A flood control project has a construction cost during the first year of $10 mil

Question

A flood control project has a construction cost during the first year of $10 million, during the second year $6 million, and during the third year $2 million. It is completed at the end of the third year and thereafter incurs an annual operating cost of $200,000 per year. Benefits from the project also begin during the fourth year and are valued at $1.5 million in that year, growing at 2% compound rate of increase out to the planning horizon analyze period) of 50 years. The interest rate is taken to be 6%. Carefully draw the cash flow diagram. What is the present worth of the cost? What is the present worth of benefit? Is this viable project economically?

Explanation / Answer

Cash flow in year 1 = -10 million

Cash flow in year 2 = -6 million

Cash flow in year 3 = -2 million

negative sign shows cash outflow

Cash flow in year 4 = 1.5-0.2 = 1.3 million

Cash flow in year 5 = 1.5*(1+0.02) - 0.2 = 1.51 million

Cash flow in year 5 = 1.5*(1+0.02)2 - 0.2 = 1.54 million

Cost in first year of the project = $10 million

Cost in second year of the project = $6 million

Cost in third year of the project = $2 million

From fourth year, annual operating cost till year 50 = $200,000 i.e. 0.2 million

Present value of annuity = P*[(1-(1+r)-n)/r], where P is annual payment, r is interest rate, and n is time period

So this annuity will start in year 4, therefore n = 50-3 = 47 years

In this case, annual payment is annual operation cost

Now, P = $0.2 million

n = 47

r = 6%

Now putiing these value in the equation

Value of operating cost in beginning of year 4 = 0.2*[(1-(1+0.06)-47)/0.06] = 3.12 million

Now we will calculate the present value of all the cost including operating cost

Present value = paymentn/(1+r)n, where paymentn is payment in year n, r is interest rate and t is time period

In this case, to calculate the present worth of total cost we need to takle sum of present value of all costs

Present worth = Payment1/(1+r)1+ Payment2/(1+r)2 + Payment3/(1+r)3+ Payment3/(1+r)3

In this case, we have calculated the value of all operating cost in starting of year 4 i.e. end of year 3, so we will discount that value for three years while calculating present worth of cost

Present worth = 10/(1+0.06)1 + 6/(1+0.06)2 + 2/(1+0.06)3 + 3.12/(1+r)3

Benefit in year 4 = $1.5 million

This benefit will grow by 2% annually till year 50

Present Value formula of growing annuity = p/(r-g)[1-((1+g)/(1+r))n], where P is first payment, r is interest rate, g is growth rate, and n is time period

In this case, g = 2%

r = 6%

This annuity will start in year 4, therefore n = 50-3 = 47 years

n = 47 years

Now putiing these value in the equation

Value of benefit in in beginning of year 4 = 1.5/(0.06-0.02)[1-((1+0.02)/(1+0.06))47] = $31.35 million

Now we will calculate the present value of this value to calculate present worth

As in previous case, n = 3 in this case as the calculated value is at beginning of year 4 i.e. end of year 3

Present worth = 31.35/((1+0.06)^3) = $26.32 million

So present worth of benefits = $26.32 million

To see if project is viable economically, we need to subtract present worth of cost from present worth of benefits, and if it is greater than 0 than project is viable

Present worth of benefits - present worth of cost = 26.32-19.07 = $7.25 million

It is greater than zero, so project is viable economically.

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