Loans

In finance, a loan is the transfer of money by one party to another with an agreement to pay it back. The recipient, or borrower, incurs a debt and is usually required to pay interest for the use of the money.

The document evidencing the debt (e.g., a promissory note) will normally specify, among other things, the principal amount of money borrowed, the interest rate the lender is charging, and the date of repayment. A loan entails the reallocation of the subject asset(s) for a period of time, between the lender and the borrower.

The interest provides an incentive for the lender to engage in the loan. In a legal loan, each of these obligations and restrictions is enforced by contract, which can also place the borrower under additional restrictions known as loan covenants. Although this article focuses on monetary loans, in practice, any material object might be lent.

Acting as a provider of loans is one of the main activities of financial institutions such as banks and credit card companies. For other institutions, issuing of debt contracts such as bonds is a typical source of funding.

Amortization Loan

In banking and finance, an amortizing loan is a loan where the principal of the loan is paid down over the life of the loan (that is, amortized) according to an amortization schedule, typically through equal payments.

Similarly, an amortizing bond is a bond that repays part of the principal (face value) along with the coupon payments. Compare with a sinking fund, which amortizes the total debt outstanding by repurchasing some bonds.

Each payment to the lender will consist of a portion of interest and a portion of principal. Mortgage loans are typically amortizing loans. The calculations for an amortizing loan are those of an annuity using the time value of money formulas and can be done using an amortization calculator.

An amortizing loan should be contrasted with a bullet loan, where a large portion of the loan will be paid at the final maturity date instead of being paid down gradually over the loan's life.

An accumulated amortization loan represents the amount of amortization expense that has been claimed since the acquisition of the asset.

Effects

Amortization of debt has two major effects:

Effect	Description
Credit risk	First and most importantly, it substantially reduces the credit risk of the loan or bond. In a bullet loan (or bullet bond), the bulk of the credit risk is in the repayment of the principal at maturity, at which point the debt must either be paid off in full or rolled over. By paying off the principal over time, this risk is mitigated.
Interest rate risk	A secondary effect is that amortization reduces the duration of the debt, reducing the debt's sensitivity to interest rate risk, as compared to debt with the same maturity and coupon rate. This is because there are smaller payments in the future, so the weighted-average maturity of the cash flows is lower.

Amortization Method

For the amortization method, the borrower repays the lender by a series of payments at regular intervals. Each payment is applied first to interest due on the outstanding balance at the time just before the payment is made to pay the interest, and after deducting the amount of interest from each payment, the amount left in each payment is going as the principal repayment to reduce the loan balance (i.e. how much the borrower owes). Payments are made to reduce the loan balance to exactly zero.

Amortization of level payment

The series of payments made by borrower is level in this subsection, and payments form annuity-immediate in our discussion ^[1].

To illustrate this, consider the following diagrams.

Borrower's perspective:

   L     R     R  ...  R  ...     R
   ↑     ↓     ↓       ↓          ↓
---|-----|-----|-------|----------|---
   0     1     2  ...  k  ...     n

Lender's perspective:

   L     R     R  ...  R  ...     R
   ↓     ↑     ↑       ↑          ↑  
---|-----|-----|-------|----------|---
   0     1     2  ...  k  ...     n

in which

↑ means the amount is received, ↓ means the amount is paid;
[math]L[/math] is the amount borrowed (i.e. the amount of loan);
[math]n[/math] is the number of payments;
[math]R[/math] is the level payment made by the borrower (this is return from the lender's perspective).

Let [math]B_k[/math] be the outstanding balance at time [math]k[/math], just after the [math]k[/math]th payment ([math]B_0=L[/math], which is the initial balance). Let [math]i[/math] be the effective interest rate during each interval for payments. We have the following results:

Proposition (Recursive method to determine outstanding balance (level payment))

[[math]]B_{k+1}=(1+i)B_k-R[[/math]]

.

Show Proof

First, [math]B_k[/math] will accumulate to [math]B_k(1+i)[/math] from time [math]k[/math] to [math]k+1[/math].
The interest due on [math]B_k[/math] is [math]iB_k[/math].
So, the reduction of outstanding balance from the payment of [math]R[/math] at [math]t=k+1[/math] is [math]R-iB_k[/math].
It follows that the outstanding balance at time [math]k+1[/math] is [math]B_k-(R-iB_k)=B_k(1+i)-R[/math].

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Proposition (Fundamental relationship between amount of loan and payments)

[[math]]L=Ra_{\overline n|i}[[/math]]

.

Show Proof

Using the above recursive method, [math]B_1=B_0(1+i)-R=L(1+i)-R[/math];
[math]B_2=B_1(1+i)-R=L(1+i)^2-R(1+i)-R[/math];
[math]B_3=L(1+i)^3-R(1+i)^2-R(1+i)-R[/math];
...
[math]B_n=L(1+i)^n-R(1+i)^{n-1}-R(1+i)^{n-2}-\dotsb-R[/math].
Since [math]B_n=0[/math] for amortization method (the loan balance is reduceed to zero at the end by definition), we have

[[math]] \begin{align} && L(1+i)^n-R(1+i)^{n-1}-R(1+i)^{n-2}-\dotsb-R&=0\\ &\Rightarrow& L(1+i)^n&=R(1+i)^{n-1}+R(1+i)^{n-2}+\dotsb+R\\ &\Rightarrow& L(1+i)^nv^n&=R(1+i)^{n-1}v^n+R(1+i)^{n-2}v^n+\dotsb+Rv^n\\ &\Rightarrow& L&=Rv+Rv^2+\dotsb+Rv^n\overset{\text{ def }}=Ra_{\overline n|i}.\\ \end{align} [[/math]]

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Proposition (Prospective method to determine outstanding balance (level payment))

[[math]]B_k=Ra_{\overline {n-k}|i}[[/math]]

.

Show Proof

[[math]] \begin{align} B_k&=L(1+i)^k-R(1+i)^{k-1}-R(1+i)^{k-2}-\dotsb-R\\ &=(1+i)^k(Rv+Rv^2+\dotsb+Rv^n)-R(1+i)^{k-1}-R(1+i)^{k-2}-\dotsb-R\\ &=\cancel{R(1+i)^{k-1}+R(1+i)^{k-2}+\dotsb+R(1+i)^{k-(k-1)}+R(1+i)^{k-k}}+R(1+i)^{k-(k+1)}\dotsb+R(1+i)^{k-n}\cancel{-R(1+i)^{k-1}-R(1+i)^{k-2}-\dotsb-R}\\ &=R(1+i)^{-1}+R(1+i)^{-2}+\dotsb+R(1+i)^{-(n-k)}\\ &=Rv+Rv^2+\dotsb+Rv^{n-k}\\ &=Ra_{\overline {n-k}|i}. \end{align} [[/math]]

.

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Proposition (Prospective method to determine outstanding balance (level payment))

[[math]]B_k=L(1+i)^k-Rs_{\overline k|i}.[[/math]]

Show Proof

From the proof of fundamental relationship between [math]L[/math] and [math]R[/math], we have

[[math]] \begin{align} B_k&=L(1+i)^k-R(1+i)^{k-1}-R(1+i)^{k-2}-\dotsb-R\\ &=L(1+i)^k-(1+i)^k(Rv+Rv^2+\dotsb+Rv^k)\\ &=L(1+i)^k-Rs_{\overline k|i}. \end{align} [[/math]]

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Now, we consider the amount of interest and principal repayment in each payment made by borrower.

Proposition (Splitting an installment into principal and interest repayments (level payment))

Let [math]P_k[/math] be the principal repaid in the [math]k[/math]th installment (i.e. [math]k[/math]th payment made by borrower), and [math]I_k[/math] be the amount of interest paid in the [math]k[/math]th installment. Suppose the installments made by borrower is level, and each of them equals [math]R[/math]. Then,

[[math]]P_k=Rv^{n-k+1},\quad I_k=R-P_k.[[/math]]

Show Proof

First, by definitions, [math]R=P_k+I_k[/math] because the installment is first deducted by the interest due ([math]I_k[/math]), and the remaining amount ([math]R-I_k[/math]) is used to repay principal. Therefore, [math]I_k=R-P_k[/math].
It remains to prove the formula for [math]P_k[/math]. By definition, [math]I_k=iB_{k-1}[/math] because the interest is due on the outstanding balance (before the [math]k[/math]th installment).

[[math]]P_k=R-I_k=R-iB_{k-1}=R-i\underbrace{Ra_{\overline {n-k+1}|}}_{\text{prospective}}=R\left(\cancel 1-\cancel i\left(\frac{\cancel 1-v^{n-k+1}}{\cancel i}\right)\right)=Rv^{n-k+1}[[/math]]

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After splitting each installment, we can make an amortization schedule which illustrates the splitting of each repayment in a tabular form. An example of amortization schedule is as follows:

Amortization schedule for a loan of [math]a_{\overline n|}[/math] repaid over [math]n[/math] periods at rate [math]i[/math]
Period	Payment	Interest paid	Principal repaid	Outstanding loan balance
0	0	0	0	[math]a_{\overline n\|}[/math] (prospective)
1	1	[math]i\underbrace{a_{\overline n\|}}_{B_0}=\underbrace{1}_R-\underbrace{v^n}_{P_1}[/math]	[math]\underbrace{v^n}_{1(v^{n-1+1})}[/math]	[math]\underbrace{a_{\overline n\|}}_{B_0}-\underbrace{v^n}_{P_1}=\underbrace{a_{\overline {n-1}\|}}_{\text{prospective}}[/math]
2	1	[math]ia_{\overline {n-1}\|}=1-v^{n-1}[/math]	[math]v^{n-1}[/math]	[math]a_{\overline {n-1}\|}-v^{n-1}=a_{\overline {n-2}\|}[/math]
...	...	...	...	...
[math]k[/math]	1	[math]ia_{\overline {n-k+1}\|}=1-v^{n-k+1}[/math]	[math]v^{n-k+1}[/math]	[math]a_{\overline {n-k+1}\|}-v^{n-k+1}=a_{\overline {n-k}\|}[/math]
...	...	...	...	...
[math]n-1[/math]	1	[math]ia_{\overline 2\|}=1-v^2[/math]	[math]v^2[/math]	[math]a_{\overline 2\|}-v^2=a_{\overline 1\|}[/math]
[math]n[/math]	1	[math]ia_{\overline 1\|}=1-v[/math]	[math]v[/math]	[math]a_{\overline 1\|}-v=0[/math]
Total	[math]n[/math]	[math]n-a_{\overline n\|}[/math]	[math]a_{\overline n\|}[/math]	not important

(You may verify the recursive method to determine outstanding balance using this table, e.g. [math]a_{\overline n|}(1+i)-1=\ddot a_{\overline n|}-1=a_{\overline {n-1}|}[/math])

It can be seen that total payment ([math]n[/math]) equals total interest paid ([math]n-a_{\overline n|}[/math]) plus total principal repaid ([math]a_{\overline n|}[/math]), and each payment equals the interest paid plus principal repaid in the corresponding period (read horizontally), as expected, because the payment is either used for paying interest, or used for repaying principal.

It can also be seen that the total principal repaid equals the amount of loan (i.e. outstanding loan balance in period 0) ([math]a_{\overline n|}[/math]), as expected, because the whole loan is repaid by the payments in [math]n[/math] periods.

Amortization of non-level payment

In this subsection, we will consider amortization of non-level payment. The ideas and concepts involved are quite similar to the amortization of non-level payment.

Borrower's perspective:

   L    R_1   R_2 ... R_k  ...   R_n
   ↑     ↓     ↓       ↓          ↓
---|-----|-----|-------|----------|---
   0     1     2  ...  k  ...     n

Lender's perspective:

   L    R_1   R_2 ... R_k  ...   R_n
   ↓     ↑     ↑       ↑          ↑  
---|-----|-----|-------|----------|---
   0     1     2  ...  k  ...     n

in which [math]R_1,R_2,\ldots,R_n[/math] are non-level payments, and the other relevant notations used in amortization of level payment have the same meaning.

Because the payments are now non-level, we need formulas different from that for the amortization of level payment to determine amount of loan and outstanding balance at different time, and to split the payment into interest payment and principal repayment. They are listed in the following.

Proposition (Relationship between amount of loan and payments (non-level payment))

[[math]]L=R_1v+R_2v^2+\cdots+R_nv^n.[[/math]]

Show Proof