Model flat-rate depreciation with a recurrence relation

A $4,000 asset loses 10% of its purchase price each year. Is the second-year loss smaller than the first?

Move through years 0 to 5 at 10%. Then change the rate to 0% and 15%. Compare yearly loss, total loss and remaining value

At 10%, what is the value after three years?

Subtract a fixed amount

P is the purchase price, r is the annual percentage number and D is the positive loss per year. The signed change is −D.

1. 1Loss Use a percentage of the purchase price \(\displaystyle D=Pr/100\)
2. 2Update Subtract D once per year \(\displaystyle V_{n+1}=V_n-D\)

Given values

\(\displaystyle P=4000,\quad r=10\)

Find D for $4,000 at 10% per year

Given values

\(\displaystyle V_0=4000,\quad D=400\)

Find the value after one year

Given values

\(\displaystyle V_1=3600,\quad D=400\)

Find the value after two years

Given values

\(\displaystyle D=400,\quad n=3\)

Find the total value lost after three years

Initial value

\(\displaystyle V_0=4000\)

Amount per step

\(\displaystyle D=400\)

Index

\(\displaystyle n=3\)

Find the value remaining after three years

Which model matches a fixed yearly loss of $400?

Check the model’s useful range

A linear rule can eventually predict zero or a negative asset value. Do not treat a negative value as a realistic resale value. The practice questions stay above zero

1. 1Write it yourself An asset costs $4,000 and loses $400 in value each year. Write the initial condition and explicit rule on paper. Check: V₀ = 4000 and Vₙ = 4000 − 400n