Calculation help for portfolio management

If A retired, and receive fixed pension income of $100,000 a year and will not increase with inflation. A’s annual expenses will total $180,000 initially and will increase with inflation of 2% annually. He now has 1,000,000 and he expects he will live 20 years and he expects he can donate $1,000,000 if he die. What is the return requirement?

9.2%…but dnt ask me how to calcuate,…

Probably a flaw in the reasoning here , but what the heck:

Initial requirement is 8% ( ( 180,000 - 100,000 ) / 1,000,000) . The expenses rise at 2% p.a. So 2% of 180,000 is 3,600 or 0.36 % of 1,000,000 . It may not be incorrect to say that the return requirement is 8% + 0.36 % = 8.36 % or (1.08 * 1.0036 ) - 1 = 8.39 %

PV = 1,000,000 FV = -1,000,000

PMT = -(80,000)(1.02) N = 20 CPT -> i/y


I think when it says initial expense _will b_e we can assume it is 180,0000 not 183,600 . Am I right ?

What is the answer Francis?

Is there an actual CFAI answer to this? in order to calculate you would need to incorporate the increase in inflation for expenses less the fixed pension payment each year, so the actual required return would change y/y. The average of all these returns would be the actual answer, but you would need to incorporate each cash flow independently to find the answer. for example:

the CF for the first year’s expenses of 183,600 less fixed pension of 100,000 is 83,600 which translates to a return of 8.36%

The CF for the next year’s expenses of 187,272 less fixed pension of 100,000 is 87,272 which translates to a return of 8.73%

because all expenses grow at inflation, but a portion is paid through a fixed pension, I don’t see how a simple formula could capture the necessary return needed.

since the 2% is applied to the total expenses but the portion actually paid by the portfolio is only approx 44% (currently) of the total, the inflation rate addition would need to be higher than the actual 2% (maybe something like 2% * 1.56 = 3.12%?) so simple addition is 8% + 3.12% = 11.12%

Maybe I’m overthinking this.

You are right FinNinja . The expenses grow geometrically at 2 pct so it is not a simple calculation . But I had another idea , what Is we look at it as two return streams , one which is classical CFA , I.e. no pension income , initial value 1,000,000 final value 1,000,000 , pmt 183600 , N 20 years , inflation 2 % . Then that return is 18.36 % plus 2% inflation in classical CFA style. . Now include a second return stream which is paying you 100,000 per year on the same base , but without inflation adjustment . That gives 10% per year. The two return streams are each independent , in correlated , weighted at 100% each , so you get a net of 10.36 % .

So I am confused which is the right way? What is the flaw in my reasoning ?

183600? or 83600?

183600 because that is the independent expense stream we are used to in private wealth management classical CFA style .

classic CFA style takes income - expenses -> income = 100, expense = 183.6 so you would have a stream of 83.6 that you needed each year,

or am I missing sth?

The 2nd cashflow does not make sence. If you seperate it, the 2nd one just a cash inflow 100k anually in 20 years. PV=0 and FV=0. How Y become 10%? (the income is not generated by 1mil portfolio)

But classical CFA usually ( never seen it otherwise) lets the income too rise at same inflation rate as expense , leaving the return requirement essentially based on single stream. in this example that would be 83600 )

Here we have one growing at 2% classically , and another growing at 0%. So to make sense of it as a single number , you should do two independent ( mental accounting ??? ) calculations and combine the result

PV=-1m , FV=+1m,PMT=0.1m , N=20, I/Y=10%

I like where your heads at Jana, however I think the return for the 183600 needs to be adjusted continually at inflation. So the 18.36% real return would be 1.1836*1.02 - 1 = 20.73% nominal; then reducethat return by the 10% from the fixed pension = 10.73%. What do you say?


because you used 1m PV portfolio to fund the living expense (18360- showed as cash outflow PMT) and left the remain 1mil as FV, the first calculation is correct.

the 2nd calculation is wrong. 100k is not generated by 1mil port. It is other income (pension) and not related to the portfolio.

It can be expressed as PV=0, PMT =100k, FV=?, Y=?

Because you left all 1ml as FV in 1st cal, the second cal FV should be 0. And it does not make sense in any way…

What’s the answer?

Does an approximation exist in the calculation? The annual payment from the portfolio is changing:

1st year: 80,000 2nd yaer: 83,600 3rd year: 87,272 …

If we have to give an answer, it is something like

Ri = 1,0836*1.02^i -1


Promise this is my final screwed up post , hopefully entertaining !

Still not willing to give up my two stream theory !

Ok lets say I still look at it as two streams , one generates income of 0.1836 million , with present value 1m , final value 0m , N=20 and another that absorbs income of 0.1m with present value of 0 , final value of 1m ( because that is what we want it to be ) , N=20


1st stream : PV=1m, PMT= (0.1836m) , N=20 , FV = 0 CPT I/Y = 17.6486 %

2nd stream: PV = 0m , PMT=0.1m , N=20 , FV=1m , CPT I/Y= Negative 8.189 pct

makes sense because instead of generating income we are absorbing pension income to reduce the final value to only 1m , instead of 2m it would have reached otherwise.

Now if we add each term and add inflation in classical CFA i.e. 17.6486 - 8.189 + 2 = 11.46% approximately , which is very close to Fin’s number of 11.12% . Ha !