menu
3.X
zoom
Loan operations
X Unit 4 Slide 1
Summary
4.1 Introduction to loan operations
4.2 Single-payment loans
4.3 Annuity-amortized loans
4.4 Changes in the conditions of the loan contract
4.5 The Annual Percentage Rate (APR) of a loan
Prepara i tuoi esami
Studia grazie alle numerose risorse presenti su Docsity
Ottieni i punti per scaricare
Guadagna punti aiutando altri studenti oppure acquistali con un piano Premium
Guide e consigli
Prepara i tuoi esami
Studia grazie alle numerose risorse presenti su Docsity
Cerca documenti
Prepara i tuoi esami con i documenti condivisi da studenti come te su Docsity
Cerca la tua università
Trova i documenti specifici per gli esami della tua università
Video Corsi
Preparati con lezioni e prove svolte basate sui programmi universitari!
Quiz
Rispondi a reali domande d’esame e scopri la tua preparazione
Docsity AINEW
Riassumi i tuoi documenti, fagli domande, convertili in quiz e mappe concettuali
Maturità 2026
Studia con prove svolte, tesine e consigli utili
Esplora domande
Togliti ogni dubbio leggendo le risposte alle domande fatte da altri studenti come te
Argomenti di studio
Esplora i documenti più scaricati per gli argomenti di studio più popolari
Ottieni i punti per scaricare
Guadagna punti aiutando altri studenti oppure acquistali con un piano Premium
Loan Operations: Single-Payment Loans vs. Annuity-Amortized Loans, Slide di Matematica Finanziaria
An introduction to loan operations, focusing on single-payment loans and annuity-amortized loans. Single-payment loans require the repayment of the entire principal sum in a single payment at the end of the loan term, while annuity-amortized loans have scheduled periodic payments that consist of both principal repayment and interest payment. the differences between these two types of loans, their characteristics, and how to calculate the unpaid balance and interest payments.
Tipologia: Slide
2014/2015
1 / 25
Questa pagina non è visibile nell’anteprima
Non perderti parti importanti!
Documenti correlati
Anteprima parziale del testo
Scarica Loan Operations: Single-Payment Loans vs. Annuity-Amortized Loans e più Slide in PDF di Matematica Finanziaria solo su Docsity!
X Unit 4 (^) Loan operations Slide 1
Summary
4.1 Introduction to loan operations
4.2 Single-payment loans
4.3 Annuity-amortized loans
4.4 Changes in the conditions of the loan contract
4.5 The Annual Percentage Rate (APR) of a loan
X Unit 4 (^) Loan operations Slide 2
In a loan contract, a borrower receives initially an amount of money (called principal) from a lender, and is obligated to repay an equal amount of money to the lender at a later time (amortization of the loan), plus an additional sum of money, which is the interest of the loan.
Depending on how the lender repays the principal, we can differentiate between two types of loans:
a) Single-payment loans: these loans require the payment of the entire principal sum and the end of its duration. The interests, however, can be paid periodically or not.
b) Loans repaid by annuities: the principal of the loan is repaid by an annuity. The interest is then also paid in an annuity, being computed over the unpaid balance.
4.1 Introduction to loan operations
X Unit 4 (^) Loan operations Slide 4
4.3 Annuity-amortized loans
Generalities
Annuity-amortized loans or, simply, amortized loans are loans with scheduled periodic payments or installments that are composed of two parts: the principal repayment and the interest payment.
The principal repayment ( is the amount of money the borrower pays which reduces the outstanding loan amount.
The interest payment ( is the amount of money the borrower pays in concept of interests, and it will be computed over the outstanding loan amount.
The total amount paid by the lender or installment ( , therefore, is the sum of the principal repaiment and the interest payment.
X Unit 4 (^) Loan operations Slide 5
4.3 Annuity-amortized loans
Generalities
- (^) When all the principal repayments are paid, the loan will be fully amortized. Therefore, the sum of all the principal repayments is equal to the principal:
- (^) We can also compute the unpaid balance at a given moment t (that is, the part of the principal which has not been paid yet) as the unpaid balance at the previous period minus the principal repayment of the period:
We will denote by the outstanding loan amount (the unpaid balance) at the moment t. Therefore, denotes the amount of money the lender receives from the borrower (that is, the principal of the loan). Given that the principal repayment ( is the amount paid by the borrower to reduce the outstanding amount, we can conclude that:
- (^) Therefore, the unpaid balance at a given moment t can be also calculated as the principal of the loan minus all the paid principal repayments or, alternatively, as the sum of all the principal repayments to be paid in the future
X Unit 4 (^) Loan operations Slide 7
4.3 Annuity-amortized loans
Generalities
The unpaid balance of a loan at any moment k can be calculated as the sum of the discounted value of all the pending installments:
Rule of financial equivalence:^ Rule of financial equivalence:
X Unit 4 (^) Loan operations Slide 8
4.3 Annuity-amortized loans
Generalities
Usually, loans are amortized by immediate annuities, i.e., the payment of the principal starts in the next period to the collection of the principal by the lender. However, it is possible to agree the payment of the principal by a deferred annuity. The period between the granting of the loan and the first payment of principal is called the grace period. There are two types of grace period: Interest-only grace period : during the grace period the lender only pays interests. 0 1 2 … c c+1 … n
No-payment grace period : during the grace period, the lender will make no payment, but the interests will be compounding. 0 1 2 … c c+1 … n
X Unit 4 (^) Loan operations Slide 10
4.3 Annuity-amortized loans
l principal payment method (straight line amortizati
The equal principal payment method (or straight line amortization method) consists on the payment of the same amount of principal repayment during the life of the loan. To compute the amortization schedule, we can follow these steps:
- First, compute the constant principal repayment by dividing the principal of the loan by the number of principal repayments:
- We can now compute all the values of as follows:
- A, the interest payment can be calculated:
- Finally, the total amount to be paid is the sum of the principal repayment and the interest payment:
- A, the interest payment can be calculated:
- We can now compute all the values of as follows:
X Unit 4 (^) Loan operations Slide 11
4.3 Annuity-amortized loans
l principal payment method (straight line amortizati
Example: compute the amortization schedule of a loan of 600,000€ that will be amortized in three equal annual principal payments. The rate of interest is 1,5% per year.
Year C^ A I a
(^1) 600.000 200.000 600.000 x 1,5%=9.000 (^) 209.
(^2) 400.000 200.000 400.000 x 1,5%=6.000 206.
(^3) 200.0000 200.000 200.000 x 1,5%=3.000 203.
X Unit 4 (^) Loan operations Slide 13
Equal installments method
4.3 Annuity-amortized loans
Once the unpaid balance is computed, we can also calculate the interest payment for any period t :
The principal repayment can be computed as the difference between the installment and the interest payment:
However, we can also study the behavior of those principal repayments in the equal installments method. Considering two consecutive principal repayments for the moments t and t+1 :
That is, the principal repayments vary in a geometric fashion with a growth rate of the rate of interest. Consequently, the principal repayments during the first stages of the loan are smaller than the principal repayments during the last stages of the loan. Consequently, as the installment is constant, the interest payments will be higher at the first stages and smaller at the last stages.
X Unit 4 (^) Loan operations Slide 14
Equal installments method
4.3 Annuity-amortized loans
Example: Consider a loan of 300,000€ of principal that will be amortized in 240 monthly installments, being the nominal interest rate of 4% compounded monthly.
a) Calculate the constant installment:
b) Calculate the unpaid balance after paying the twelveth installment
c) Calculate the sum of interests paid during the five first years.
X Unit 4 (^) Loan operations Slide 16
4.3 Annuity-amortized loans
Equal installments method
Excel functions for equal installments loans
=PAGO(rate of interest; # installments; principal)
Function PAGO can be used to estimate the installment of a equal installments loan.
=PAGOPRIN(rate of interest; period; # installments; principal) Function PAGOPRIN can be used to estimate the principal repayments of a equal installments loan.
=PAGOINT(rate of interest; period; # installments; principal) Function PAGOINT can be used to estimate the interest payments of a equal installments loan.
X Unit 4 (^) Loan operations Slide 17
4.4 Changes in the conditions of the loan contract Next, we will study how to recompute the amortization schedule of a loan when some changes in the initial conditions happen. Specifically, we will study the changes in the rate of interest and the prepayment of principal.
Although the examples we have seen so far are fixed interest loans, the floating interest loans are quite frequent nowadays. In a floating interest loan , the interest rate of the loan is referred to an interest index (such as the EURIBOR or the LIBOR). Thus, the interest of the loan is usually equal to the value of that index plus a certain spread. With this agreement, the interest rate of the loan will be similar to the normal rates of interest in the financial markets in any moment. Usually, the floating interest loans include an interest revision plan. In that plan, the borrower and the lender agree the period for revision (a year, a half-year, a quarter…). Thus, at the beginning of every period, the interest rate for that period is calculated as the value of the reference index plus the spread. That interest rate will work as a fixed rate during that revision period, but it may change for the next revision period.
Albeit these changes can be applied to any type of loan, we will study their application to the equity installments loan, because it is the most common in practice.
Changes in the interest rate
X Unit 4 (^) Loan operations Slide 19
4.4 Changes in the conditions of the loan contract
Prepayment of principal
Loan contracts usually include a clause that allows the lender to make non-periodical payments to reduce the unpaid balance. These payments are known as “prepayment of principal”. However, the borrowers usually charges a commission when the lender makes such prepayments.
Given that the prepayments of principal modify the unpaid balance, it is necessary to recalculate the amortization schedule. There are two options for recalculating it:
- (^) Option 1: keeping the same installment, but modifying the maturity : the lender can decide to keep paying the same installment but, as the unpaid balance is smaller after the prepayment of principal, the number of installments required to fully payback the loan will be lower.
- (^) Option 2: keeping the maturity, but reducing the installment : in this case, the lender decides to keep paying the same number of installments but, as the unpaid balance is smaller, the installments to be paid will be lower.
For both cases, we will use the financial equivalence to calculate the new conditions of the loan.
X Unit 4 (^) Loan operations Slide 20
4.4 Changes in the conditions of the loan contract
Prepayment of principal
Example: On March 31st 2015, a lender got a loan of a principal of 300,000 € to be amortized in 240 equal monthly installments. The nominal interest rate is 4.5% compounded monthly. On March 31st 2016, after paying the installment corresponding to that month, the lender decided to make a prepayment of principal of 10,000€. Calculate the following:
a) The new number of installments if the installment remains the same. b) The new value of the installment if the number of installments remains unchanged.
Solution : We begin calculating the original installment and the unpaid before and after the prepayment. Interest: Installment:
Unpaid balance before the prepayment: Unpaid balance after the prepayment: If the installment remains the same:
If the number of installments remains unchanged: