Payments of 300, 500 and 700 are made at the end of years five, six, and eight, respectively

1. Point in time : X

Method of equated time (weighted average time) = [∑ (Payments * year) ] / Single Payment

X   = [ $300 * 5 + $500 * 6 + $700 * 8 ] / $1500

= 6.733

2. Exact point in time : Y

Exact point of time depends on the interest rate. Here the Annual effective interest rate is 4%. Single payment is $1500.

For the exact point of time single payment present value is equal to all payments present value. So, the following equation is formed :

$1500 * PVIF ( 4% , n ) = $300 * PVIF ( 4% , 5 ) + $500 * PVIF ( 4% , 6 ) + $700 * PVIF ( 4% , 8 ) [ PVIF = Present value interest factor = 1 / (1+r)ⁿ , where r = interest rate and n = number of period ]

$1500 * 1 / (1 + 0.04)ⁿ = $300 * 0.8219 + $500 * 0.7903 + $700 * 0.7307 [ Values taken directly from PVIF table ]

$1500 / (1 + 0.04)ⁿ = $246.57 + $395.15 + $511.49

$1500 / (1 + 0.04)ⁿ = $1153.21

1 / (1 + 0.04)ⁿ = $1153.21 / $1500

1 / (1 + 0.04)ⁿ = 0.7688

n = 6.707      (from working note)

so, Y = 6.707

then X + Y = 6.733 + 6.707 = 13.44

Working Note:

PVIF ( 4%, 6) = 0.7903

PVIF ( 4%, 7) = 0.7599

from unit method; 0.0304 ( 0.7903 - 0.7599 ) decrease when difference is 1 ( 7 - 6 )

0.0215 ( 0.7903 - 0.7688 ) decreases when difference is 0.707 [ ( 1 / 0.0304 ) * 0.0215 ]

so, PVIF ( 4%, 6.707 { 6 + 0.707 } ) = 0.7688