guide:1bdf94e2b2

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 .

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

Omitted since the main idea is identical to the proof for the level payment version.

■

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

[[math]]B_k=R_{k+1}v+R_{k+2}v^2+\cdots+R_nv^{n-k}.[[/math]]

Show Proof

Omitted since the main idea is identical to the proof for the level payment version.

■

Proposition (Retrospective method to determine outstanding balance (non-level payment))

[[math]]B_k=L(1+i)^k-R_1(1+i)^{k-1}-R_2(1+i)^{k-2}-\cdots-R_k.[[/math]]

Show Proof

Omitted since the main idea is identical to the proof for the level payment version.

■

Proposition (Retrospective method to determine outstanding balance (non-level payment))

[[math]]B_k=L(1+i)^k-R_1(1+i)^{k-1}-R_2(1+i)^{k-2}-\cdots-R_k.[[/math]]

Show Proof

Omitted since the main idea is identical to the proof for the level payment version.

■

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

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

Show Proof

Omitted since the main idea is identical to the proof for the level payment version.

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

[[math]]I_k=iB_{k-1},\quad P_k=R_k-I_k.[[/math]]

Show Proof

• [math]I_k=iB_{k-1}[/math]: It follows from definition of [math]I_k[/math].
• [math]P_k=R_k-I_k[/math]: It follows from [math]R_k=P_k+I_k[/math] which is true by definition.

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Amortization of payments that are made at a different frequency than interest is convertible

In this situation, we can obtain the amount of loan, outstanding balance, and principal repaid and interest paid in a payment, by calculating the equivalent interest rate that is convertible at the same frequency at which payments are made. Then, the previous formulas can be used directly, at this equivalent interest rate. This method is analogous to the method for calculating the annuity with payments made at a different frequency than interest is convertible.

Sinking Fund

A sinking fund is a fund established by an economic entity by setting aside revenue over a period of time to fund a future capital expense, or repayment of a long-term debt.

In modern finance, a sinking fund is, generally, a method by which an organization sets aside money over time to retire its indebtedness. More specifically, it is a fund into which money can be deposited, so that over time preferred stock, debentures or stocks can be retired.

Definition (Sinking Fund Method)

For sinking fund method, all principal (i.e. amount of loan) is repaid by the borrower in a single payment at maturity. Interest due on the principal is paid at the end of each period and a deposit is made into a sinking fund at the end of each period (same amount of deposit for each of the time points), so that the accumulated value of sinking fund equals the amount of principal at maturity.

Borrower's perspective:

Loan repayment:
   L     Li    Li     ...         Li    L
   ↑     ↓     ↓                  ↓     ↓
---|-----|-----|------------------|-----|---
   0     1     2      ...        n-1    n
   \    / \   /                   \    /
    \  /   \ /       ...           \  / 
     i      i                        i       rate

Sinking fund:
         D     D      ...         D     D  L
         ↓     ↓                  ↓     ↓ ↗
---|-----|-----|------------------|-----|---
   0     1     2      ...        n-1    n
   \    / \   /                   \    /
    \  /   \ /       ...           \  / 
     j      j                        j       rate

Lender's perspective: (Lender do not know how the borrower repays the loan, so sinking fund is not shown)

Loan repayment:
   L     Li    Li     ...         Li    L
   ↓     ↑     ↑                  ↑     ↑ 
---|-----|-----|------------------|-----|---
   0     1     2      ...        n-1    n
   \    / \   /                   \    /
    \  /   \ /       ...           \  / 
     i      i                        i       rate

in which

• [math]L[/math] is the amount borrowed
• [math]n[/math] is the number of payment periods
• [math]i[/math] is the effective interest rate paid by borrower to lender
• [math]j[/math] is the effective interest rate earned on the sinking fund (which is usually strictly less than [math]i[/math] in practice)
• [math]D[/math] is the level sinking fund deposit

Let [math]R[/math] is the level payment made by borrower at the end of each period, which equals [math]D+[/math] interest paid to lender, i.e. [math]R=Li+D[/math].

By definition of sinking fund method, [math]L=Ds_{\overline n|j}[/math] because the accumulated value of sinking fund equals amount of loan at maturity.

Using these two equations, we can have the following result.

Proposition (Relationship between each payment made by borrower and amount of loan in sinking fund method)

[[math]]R=L\left(i+\frac{1}{s_{\overline n|j}}\right).[[/math]]

Show Proof

Because [math]L=Ds_{\overline n|j}\Rightarrow D=\frac{L}{s_{\overline n|j}}[/math]

[[math]]R=Li+D=Li+\frac{L}{s_{\overline n|j}}=L\left(i+\frac{1}{s_{\overline n|j}}\right).[[/math]]

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Recall that [math]\frac{1}{a_{\overline n|i}}=i+\frac{1}{s_{\overline n|i}}[/math]. We can observe that a similar expression compared with the right hand side appears in above equation ([math]i+\frac{1}{s_{\overline n|j}}[/math]). In view of this, we define

[[math]]\frac{1}{a_{\overline n|i\& j}}=i+\frac{1}{s_{\overline n|j}}.[[/math]]

(we use '[math]i\& j[/math]' because the right hand side involves both [math]i[/math] and [math]j[/math].) Then, if the amount of loan is 1, then the payment made by borrower at the end of each period is [math]\frac{1}{a_{\overline n|i\& j}}[/math].

Naturally, we would like to know what [math]a_{\overline n|i\& j}[/math] equals. We can determine this as follows:

[[math]] \begin{align} \frac{1}{a_{\overline n|i\& j}}&=i+\frac{1}{s_{\overline n|j}}\\ &=\left(\frac{1}{a_{\overline n|j}}-j\right)+i\qquad \text{because }\frac{1}{a_{\overline n|j}}=\frac{1}{s_{\overline n|j}}+j\\ &=\frac{1}{a_{\overline n|j}}+(i-j)\\ &=\frac{1+(i-j)a_{\overline n|j}}{a_{\overline n|j}}\\ \Rightarrow a_{\overline n|i\& j}&=\frac{a_{\overline n|j}}{1+(i-j)a_{\overline n|j}}. \end{align}[[/math]]

(The right hand side also involve [math]i[/math] and [math]j[/math], as expected, because the reciprocal of an expression involving [math]i[/math] and [math]j[/math] should also involve [math]i[/math] and [math]j[/math]) In particular, if [math]i=j[/math], [math]a_{\overline n|i\& j}=a_{\overline n|i}=a_{\overline n|j}[/math] as expected, and

[[math]]R=Li+D=L\left(i+\frac{1}{s_{\overline n|i}}\right)=\frac{L}{a_{\overline n|i}}.[[/math]]

Therefore, each level payment made by borrower in the sinking fund method is the same as the level payment in the amortization method, because [math]L=Ra_{\overline n|i}[/math] in amortization method of level payment.

Using this notation, we can express the relationship between [math]R[/math] and [math]L[/math] as follows:

[[math]]R=\frac{L}{a_{\overline n|i\& j}}=\frac{L(1+(i-j)a_{\overline n|j})}{a_{\overline n|j}}[[/math]]

General References

• Wikibooks contributors. "Financial Math FM/Loans,". Wikibooks. Wikibooks. Retrieved 5 November 2023.

Wikipedia References

• Wikipedia contributors. "Loan". Wikipedia. Wikipedia. Retrieved 5 November 2023.