Introduction
Several types of financial situations have so far been explored and have been modelled using recurrence relations. It is helpful to summarise each of these relations by recalling that they can fall into one of three types; an arithmetic sequence, a geometric sequence, or a combination of both.
Ideas
Summary of recurrence relations for financial situations
Arithmetic recurrence relations
Flat rate and unit cost depreciation can be modelled using the following recurrence relation: V_{n+1}=V_n-d,\,V_0=P, where V_{n+1} is the value of the asset after n+1 time periods, d is the amount of depreciation per time period, and P is the initial value of the asset.
Interest on loans and investments can be modelled using the following recurrence relation:V_{n+1}=V_n+d,\,V_0=P, where V_{n+1} is the value of the investment or loan after n+1 time periods, d is the amount added per time period, calculated as a percentage of the principal, and P is the initial value of the investment or loan (the principal value).
Geometric recurrence relations
In modelling compound interest with a recurrence relation, the following can be used: V_{n+1}=R\times V_n,\,V_0=P, where V_{n+1} is the value of the loan or investment after n+1 time periods, R is equal to 1+\dfrac{r\%}{100}, usually expressed as a decimal, where r\% is the interest rate. P is the initial (or principal) amount.
In modelling reducing balance depreciation with a recurrence relation, the following can be used: V_{n+1}=r\times V_n,\,V_0=P, where V_{n+1} is the value of the asset after n+1 time periods, r is the remaining value of the asset after each time period, usually expressed as a decimal. P is the initial value of the asset.
Combination of arithmetic and geometric recurrence relations
Interest on an annuity investment can be modelled using the following recurrence relation: V_{n+1}=RV_n-d,\,V_0=P, where V_{n+1} is the value of the investment or loan after n+1 time periods, R is equal to 1+\dfrac{r\%}{100}, usually expressed as a decimal, where r\% is the interest rate, d is the amount subtracted per time period, and P is the initial value of the investment (the principal value).
Interest on a reducing balance loans or annuity can be modelled using the following recurrence relation:
Interest on an annuity investment can be modelled using the following recurrence relation: V_{n+1}=RV_n+d,\,V_0=P, where V_{n+1} is the value of the investment after n+1 time periods, R is equal to 1+\dfrac{r\%}{100}, usually expressed as a decimal, where r\% is the interest rate, d is the amount added per time period, and P is the initial value of the investment (the principal value).
Flat rate and unit cost depreciation, and interest on loans and investments can be modelled using the arithmetic recurrence relation.
Compound interest and reducing balance depreciation can be modelled using the geometric recurrence relation.
Reducing balance loans, annuity, and annuity investment can be modelled using the combination of arithmetic and geometric recurrence relation.
Finance application summaries for assets, investments, and loans
Having summarised the recurrence relations for each of the financial situations, it is also helpful to remember the key differences in entering information into the CAS calculator finance application. Each of the situations are presented below.
| N | \text{Total number of payments} | |
|---|---|---|
| I\% | \text{Annual interest rate} | |
| PV | -ve | \text{Since asset was paid for} |
| PMT | 0 | \text{No payments are made} |
| FV | 0 \text{ or} +ve | 0\text{ if it loses all its value, or positive if some value is left} |
| P/Y | \text{Number of payments per year} | |
| C/Y | \text{Number of compounding periods per year} | |
| \text{Format} | \text{Compound at END} |
| N | \text{Total number of payments} | |
|---|---|---|
| I\% | \text{Annual interest rate} | |
| PV | -ve | \text{Bank gives you money} |
| PMT | 0 | \text{No payments are made} |
| FV | +ve | \text{The amount owing} |
| P/Y | \text{Number of payments per year} | |
| C/Y | \text{Number of compounding periods per year} | |
| \text{Format} | \text{Compound at END} |
| N | \text{Number of payments} | |
|---|---|---|
| I\% | \text{Annual interest rate} | |
| PV | -ve | \text{Since you give money to the bank} |
| PMT | 0 | \text{No payments are made} |
| FV | +ve | \text{The value of the investment} |
| P/Y | \text{Number of payments per year} | |
| C/Y | \text{Number of compounding periods per year} | |
| \text{Format} | \text{Compound at END} |
| N | \text{Number of payments} | |
|---|---|---|
| I\% | \text{Annual interest rate} | |
| PV | +ve | \text{Bank lends money to you} |
| PMT | -ve | \text{Payment is made to the bank} |
| FV | 0 \text{ or} -ve | 0\text{ if fully paid at end of loan, or negative if some is left} |
| P/Y | \text{Number of payments per year} | |
| C/Y | \text{Number of compounding periods per year} | |
| \text{Format} | \text{Compound at END} |
| N | \text{Number of payments} | |
|---|---|---|
| I\% | \text{Annual interest rate} | |
| PV | -ve | \text{You give money to the bank} |
| PMT | +ve | \text{Bank pays you} |
| FV | 0 \text{ or} +ve | 0\text{ if it loses all its value, or positive if some value is left} |
| P/Y | \text{Number of payments per year} | |
| C/Y | \text{Number of compounding periods per year} | |
| \text{Format} | \text{Compound at END} |
| N | 1 | \text{Since each period is the same} |
|---|---|---|
| I\% | \text{Annual interest rate} | |
| PV | -ve | \text{You give money to the bank} |
| PMT | +ve | \text{Bank pays you} |
| FV | +ve | \text{Equal to PV in magnitude as it is in perpetuity} |
| P/Y | \text{Number of payments per year} | |
| C/Y | \text{Number of compounding periods per year} | |
| \text{Format} | \text{Compound at END} |
| N | \text{Number of payments} | |
|---|---|---|
| I\% | \text{Annual interest rate} | |
| PV | -ve | \text{You give money to the bank} |
| PMT | +ve | \text{Bank pays you} |
| FV | 0 \text{ or} +ve | 0\text{ if used up, or positive if money left over} |
| P/Y | \text{Number of payments per year} | |
| C/Y | \text{Number of compounding periods per year} | |
| \text{Format} | \text{Compound at END} |
We should be careful with the signs and values of PV,\,PMT, and FV, as they are the ones that are constantly changing depending on the situation presented. Only in perpetuity, the value of N should be set to 1.