Private Wealth Management puzzles, solved step by step

51%. The simplest route is the opposite event. Each has a 70% chance of not reaching 90, so the chance that neither does is 0.7 times 0.7, which is 49%. At least one reaching 90 is everything else: 100% minus 49%, or 51%. For a couple, a plan that ends at 90 fails more often than it holds.

Why not add the two 30% chances?

Think of two friends each with a 30% chance of turning up to dinner. Adding gives 60%, but that double counts the evenings when both arrive. With more people the same adding would soon pass 100%, which is impossible. For an at-least-one question, compute the chance that nobody does it and subtract from one. Neither reaching 90 has chance 0.49, so at least one does with chance 0.51.

What does 51% change in a retirement plan?

It changes the horizon. A couple's money has to last for the longer of two lives, not the average life, and the longer of two lives is usually well past either single estimate. If each spouse alone has a 30% chance of reaching 90, a plan that runs out at 90 leaves the surviving spouse short in about half of all outcomes. That is why planners model the joint horizon and a longevity bufferMoney or income set aside to cover the years if the client, or the surviving spouse, lives longer than the planning age. rather than a single life expectancy.

Name the assumption. Spouses are not truly independent: they share a household, a diet and often a doctor, and some studies of couples suggest their lifespans move together. The 30% figure itself would come from a mortality table that has to be checked for the client's circumstances. The puzzle is about the method, not the actuarial figure.

No: for a 20-year life the return is about 3.4% a year. He pays 100 and receives 7 a year for 20 years, 140 in total, of which 100 is his own capital coming back and only 40 is return. The rate that makes 20 payments of 7 worth 100 today is 3.44%. The payout rate is not a yield, because the capital is never returned.

Why is a 7% payout not a 7% return?

Imagine lending a friend Rs 1 lakh and being repaid Rs 7,000 a year until one of you dies, with nothing more after that. The Rs 7,000 is partly interest and partly your own Rs 1 lakh coming back in slices. An annuity payout rate includes the return of the buyer's own capital, because nothing is paid back at death; only the interest part is return. A bank deposit paying 7% hands back the Rs 1 lakh at the end as well; the annuity does not.

How does the return change with how long he lives?

Lifespan is the whole trade. If he dies after 10 years the return is about -6.0%, negative, because he gets back only 70; at 20 years it is 3.4%; at 30 years it reaches 5.7%. An annuity is insurance against living long, priced so that the insurer wins on clients who die early. That is its purpose, and the client should buy it for longevity protectionIncome that continues however long the client lives, so the risk of outliving savings passes to the insurer., not for yield.

Add what a private banker would. Annuity income is usually taxed as income in the year received, and payouts are fixed in rupees, so inflation erodes them. The 7% here is an illustration; real annuity rates depend on age, the option chosen and the current rate environment, and have to be taken from a current quote.

His rupees stay flat and their value falls by more than half. Taking the full 8% leaves the capital at Rs 1 crore every year. But prices rise 6% a year, so after 15 years they are 2.40 times higher. His capital is then worth about Rs 41.7 lakh in today's rupees, and his Rs 8 lakh income buys what Rs 3.34 lakh buys now. Only the real return, about 2%, was safe to spend.

Where does the damage come from if he never touches the capital?

A landlord who spends every rupee of rent and never raises it feels fine for a year or two. Fifteen years later the same rent buys half the groceries, and the house has not grown to compensate. Of the 8% return, 6 points are not income at all: they are the amount the capital must grow just to hold its purchasing power. Spending the full nominal return spends that inflation cushion too.

What could he safely spend instead?

Only the real return. The capital has to grow 6% a year to stand still, so of the Rs 8 lakh, only about Rs 2 lakh is spendable in the first year, and that withdrawal can rise with inflation each year after. The sustainable withdrawal is the return minus inflation, roughly 1.9% in real terms, not the headline 8%. That is a hard conversation, because the retiree's income drops from Rs 8 lakh to Rs 2 lakh on day one, but it is the honest arithmetic.

The limits: most retirees also spend some capital over a finite lifetime, so the true plan sits between these two extremes. And real portfolios do not earn a smooth 8%; a few bad years early in retirement do far more damage than the same years later.

About age 70.4 with no discounting, and about age 78.7 at 7%. Rs 50 lakh divided by Rs 40,000 is 125 months, a little over ten years. At 7% a year each later payment is worth less today, and the present value of the pension only reaches Rs 50 lakh after about 225 months. So the pension wins only if the retiree, or whoever it continues to, lives beyond the late seventies.

Why is the undiscounted answer too generous to the pension?

A friend offers you Rs 1,000 today or Rs 100 a month for a year. Even though Rs 1,200 is more, you would think about what the Rs 1,000 could do meanwhile. A rupee of pension received at 75 is worth less than a rupee in hand at 60, because the lump sum could have been invested for those fifteen years. Discounting at the rate the lump sum could earn puts both choices on the same footing.

What decides the choice beyond the break-even age?

Three things the formula does not see. Longevity: the pension is insurance against living long, which is exactly the risk a lump sum cannot cover; a life table for the client's age and health says how likely age 79 is. The rate matters most: the higher the return the client could earn on the lump sum, the later the break-even, and at 10% a year the pension's present value never reaches Rs 50 lakh at all, however long he lives. Then the details: whether the pension continues to a spouse, whether it rises with inflation, and how each option is taxed.

The ceiling explains the last point. At 7% the whole infinite stream is worth Rs 40,000 / (0.07 / 12), about Rs 68.6 lakh; at 10% it is worth Rs 48 lakh, less than the lump sum. Check the payer's strength too: a pension is only as good as the promise behind it.

Twenty years of spending, Rs 60 lakh short of 25 times. Rs 1 lakh a month is Rs 12 lakh a year, and Rs 2.4 crore over Rs 12 lakh is 20. Twenty-five times spending would be Rs 3 crore. Put another way, the client would withdraw 5% of the corpus in year one, against the 4% that the 25 times marker implies.

Why express a corpus in years of spending?

A client hears Rs 2.4 crore and feels rich; a client who hears twenty years of your current life hears a question: and after that? Dividing the corpus by annual spending turns an abstract balance into a length of time, which is what a retirement decision is actually about. It also removes the units: the same arithmetic works for a Rs 50 lakh corpus or a Rs 50 crore one.

Where does 25 times come from, and how firm is it?

Twenty-five times spending is the same thing as withdrawing 4% a year, the 4% ruleA planning guideline from US research by William Bengen in the 1990s: withdraw 4% of the starting portfolio, raised each year for inflation, and the money historically lasted about 30 years. from US retirement research. It is a starting point, not a law, because it came from one country's market history and a 30 year retirement. Higher inflation, longer lives and early retirement all argue for a larger multiple, and planners in India often use one. Say the marker, then say its limits.

Does twenty years of spending mean the money runs out in twenty years?

Only if the corpus earns exactly inflation. If it earns 2% a year above inflation and the client withdraws Rs 12 lakh in today's money at the start of each year, the money lasts about 25 years. Years of spending is a zero-real-return yardstick, so it is conservative when real returns are positive and optimistic when they are negative. That is why it is the right first number to say, and the wrong last one.

Bad year first ends at about Rs 56.1 lakh; bad year last ends at Rs 70 lakh. Left untouched, both pots would reach Rs 117.1 lakh whatever the order. With Rs 10 lakh withdrawn every year, an early 20% fall shrinks the pot the good years grow from, so each good year earns Rs 6 to 7 lakh instead of Rs 10 lakh. Same average return, Rs 13.9 lakh apart.

Why does order not matter without withdrawals?

Multiplication does not care about order: 0.8 x 1.1 x 1.1 x 1.1 x 1.1 is the same as 1.1 x 1.1 x 1.1 x 1.1 x 0.8, and both give 1.1713. A pot left alone ends at Rs 117.1 lakh either way. Withdrawals turn the problem from multiplication into multiplication and subtraction, and subtraction does care about order. Taking a fixed Rs 10 lakh from a pot that has just fallen removes a bigger share of it, and that share never gets to recover.

Where exactly does the Rs 14 lakh go?

Follow the rupees. Both retirees lose Rs 20 lakh in their bad year, because in both cases the fall hits a Rs 100 lakh pot. The difference is what the four good years earn: bad-last grows Rs 100 lakh by 10% four times, Rs 40 lakh in all, while bad-first grows a pot that starts at Rs 70 lakh and keeps shrinking, earning only about Rs 26.1 lakh. Each good year earns less than the Rs 10 lakh he withdraws, so the pot keeps falling even in good years.

This is sequence-of-returns risk, and it is why the years just before and just after retirement matter most. Planners respond with a cash or short-bond bucket that funds a few years of withdrawals, so the retiree is not forced to sell growth assets after a fall. Say the mechanism; the right size of such a bucket depends on the client.

About 47 years, nearly double the 25. At 3% real the corpus earns Rs 9 lakh in the first year, three quarters of the Rs 12 lakh drawn, so it shrinks slowly. Solving the annuity, 3 crore = 12 lakh x (1 - 1.03 to the minus n) / 0.03, gives n of about 46.9 years with year-end withdrawals. It would last for ever only at Rs 400 lakh.

Why does 3% nearly double the corpus's life?

Think of a water tank you draw 12 buckets a year from, while a tap refills 9 buckets. The tank drains at 3 buckets a year at first, not 12. A real return refills the corpus each year, and early on the refill covers most of the spending, so a small return stretches a corpus far more than intuition expects. As the corpus shrinks the refill shrinks too, which is why the curve steepens near the end.

How do you check the answer without solving anything?

Use the perpetuity as a ceiling. Rs 12 lakh a year for ever at 3% needs 12 / 0.03 = Rs 400 lakh. The client holds three quarters of the for-ever corpus, so the money runs out, but late: the answer must sit well above 25 years and below infinity. Nearly 47 years passes that test.

Then say the limitation. A steady 3% real return does not exist; the order of good and bad years matters, and a bad run early in retirement shortens the life of the corpus even if the average holds. Use 47 years as the arithmetic, not a promise.

About Rs 48 lakh more, 11.6% on top of the 30-year corpus. At a 2% real return Rs 18 lakh a year for 30 years needs Rs 411 lakh today, and for 35 years Rs 459 lakh. The five extra years are Rs 90 lakh of spending, but they come last, so they cost only about Rs 48 lakh today.

How do you price 30 and 35 years of spending?

A retirement corpus is the price today of a stream of future spending. Paying for a child's school fees years ahead costs less today than the fees themselves, because the money set aside earns something while it waits. The corpus needed is the present value of every year's withdrawal at the real return, so later years cost less today than earlier ones.

Why is the extra much less than Rs 90 lakh, and why is that not the whole story?

The years 31 to 35 are 30 to 34 years away, and at 2% real each rupee then costs only about 55 paise today. Adding years at the end of retirement is cheaper than it looks, which is why the corpus rises about 12% for a 17% longer retirement. The lower the real return, the closer the extra gets to the full Rs 90 lakh.

But if the five extra years come from retiring five years earlier rather than living longer, the client is hit twice: he needs the bigger corpus, and he has five fewer years of saving and growth to build it. That double effect, not the Rs 48 lakh alone, is what makes early retirement expensive. The 2% real return is an assumption; at 0% the extra would be the full Rs 90 lakh.