Jack Lin’s Retirement
You work for w years, saving a proportion zPROP of pay. You
agree to die after p years of pension. After-savings pay is (1zPROP)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 rate ireal relative to price where
(1+ i
real
)=
(1+𝑖 𝑛𝑜𝑚𝑖𝑛𝑎𝑙 )
(1+𝑓𝑝𝑟𝑖𝑐𝑒 )
(Ret-01)
We are assuming that after retirment, Jack requires a
pension constant in purchasing power so the pension is
indexed to 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+ i
rel to pay
)=
(1+𝑖 𝑛𝑜𝑚𝑖𝑛𝑎𝑙 )
(1+𝑓𝑝𝑎𝑦 )
(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 ) RRAT X,
indexed to price inflation, and hence you needs a lump sum
available at retirement of:
(1 - zPROP ) R RAT X a p (at rate ireal )
At retirement you have accumulated
zPROP X 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:
zPROP X 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
𝑔
zPROP = 𝑤+𝑔
Your Retirement 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+𝑖)
(1+ irel to price ) = (1+𝑟𝑝𝑟𝑖𝑐𝑒)
(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 =1.004807692 1/12 -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 or employer)
(Note: www has many retirement calculators, some of them
misleading. A pretty good one is at www.canadalife.com,
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 longterm 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 =1.004926108 1/12 -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.