Jack Lin’s Retirement
You work for w years, saving a proportionzPROP of pay. You agree to die after p years of pension. After-savings pay is(1-zPROP)of pay.
In an actual US or Canadian situation, you can guarantee about 2% per annum real interest by buying inflation-indexed government bonds, though less if tax is payable on the interest and if you are paying investment management expenses.
Calculate your required savings rate zPROP.
(This page not on exams) Comments on zPROP
We looked previously at the situation of zero real interest rate. So interest didn’t help saving. If working 25 to 55 (w=30) and then retired from 55 to 85 (p=30), then each year of work pays for a year of work-life and a year of retirement so we should all be saving half our salary. Not many people are saving that much. Bear in mind:
1. For the last 50 years most people in North America who have had steady careers have been partially covered by a employer pension plan. Nowadays not so much; in particular in the private sector, defined benefit plans are often being replaced by defined contribution plans, often less generous and with few guarantees.
2. Government pensions for those retiring at 65, might give pensions totaling about one-third of pay for a low-paid person, less for a high-paid person. Maximum total if 35 years employd at above average wage in Canada is about $16,000 per annum government pension.(Canada Pension Plan $10,000 and Old Age Security $6,000)
3. Pension contribution rates for defined benefit plans, calculated by consulting actuaries, typically assume that resignations and dismissals will much reduce the number of people actually getting full retirement pensions. But do you want to be forced to work for the same employer for a lifetime? Also, often zero after-retirement inflation increases assumed
Let’s assume that the investments, after price inflation fprice,earn a real rateireal relative to price where
(1+ ireal) = (Ret-01)
We are assuming that after retirment, Jack requires a pension constant in purchasing power so the pension is indexedto prices. (May assume national average pay increases a couple of percent per annum faster than prices every year, and maybe that you get pay rises a little faster than the government-reported national average pay)
Let’s assume that the investments, after wage inflation fwage,earn a real rate irel to pay where
(1+ irel to pay) = (Ret-02)
Lets say that at retirement you are earning X per year and require to replace a ratio Rrat of your pre-retirement living standard. So you need a pension of (1 - zPROP)RRATX, indexed to price inflation, and hence you needs a lump sum available at retirement of:
(1 - zPROP) R RAT X a p(at rateireal)
At retirement you have accumulated
zPROPX ACCUM(12)w(at rate irel to wages)
Jack’s Retirement Method 3 With Non-Zero Interest (Cont)
To make this match with the accumulation, need:
zPROPX ACCUM(12)w (at rate irel to wages)
= (1- zPROP)X a(12)g (at rate irel to prices) (Jack 3-03)
zPROP= [a(12)g (at rate irel to prices)]
/ [ACCUM(12)w (at rate irel to wages)+ a(12)g (at rate irel to prices)]
(Jack 3-04)
Note that a special case, if both interest rates are zero, is again
zPROP=
YourRetirement Example 1 (Page 1)
You look at yields-to-maturity on Government of Canada Real Return bonds and the usual nominal bonds and at historical statistics on Consumer Price Index and wages. You decide that the following assumptions make sense long-term for you:
Investment return (gross of inflation): 4.5 % per annum effective
Price rises (inflation): 2.5% per annum effective
Salary rises (gross of inflation):
National average 3.5 % per annum effective
You personally till retirement 4.0% per annum effective
(1+ irel to price ) = (Jack 3-01)
= 1.045/1.025
= 1.019512 (very close to 2%=4.5%-2.5%)
i(12) rel to price/12 = 1.0195121/12 -1
= 0.001626016 (per month)
(1+ irel to your pay) = 1.045/1.040
= 1.004807692 (close to 0.5%=4.0%-3.5%)
i(12) rel to your wage /12 =1.0048076921/12 -1
= 0.00039976 (per month)
Your Retirement Example 1 (Page 2)
You look at yields-to-maturity on Government of Canada Real Return bonds and the usual nominal
zPROP=[ a(12)30 (at 0.001626016 pm)]
/ [ACCUM(12)35(at 0.00039976 pm)+ a (12)30 (at 0.001626016 pm )]
Your Retirement Example 1 (Page 3)
You want to work ages 25-60, w=35, then die at 90, g=30.
zPROP= [ a (12)g (at rate irel to prices)]
/ [ACCUM(12)w (at rate irel to your wage)+ a (12)g (at rate irel to prices)]
(Jack 3-04)
= [ a(12)30 (at 0.001626016 per month)]
/ [ACCUM(12)35 (at 0.00039976 )+ a (12)30 (at at 0.001626016 )]
= (272.342/12) / [(457.218/12) + (272.342/12)]
= 0.3733
(i.e. 37.33% of salary, including from government oremployer)
(Note: www has many retirement calculators, some of them misleading. A pretty good one is at , though it lets you assume excessively optimistic investment return rates like 10% per annum – would be nice! The AARP calculator, which can be found by Googling ‘AARP retirement calculator’, gives better guidance on choice of assumptions but doesn’t allow salary rises at a different rate from price inflation. Maybe transfer AARP assumptions to one of the other calculators such as Canada Life’s)
Your Retirement Example 2 Using Real Rates (Page 1)
You decide that the following assumptions make sense long-term for you:
Real Investment return (rel to prices): 2.0% per annum effective
Price rises (inflation): No assump needed
Real salary rises f (relative to price inflation):
National average No assump needed
You personally till retirement 1.5% per annum effective
(1+ irel to price ) = 1.02
i(12) rel to price/12 = 1.021/12 -1
= 0.001652 (per month)
(1+ irel to your wage ) = (1+ irel to price ) / (1+f)
= (1.02)/(1.015)
= 1.004926108
i(12)rel to your wage /12 =1.0049261081/12 -1 = 0.000409585 pm
ACCUM (12)35(at 0. 0.000409585 pm)
= (1/12) [1+(1.02/1.015)1/12 +(1.02/1.015)2/12+.+ (1.02/1.015)(12*35-1)/12]
= (1/12) [1+(1.02/1.015)1/12 +(1.02/1.015)2/12+..+(1.02/1.015) (12*35-1)/12]
= (1/12) [(1.02/1.015) 35 -1)/((1.02/1.015)1/12-1))
= (1/12)[1.00040958535*12 -1 ]/0.000409585 = 458.188/12
Your Retirement Example 2 Using Real Rates (Page 2)
zPROP= [ a(12)30 (at 0.001652 pm)]
/ [ACCUM(12)35(at 0. 0.000409585 pm)+ a (12)30(at 0.001652 pm )]
= (271.193/12) [ (458.188/12) +(271.193/12)]
= 0.37181
So savings rate of 37.18% of salary, roughly the same as the previous example because the assumptions are roughly equivalent. But our use of assumptions in Example 2 of a real rate of investment return and a real rate of pay rises is better than having to make three (semi-guessed) assumptions - we avoid a specific assumption about future price inflation.