Construct and use reducing-balance depreciation models

An asset depreciates by 15% each year under the reducing-balance method. Which multiplier should be used?

For the $10,000 asset, set the annual rate to 5%, 15% and 30%. Compare the two losses. The rate is unchanged between years, but the base on which it acts is smaller

At a fixed rate in this explorer, what happens from year 1 to year 2?

Construct both equations

An $8,000 asset loses 15% of its current value each year. It retains 85%, so the forward recurrence multiplies the current value by 0.85. The initial equation is needed as well

1. 1Keep the remaining fraction For annual depreciation r%, subtract the rate fraction from 1. Here the rate is strictly between 0% and 100% \(\displaystyle R=1-\frac{r}{100}\)
2. 2State the complete recurrence Use the purchase value at index 0 and multiply the current value at each yearly step \(\displaystyle \begin{gathered}V_0=P\\V_{n+1}=RV_n\end{gathered}\)

An asset begins at $12,000 and loses 20% of its current value each year. Its first two values are $9,600 and $7,680. Which amount is the loss during year 2 alone?

Name the interval before subtracting

An asset bought for $12,000 is worth $9,600 after year 1 and $7,680 after year 2. A value is the amount remaining. A one-year loss compares consecutive values; total loss compares the final value with the purchase price. Both losses are positive in this decreasing model

1. 1Distinguish value, one-year loss and total loss A loss is a positive decrease in asset value, not interest earned. Subtract full-precision values before rounding the result \(\displaystyle \begin{gathered}L_n=V_{n-1}-V_n\\L_{\mathrm{total}}=P-V_n\end{gathered}\)

Purchase value ($)

\(\displaystyle P=9600\)

Annual depreciation rate

\(\displaystyle r=25\%\)

A different asset depreciates under the reducing-balance method. Find the loss during the second year alone

Keep the ideal model unrounded

For a $100 starting value with multiplier 0.898, the exact second value is 80.6404 dollars, not 80.64. Use the exact model again for the third value. Displaying cents is not the same as rounding the model at each step

1. 1Report once The 72.42 is a rounded report. A model that rounds every yearly update to cents instead gives 72.41 here; that is a different convention

A question asks for a loss to the nearest cent using a full-precision model. Which calculation follows that instruction?

Under the same 20% annual rate and $12,000 purchase price, why are reducing-balance and flat-rate losses the same in year 1 but different in year 2?

Compare the same asset over the same time

At $12,000 and 20% per year, three reducing-balance steps leave $6,144. A fixed $2,400 annual loss leaves $4,800 after the same three years. Both are positive over this horizon. Do not continue the flat model into negative resale values

1. 1Compare like-for-like models Flat-rate depreciation removes the same dollar amount based on the purchase price each year. State the rate, horizon and any cent-rounding convention before comparing

Recover a recurrence from an exact table

If a stated constant reducing-balance model gives $16,000 initially and exactly $14,000 one year later, the multiplier is 0.875. The rate removed is 12.5%, not 87.5%. The constant-model assumption is essential

Do two observed asset values prove that every later value will follow one fixed multiplier?

The ideal model approaches zero without reaching it

With a positive initial value and a multiplier strictly between zero and one, every finite modelled value is positive. The values tend to zero as the number of years increases. A cents display may eventually round to zero, but that does not mean the exact model has reached zero or that a real asset has no salvage value

1. 1State the model limits A constant percentage is an assumption, not a guarantee of resale prices. Repairs, market changes, disposal and a salvage floor may require another model or a restricted time interval

A model is fitted for the first five years, after which the asset is sold for parts. What is the sound interpretation?

Write your own depreciation model

Ungraded written task. Choose an asset, a positive purchase value and a plausible annual reducing-balance rate strictly between 0% and 100%. Define V and n, state your constant-rate assumptions and a sensible time horizon, and write both recurrence equations. Calculate the first two values, the second-year loss and the total loss after two years. Explain one reason an actual resale value could differ. This is self/teacher review, not an automatically graded investigation

One possible chosen-asset model

Example only: model an $18,000 delivery vehicle at 12% reducing-balance depreciation annually for the next two years, assuming a fixed rate with no capital improvements or disposal. Let V(n) be dollars after n completed years. The second-year loss is $1,900.80; total loss after two years is $4,060.80. Market conditions or the vehicle’s condition may make its resale value differ

Write a fair model comparison

Ungraded written task. Choose one purchase price, an annual rate and a short whole-year horizon for which both a reducing-balance model and a flat-rate model stay positive. Define both complete recurrences, compare the final values and explain why the later annual losses differ. State whether any fixed annual dollar loss is rounded to cents. The explanation is self/teacher-reviewed, not automatically scored by entering a flat-rate value

One possible equal-rate comparison

Example only: a $20,000 asset at 15% annually for three years. The fixed annual flat-rate loss is exactly $3,000. Both models start together and lose $3,000 in year 1; the second reducing-balance loss is $2,550. After three years reducing-balance leaves $12,282.50 and flat-rate leaves $11,000. The reducing-balance value is higher because later losses act on a smaller base