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Note that formula (8.21) produces the proper values for Bq and B. Then the depreciation charge is

(1 +


It is evident from formula (8.22) that the sinking fimd method produces i depreciation charges which increase over the life of the asset. This may or may \ not be a reasonable pattern of depreciation charges, depending upon the nature of the asset. For example, it may produce a reasonable pattern of depreciation: charges for an office building which depreciates slowly at first and then more rapidly later on. It probably would not produce a reasonable pattern of depreciation charges for an automobile.

However, it should be stressed diat the choice of depreciation method often is not motivated by an analysis of which method produces the most realistic book values for the asset. A more important consideration for many individuals and firms is to use an allowable method that is most favorable from a tax perspective. An individual or firm that wants to have high tax deductions in the early years of the life of the asset would not select the sinking fiind metiiod.

It should not be thought that the sinking fund method actually requires the use of a sinking fimd. It is merely a method of calculating book values and depreciation charges. A sinking fund may or may not actually be accumulated to replace the loss of capital.

The second, and simplest, method is the straight line method. This mediod is widely used in practice because of its simplicity. In this method the depreciation charge is constant, so that

Dt =


A + - S. n

Practical applications 273 (8.24)

t fjote that formula (8.24) produces the proper values for Bq and B . It is I interesting to note that the straight-line method is a special case of the sinking-fund method in which = 0.

The third method is the declining balance method, the constant percentage method, or the compound discount method. This method produces depreciation charges which decrease throughout the life of the asset, as opposed to the sinking fund method in which they increase and the straight line method in Which they are constant.

The declining balance method is characterized by the fact that the ilepreciation charge is a constant percentage of the book value at the beginning of the period, i.e.

D, = d- B,.] . (8.25)

Now D, = B,.j -B, so that

B, = B, i(l -d) lind since this is true for all t we have

= BqH -

= 1

- d)

= Bi(l -

= A{\

= 2(1 -

= Ail

= B,-l(l

- d)

= (1


B =

Vld - d) fSince A and S are given, d can be found by

Ail - d)" = S S A

= Ail - d)" = S.

i\ -d)" = I

As a result the book values are linear

time must equal the initial value of the asset less the amount in the sinking bsxi i.e.












3. Using formula (8.27) we obtain

d = 1 -(.l)-2 = 1 - .631 = .369 and using formulas (8.25) and (8.26) we have










as (8.29) and (8.30) we have











Example 8.10 Rework Example 8.9 using the 200% declining balance


Using formula (8.28), we have










Note that d5 = 296 is an arbitrary amount needed to make b5 = 1000 exacdy. If 5 had been ignored, 5 would have been computed as .4(1296) = 518.40.


This section discusses another type of analysis used in connection with fixed assets. An issue of considerable importance in practice is the comparison of the costs of alternate possible fixed assets. There are three costs involved in owning a fixed asset:

1. Loss of interest on the original purchase price, since that money could have been invested elsewhere at interest.

2. Depreciation expense.

3. Maintenance expense.

The periodic charge of an asset is defined to be the cost per period of owning the asset. If we let H be the periodic charge and M be the periodic maintenance expense, then the fundamental equation of value is

H = Ai +

+ M.



Formula (8.31) is quite analogous to formula (8.19). The term Ai is the loss

- s

of interest on the original purchase price. The term - is the periodic

depreciation expense. The term M is the periodic maintenance expense. Recall that in formula (8.19) the term M is not needed, since R is net of expenses.

The capitalized cost of an asset is defined to be the present value of the periodic charges forever, i.e. the present value of a pe etuity for the amount of the periodic charge. The capitalized cost can be looked upon as the present value of maintaining an identical asset in operation indefinitely. Denoting the capitalized cost by K, we have

This is easily confirmed by general reasoning. Under die straight line method over nve years the depreciation factor would be 20% per year (ignoring salvage value). The 200% declining balance factor must be twice that amount, i.e. 40%. Using formula (8.25) and (8.26) we have

2. Using formulas (8.23) and (8.24) we have

return it to the second party. The process of buying back the security is often called "covering the short."

The calculation of yield rates for short sale transactions presents some unanticipated difficulties. Consider the situation in which an investor sells a stock short for $1000 and buys it back for $800 at the end of one year. Clearly, a $200 profit has been made, but what is the yield rate?

An equation of value might appear to be

1000(1 + 0 = 800.

However, this equation produces a yield rate of i = -20% which is clearly unreasonable since a profit has been made.

It might be tempting to try to reverse die transaction and solve the equation of value

800(1 + 0 = 1000

which at least gives a positive answer, i.e. i = 25%. However, this answer cannot be justified either. It arises as a $2 profit on an $8 investment, but there never was an $800 investment.

If, in fact, the transaction occurs exactly as stated, the yield rate does not exist. Some might prefer to say that the yield rate is infinite, since a profit was made on no investment. This point was illustrated previously in Example 5.3.

In practice, short sales normally do not occur as just illustrated. Governmental regulations in the United States required the short seller to make a deposit of a percentage of the price, e.g. 50%, at the time the short sale is made. This deposit is called the margiti and caimot be recovered by the short seller until the short position is covered. The required margin percentage may be changed from time to time by the government. The reader should note in this context the word "margin" has a totally different meaning than it had in Section 8.3 in connection with adjustable rate mortgages.

Thus, in the above illustration the short seller would have to deposit margin of $500 at the time of the short sale if die margin requirement is 50%. Now the situation is such that a valid yield rate can be computed. A $200 profit is made and $500 was deposited over a one-year period of time, so that the yield rate is 40%.

Actually the situation is slightly more complicated than just described. The short seller will be credited with interest on the margin deposit which will increase the yield rate somewhat. If the short seller is credited with 8 % interest on the margin deposit, then the amount of interest earned will be .08(500) = 40. The yield rate taking this interest into account becomes 240/500 = 48%.

The astute reader may ask about interest on the proceeds of the original short sale. In the above illustration could the short seller also earn interest on

{he $1000 proceeds from die short sale? The answer is no. Governmental regulations require that these proceeds remain in a non-interest bearing special account until the short position is covered at which time these funds will be used for the purchase necessary to cover the short position. Any positive residual is the profit on the transaction, while any negative residual is the loss on the transaction.

In practice, if the short position develops a loss, additional margin may be required prior to the position being covered. Conversely, if the position develops a profit, some of the margin may be released and can be withdrawn or used for other pu oses. These types of adjustments are controlled by governmental regulation.

One other aspect of short selling is significant. If the security in question pays dividends (e.g. on stocks) or interest (e.g. on bonds), then the short seller is required to pay these to the purchaser of the security. This will serve to decrease the yield rate realized.

For example, in the above illustration, if the stock sold short pays $60 in dividends during the year, the short sellers net profit is as follows:

Gain on short sale + 200

Interest on margin -f- 40

Dividends on stock - 60

Net profit + 180

Now the yield rate becomes 180/500 = 36%. This requirement creates a significant disincentive to sell short any securities which pay a significant amount of dividends or interest.

Short selling by itself is a speculative endeavor and should not be entered into lightly. The short seller is counting on a significant decline in the value of the security in order to come out ahead. Speculative short selling is normally done over relatively brief periods, so that it is unusual for short positions to remain open for extended periods of time.

However, investment strategies have been developed that involve combinations of long and short positions in related securities which have greatiy reduced risk and also have an excellent prospect to earn a reasonable return on investment. A generic name for these types of transactions is hedging, and many of the transactions become quite complex. In fact, occasionally sitiiations of diis type arise in which a profit is certain. Such transactions of guaranteed profitability are called arbitrage. Generally, arbitrage opportunities are fleeting since market prices react quickly to eliminate such golden opportunities.

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