Model and analyse fixed-change finance

Linear financial models

A linear financial model changes by the same fixed amount in each period. Find that amount from the initial value or usage rule, keep its sign, and apply it once for every transition

1. Identify a fixed change Simple interest and flat-rate depreciation use a fixed amount based on the initial value, while unit-cost depreciation uses a fixed amount per usage block \(\displaystyle D=\frac{r}{100}V_0\quad\text{or}\quad D=\text{cost per block}\times\text{number of blocks}\)
2. Keep the direction of change Add a fixed increase for interest and subtract a fixed decrease for depreciation \(\displaystyle V_{n+1}=V_n+D\quad\text{or}\quad V_{n+1}=V_n-D\)
3. Use the linear explicit rule Apply the same fixed change once per period from the starting value \(\displaystyle V_n=V_0+nD\quad\text{or}\quad V_n=V_0-nD\)
4. Distinguish value from change The resulting balance or asset value includes the starting value, not just the total change or one period's change \(\displaystyle V_4=2400+4(120)=2880,\quad\text{total interest}=4(120)=480\)

Worked example

A piece of equipment is worth $9,000 and loses 12% of its initial value each year under a flat-rate depreciation model. Find its value after three years

1. \(\displaystyle D=\frac{12}{100}\times9000=1080\)The fixed annual depreciation is 1,080 dollars because flat-rate depreciation uses the initial value
2. \(\displaystyle V_{n+1}=V_n-1080\)Depreciation reduces the asset value by the same amount at each transition
3. \(\displaystyle V_3=9000-3\times1080=5760\)Subtract three fixed depreciation amounts from the initial value
4. \(\displaystyle V_3=5760\text{ dollars}\)The equipment value after three years is $5,760

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