3433-vol. 6 issue 2-3.pmd - iarfc

Volume 6, Issue 2 & 3 79
asset allocations tend to overstate the withdrawal amount. The volatility of
the stock market implies that the odds of running out of money are about 50%
if one withdraws the average expected rate per year. For a 30-year horizon,
many portfolios allow a 4.5% real withdrawal rate with a failure probability of
less than 10%, as evidenced in Cooley, Hubbard, and Walz (2003), Pye (2000),
Ameriks, Veres, and Warshawsky (2001), Bengen (2001), and Tezel (2004).
Using the closed form formulas provided above, the replacement
ratios for any combination of the input variables can be calculated by using
an Excel template shown in Table 1.
Table 1
Spreadsheet Formulas for Calculating Income Replacement
Ratio
A B C D
1 INPUTS
2 Expected inflation i 2.50%
3 Expected real income growth g e
1.00%
4 Expected real return on savings r e
5.00%
5 Withdrawal rate x 5.00%
6 Savings period n 35
7 Initial income I 1
$40,000
8 Savings ratio s 10.00%
9 Additional savings ratio as a % of real growth z 20.00%
10 OUTPUTS
11 Expected income growth g = (1+g e
)(1+i)-1 g 3.53%
12 Expected return on savings r = (1+r e
)(1+i)-1 r 7.63%
13 Income prior to retirement I n
= I 1
(1+g) n-1 I n
$129,897
14 Wealth prior to retirement W n
(see line 17) W n
$1,127,803
15 Retirement income P = W n
x P $56,390
16 Income replacement ratio R = P/ I n
R 43.41%
17 W n
= (I 1
/ (g-r)) * (s*((1+g) n – (1+r) n ) + (((1+g) n –(1+g))/g – ((1+r) n –(1+r))/r)
g e
*z)
Cells D2 to D9 are user defined.
Cell D11 = D2+D3+D2*D3
Cell D12 = D2+D4+D2*D4
Cell D13 = D7 * (1+D11)^(D6-1)
Cell D14 = (D7/(D11-D12))*(D8*((1+D11)^D6-(1+D12)^D6)+(((1+D11)^D6-
(1+D11))/D11-((1+D12)^D6-(1+D12))/D12)*D3*D9)
Cell D15 = D14*D5
Cell D16 = D15 / D13
Research & Theory