An investment project costs $12,900 and has annual cash flows of $3,500 for six years.  a. What is the discounted payback period if the discount rate is zero percent?   b. What is the discounted payback period if the discount rate is 6 percent?   c. What is the discounted payback period if the discount rate is 22 percent?

a. If the discount rate is zero percent, the discounted payback period is simply the initial investment divided by the annual cash flow. So, $12,900 /$3,500 = 3.69 years.

b. If the discount rate is 6 percent, we need to calculate the present value of each year's cash flow and then add these up until we reach the initial investment. The formula for the present value is PV = CF / (1 + r)^n, where CF is the cash flow, r is the discount rate, and n is the year.

Year 1: $3,500 / (1 + 0.06)^1 =$3,301.89 Year 2: $3,500 / (1 + 0.06)^2 =$3,113.67 Year 3: $3,500 / (1 + 0.06)^3 =$2,936.86 Year 4: $3,500 / (1 + 0.06)^4 =$2,771.56

By the end of year 4, the total discounted cash flow is $3,301.89 +$3,113.67 + $2,936.86 +$2,771.56 = $12,123.98, which is less than the initial investment.

Year 5: $3,500 / (1 + 0.06)^5 =$2,617.31

By the end of year 5, the total discounted cash flow is $12,123.98 +$2,617.31 = $14,741.29, which is more than the initial investment. So, the discounted payback period is somewhere in year 5.

c. If the discount rate is 22 percent, we follow the same process as in part b, but with a higher discount rate.

Year 1: $3,500 / (1 + 0.22)^1 =$2,868.85 Year 2: $3,500 / (1 + 0.22)^2 =$2,350.28 Year 3: $3,500 / (1 + 0.22)^3 =$1,926.13 Year 4: $3,500 / (1 + 0.22)^4 =$1,578.13 Year 5: $3,500 / (1 + 0.22)^5 =$1,293.04 Year 6: $3,500 / (1 + 0.22)^6 =$1,059.05

By the end of year 6, the total discounted cash flow is $2,868.85 +$2,350.28 + $1,926.13 +$1,578.13 + $1,293.04 +$1,059.05 = $11,075.48, which is less than the initial investment. So, the discounted payback period is more than 6 years.

a. If the discount rate is zero percent, the discounted payback period is simply the initial investment divided by the annual cash flow. So, $12,900 /$3,500 = 3.69 years.

b. If the discount rate is 6 percent, we need to calculate the present value of each year's cash flow and then add these up until we reach the initial investment. The formula for the present value is PV = CF / (1 + r)^n, where CF is the cash flow, r is the discount rate, and n is the year.

Year 1: $3,500 / (1 + 0.06)^1 =$3,301.89 Year 2: $3,500 / (1 + 0.06)^2 =$3,113.67 Year 3: $3,500 / (1 + 0.06)^3 =$2,937.42 Year 4: $3,500 / (1 + 0.06)^4 =$2,772.66

By the end of year 4, the total discounted cash flow is $12,125.64, which is less than the initial investment.

Year 5: $3,500 / (1 + 0.06)^5 =$2,618.35

By the end of year 5, the total discounted cash flow is $14,743.99, which is more than the initial investment. So, the discounted payback period is somewhere in year 5.

c. If the discount rate is 22 percent, we use the same process as in part b, but with a higher discount rate.

Year 1: $3,500 / (1 + 0.22)^1 =$2,868.85 Year 2: $3,500 / (1 + 0.22)^2 =$2,350.69 Year 3: $3,500 / (1 + 0.22)^3 =$1,927.78 Year 4: $3,500 / (1 + 0.22)^4 =$1,579.65 Year 5: $3,500 / (1 + 0.22)^5 =$1,294.95 Year 6: $3,500 / (1 + 0.22)^6 =$1,061.10

By the end of year 6, the total discounted cash flow is $11,083.02, which is less than the initial investment. So, the project never pays back its initial investment if the discount rate is 22 percent.