Financial Math FM/Loans

Learning objectives

The Candidate will understand key concepts concerning loans and how to perform related calculations.

Learning outcomes

The Candidate will be able to:

• Define and recognize the definitions of the following terms: principal, interest, term of loan, outstanding balance, final payment (drop payment, balloon payment), amortization.
• Calculate:

• The missing item, given any four of: term of loan, interest rate, payment amount, payment period, principal.
• The outstanding balance at any point in time.
• The amount of interest and principal repayment in a given payment.
• Similar calculations to the above when refinancing is involved.

Introduction

In this chapter, two methods of repaying a loan will be discussed, namely Template:Colored em and Template:Colored em. In particular, for each of these two methods, we will discuss how to determine he outstanding loan balance at any point in time, and the amount of interest and principal repayment in each payment made by borrower.

Amortization method

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.

Template:Colored em's perspective:

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

Template:Colored em's perspective:

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

in which

• ↑ means the amount is Template:Colored em, ↓ means the amount is Template:Colored em;
• $L$ is the amount borrowed (i.e. the amount of loan);
• $n$ is the number of payments;
• $R$ is the level payment made by the borrower (this is return from the Template:Colored em's perspective).

• Let $B_k$ be the outstanding balance at time $k$, just after the $k$th payment ($B_0=L$, which is the initial balance).
• Let $i$ be the effective interest rate during each interval for payments.

Proof.

• First, $B_k$ will accumulate to $B_k(1+i)$ from time $k$ to $k+1$.
• The interest due on $B_k$ is $i{B}_k$.
• So, the reduction of outstanding balance from the payment of $R$ at $t=k+1$ is $R-i{B}_k$.
• It follows that the outstanding balance at time $k+1$ is $B_k-(R-i{B}_k)=B_k(1+i)-R$.

$◻$

Proof.

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

$$\begin{array}{cccc}& & L(1+i{)}^n-R(1+i{)}^{n-1}-R(1+i{)}^{n-2}-\cdots -R& =0\\ & \Rightarrow & L(1+i{)}^n& =R(1+i{)}^{n-1}+R(1+i{)}^{n-2}+\cdots +R\\ & \Rightarrow & L(1+i{)}^n{v}^n& =R(1+i{)}^{n-1}{v}^n+R(1+i{)}^{n-2}{v}^n+\cdots +R{v}^n\\ & \Rightarrow & L& =Rv+R{v}^2+\cdots +R{v}^n\stackrel{\text{ def }}{=}R{a}_{\bar{n}|i}.\\ \end{array}$$

$◻$

Proof.

• From the proof of fundamental relationship between $L$ and $R$, we have

$$\begin{array}{cc}{B}_k& =L(1+i{)}^k-R(1+i{)}^{k-1}-R(1+i{)}^{k-2}-\cdots -R\\ & =(1+i{)}^k(Rv+R{v}^2+\cdots +R{v}^n)-R(1+i{)}^{k-1}-R(1+i{)}^{k-2}-\cdots -R\\ & =\cancel{R(1+i{)}^{k-1}+R(1+i{)}^{k-2}+\cdots +R(1+i{)}^{k-(k-1)}+R(1+i{)}^{k-k}}+R(1+i{)}^{k-(k+1)}\cdots +R(1+i{)}^{k-n}\cancel{-R(1+i{)}^{k-1}-R(1+i{)}^{k-2}-\cdots -R}\\ & =R(1+i{)}^{-1}+R(1+i{)}^{-2}+\cdots +R(1+i{)}^{-(n-k)}\\ & =Rv+R{v}^2+\cdots +R{v}^{n-k}\\ & =R{a}_{\overline{n-k}|i}.\end{array}$$.

$◻$

Proof.

• From the proof of fundamental relationship between $L$ and $R$, we have

$$\begin{array}{cc}{B}_k& =L(1+i{)}^k-R(1+i{)}^{k-1}-R(1+i{)}^{k-2}-\cdots -R\\ & =L(1+i{)}^k-(1+i{)}^k(Rv+R{v}^2+\cdots +R{v}^k)\\ & =L(1+i{)}^k-R{s}_{\bar{k}|i}.\end{array}$$

$◻$

• Another method to determine outstanding balance (and also principal and interest paid in different payments) is using BA II Plus.
• Procedure:

1. Input $-R$ into PMT (if $R$ is unknown, it should be determined first).
2. Input $L$ into PV

• We can also compute PMT or PV given sufficient information.

1. Press 2ND PV
2. Press the starting payment number (for $k$th payment, press $k$) and press ENTER ↓.
3. Press the ending payment number (for $k$th payment, press $k$) and press ENTER ↓ (press the same number as the starting payment number for selecting exactly one payment ).
4. Then, outstanding balance just after the selected payment(s) is displayed (BAL=... is displayed).
5. Press ↓ and loan (or "principal") paid in the selected payment(s) is displayed (PRN=... is displayed).
6. Press ↓ and interest paid in the selected payment(s) is displayed (INT=... is displayed).

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

Proof.

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

$$P_k=R-I_k=R-i{B}_{k-1}=R-i\underset{\text{prospective}}{\underbrace{R{a}_{\overline{n-k+1}|}}}=R(\cancel{1}-\cancel{i}\left(\frac{\cancel{1}-v^{n-k+1}}{\cancel{i}}\right))=R{v}^{n-k+1}$$

$◻$

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

Amortization schedule for a loan of $a_{\bar{n}|}$ repaid over $n$ periods at rate $i$

Period	Payment	Interest paid	Principal repaid	Outstanding loan balance
0	0	0	0	$a_{\bar{n}|}$ (prospective)
1	1	$i\underset{B_0}{\underbrace{a_{\bar{n}|}}}=\underset{R}{\underbrace{1}}-\underset{P_1}{\underbrace{v^n}}$	$\underset{1(v^{n-1+1})}{\underbrace{v^n}}$	$\underset{B_0}{\underbrace{a_{\bar{n}|}}}-\underset{P_1}{\underbrace{v^n}}=\underset{\text{prospective}}{\underbrace{a_{\overline{n-1}|}}}$
2	1	$i{a}_{\overline{n-1}|}=1-v^{n-1}$	$v^{n-1}$	$a_{\overline{n-1}|}-v^{n-1}=a_{\overline{n-2}|}$
...	...	...	...	...
$k$	1	$i{a}_{\overline{n-k+1}|}=1-v^{n-k+1}$	$v^{n-k+1}$	$a_{\overline{n-k+1}|}-v^{n-k+1}=a_{\overline{n-k}|}$
...	...	...	...	...
$n-1$	1	$i{a}_{\bar{2}|}=1-v^2$	$v^2$	$a_{\bar{2}|}-v^2=a_{\bar{1}|}$
$n$	1	$i{a}_{\bar{1}|}=1-v$	$v$	$a_{\bar{1}|}-v=0$
Total	$n$	$n-a_{\bar{n}|}$	$a_{\bar{n}|}$	not important

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

It can be seen that total payment ($n$) equals total interest paid ($n-a_{\bar{n}|}$) plus total principal repaid ($a_{\bar{n}|}$), 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) ($a_{\bar{n}|}$), as expected, because the whole loan is repaid by the payments in $n$ periods. Template:Colored example Template:Colored exercise

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. Template:Colored em's perspective:

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

Template:Colored em's perspective:

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

in which $R_1,R_2,\dots ,R_n$ 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. Template:Colored proposition

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

$◻$

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

$◻$

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

$◻$

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

$◻$

Proof.

• $I_k=i{B}_{k-1}$: It follows from definition of $I_k$.
• $P_k=R_k-I_k$: It follows from $R_k=P_k+I_k$ which is true by definition.

$◻$

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. Template:Colored exercise

Sinking fund method

After discussing amortization method, we discuss another way to repay a loan, namely sinking fund method.

Template:Colored em'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

Template:Colored em'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

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

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

By definition of sinking fund method, $L=D{s}_{\bar{n}|j}$ because the accumulated value of sinking fund equals amount of loan at maturity.

Using these two equation, we can have the following theorem. Template:Colored proposition

Proof. Because $L=D{s}_{\bar{n}|j}\Rightarrow D=\frac{L}{s_{\bar{n}|j}}$ $$R=Li+D=Li+\frac{L}{s_{\bar{n}|j}}=L(i+\frac{1}{s_{\bar{n}|j}}).$$

$◻$

Recall that $\frac{1}{a_{\bar{n}|i}}=i+\frac{1}{s_{\bar{n}|i}}$. We can observe that a similar expression compared with the right hand side appears in above equation ($i+\frac{1}{s_{\bar{n}|j}}$). In view of this, we Template:Colored em $$\frac{1}{a_{\bar{n}|i\mathrm{\&}j}}=i+\frac{1}{s_{\bar{n}|j}}.$$ (we use '$i\mathrm{\&}j$' because the right hand side involves both $i$ and $j$.) Then, if the amount of loan is 1, then the payment made by borrower at the end of each period is $\frac{1}{a_{\bar{n}|i\mathrm{\&}j}}$.

Naturally, we would like to know what $a_{\bar{n}|i\mathrm{\&}j}$ equals. We can determine this as follows: $$\begin{array}{cc}\frac{1}{a_{\bar{n}|i\mathrm{\&}j}}& =i+\frac{1}{s_{\bar{n}|j}}\\ & =(\frac{1}{a_{\bar{n}|j}}-j)+i\text{because }\frac{1}{a_{\bar{n}|j}}=\frac{1}{s_{\bar{n}|j}}+j\\ & =\frac{1}{a_{\bar{n}|j}}+(i-j)\\ & =\frac{1+(i-j)a_{\bar{n}|j}}{a_{\bar{n}|j}}\\ \Rightarrow a_{\bar{n}|i\mathrm{\&}j}& =\frac{a_{\bar{n}|j}}{1+(i-j)a_{\bar{n}|j}}.\end{array}$$ (The right hand side also involve $i$ and $j$, as expected, because the reciprocal of an expression involving $i$ and $j$ should also involve $i$ and $j$) In particular, if $i=j$, $a_{\bar{n}|i\mathrm{\&}j}=a_{\bar{n}|i}=a_{\bar{n}|j}$ as expected, and $$R=Li+D=L(i+\frac{1}{s_{\bar{n}|i}})=\frac{L}{a_{\bar{n}|i}}.$$ Therefore, each level payment made by borrower in the sinking fund method is the same as the Template:Colored empayment in the amortization method, because $L=R{a}_{\bar{n}|i}$ in amortization method of level payment.

Using this notation, we can express the relationship between $R$ and $L$ as follows: $$R=\frac{L}{a_{\bar{n}|i\mathrm{\&}j}}=\frac{L(1+(i-j)a_{\bar{n}|j})}{a_{\bar{n}|j}}$$ Template:Colored exercise

If we Template:Colored em that the balance in the sinking fund could be used to reduce the amount of loan, then net amount of loan after $k$th payment is $$L-D{s}_{\bar{k}|j},$$ net amount of interest paid in the $k$th period is $$Li-j(D{s}_{\overline{k-1}|j}),$$ and principal repaid in the $k$th period is $$D{s}_{\bar{k}|j}-D{s}_{\overline{k-1}|j}=D(1+j{)}^{k-1}.$$