15 4 Find the value of the unknown quantity Po that establis

(15) 4. Find the value of the unknown quantity, Po, that establishes equivalence in the cash-flow diagram below. Suppose i-20% per year. Use an annuity factor and a uniform gradient factor in your solution. Show all work. 2,000 $1,400 $1,100 $800 5500 $500 $500 $500 $500 1123 45 6 7 8 9 10 End of Year Po\'-?

Solution

Rate of interest = 20%

1. Convert the linear gradient series into uniform series and then calculate the PW of the uniform series.

Take A1 (inflow in the first years) = 1400

Gradient (decreasing) = 300

Number of years = 3

A = A1 – G (A/G, 20%, 3)

A = 1400 – 300 (A/G, 20%, 3)

A = 1400 – 300 (0.8791)

A = 1136

PW of uniform series 1018 at the 0^th year

PW = 1136 (P/A, 20%, 3)

PW = 1136 (2.1065) = 2393------------------------------------------- (1)

2. Calculate the present worth of 500 for 5 years (4^th, 5^th, 6^th, 7^th & 8^th) at the end of 3^rd year

PV at 3^rd year = 500 (P/A, 20%, 5)

PV at 3^rd year = 500 (2.9906) = 1495

Bring the above single amount to the 0^th year by using single payment factor.

PW of 1495 at 0^th year = 1495 (P/F, 20%, 3)

PW of 1495 at 0^th year = 1495 (0.5787) = 865 ------------------      (2)

3. PW 2000 of the 10^th year by using single payment factor

PW = 2000 (P/A, 20%, 10)

PW = 2000 (0.1615) = 323 -------------------------------------------   (3)

Total Present Value at 0^th year = 2393 + 865 + 323 = 3581