly by inertia.
• About quarter of loans in delinquency.
• Friedman (1955):
Such an investment necessarily involves much risk. [...] The result is
that if fixed money loans were made, and were secured only by ex-
pected future earnings, a considerable fraction would never be repaid.
• Financing college involves three parties: student, college, lender.
• With federal student loans, the lender is the government.
• Risk is shared between the student and the government.
• The university has no risk.
• Optimal arrangement?

24. Student Loans Modeling Student Loans Conclusion
Other Contracts
• Student loans are not the only types of contracts to finance college.
• Income-share agreements:
debt whose principal is contingent on the borrower’s income.
• Similarities and differences with income-driven repayment?
• How should risk be shared among students, lenders, and colleges?
• Colleges have more information on income-potential of various degrees.
• Which contracts are optimal for financing education?

25. Student Loans Modeling Student Loans Conclusion
Conclusion
• Model of federal student loans.
• Three features: income-driven repayment, forgiveness, simple interest.
• Complete solution for first two features.
• Partial results including simple interest.
• Minimum payments for large balances, maximum for small.
• Intermediate cases more complex.
• What are optimal contracts for education financing?

26. Student Loans Modeling Student Loans Conclusion
Thank You!
Questions?
https://epubs.siam.org/doi/10.1137/22M1505840
https://epubs.siam.org/doi/10.1137/21M1392267