Handbook of Civil Engineering Calculations

for i — 8 or 10 percent, the value of the original capital at the end of the sixth year =

$8000(1.469) + $7800(1.360) + $7000(1.260) + $6200(1.166) + $2200(1.100) + $1800 =

$42,629.

2. Compute the average investment rate
   Let i' = average investment rate. Equate the original investment to $42,629 at a date 6
   years in the future, and solve for i''. Thus, $24,000(SPCA for i") = $42,629; i" = 10.1 per-
   cent.

RATE OF RETURN ON A

SPECULATIVE INVESTMENT

A firm purchased a parcel of land for $25,000 and spent $600 during the first year to im-
prove the property. (This investment for improvements should be considered a lump-sum
end-of-year payment.) The expenses for real estate tax, insurance, and maintenance to-
taled $ 1200 per year. At the end of 5 years, the firm sold the property at a price that yield-
ed $48,700 after payment of legal fees and commissions. In computing the federal income
tax, the firm deducted the ordinary expenses of holding this property from the income de-
rived from other sources. This income was subject to a 53 percent tax rate. The profit on
the sale of the land was taxed at the 25 percent capital-gains rate. What was the rate of re-
turn on the investment?

Calculation Procedure:

1. Determine the effective annual payment
   The expenses related to possession of the land served to reduce the income tax payments.
   Therefore, the effective cost of holding the property (or any similar asset) was less than
   the actual expenses. To obtain the effective annual payment, deduct the annual income
   tax saving from the annual payment related to the asset. Thus, effective annual payment =
   $1200(1.00-0.53)-$564.


2. Compute the net proceeds from the sale of the asset
   Deduct the capital-gains tax from the selling price of the asset to obtain the net proceeds.
   This is often called the effective selling price. Thus, capital gains = $48,700 - ($25,000 +
   $600) = $23,100. The capital-gains tax = $23,100(0.25) = $5775. Hence, net proceeds =
   $48,700-$5775 = $42,925.


3. Set up an equation for the rate of return
   Selecting the date at which the asset was sold as the reference date, express the value of
   every sum of money, and equate the total effective payments to the income. Thus,
   $25,000(SPCA for « = 5 years, i = ?) + $600(SPCA for n = 4 years, i = ?) + $564(USCA
   for /i = 5 years, i = ?) = $42,925.


4. Solve the rate-of-return equation, using trial values
   As a trial, set / = 10 percent, and evaluate the left-hand side of the relation in step 3. Thus,
   $25,000(1.611) + $600(1.464) + $564(6.105) = $44,597.
   Since the actual income, $42,925, was less than $44,597, the assumed rate of return is
   too high. Try 8 percent. Then $25,000(1.469) + $600(1.360) + $564(5.867) = $40,850.
   This is less than the actual income. Interpolating linearly between the two trial values
   yields / = 9.1 percent.