How to calculate effective rate of return for annuity payments and annuity receipts?

With the following variables

p = payments in
r = annual interest rate
s = accumulated amount upon payment in year x
a = annuity receipts
n = number of annuity receipts (one per year)

Formulae for the accumulation and distribution to zero

s = (p ((1 + r)^x - 1))/r

(a + (1 + r)^n (r s - a))/r = 0

Given values

p = 100000
x = 7
a = 153127
n = 8

Solve for r

(a + (1 + r)^n (p ((1 + r)^x - 1) - a))/r = 0

r = 0.07797524847117951

Effective return is 7.7975 % per annum

Check

7 payment in

s = p (1 + r)^6 + p (1 + r)^5 + p (1 + r)^4 +
    p (1 + r)^3 + p (1 + r)^2 + p (1 + r) + p = 886767.53

8 receipts from year 8 to 15

s = s (1 + r) - a = 802786.45
s = s (1 + r) - a = 712256.93
s = s (1 + r) - a = 614668.34
s = s (1 + r) - a = 509470.25
s = s (1 + r) - a = 396069.32
s = s (1 + r) - a = 273825.93
s = s (1 + r) - a = 142050.57
s = s (1 + r) - a = 0

Final balance is zero

Derivations

Formulae obtained from an annuity summation and recurrence equation:

q[n + 1] = q[n] (1 + r) - a    where   q[0] = s