Private Wealth Management puzzles, solved step by step
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- Hard
- 30
007A husband and wife are both 60. Each has a 30% chance of living to 90, independently of the other. What is the chance that at least one of them reaches 90?Wealth management
Try it first
Quick instinct: the chance at least one of them reaches 90?
Show the worked solution
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.
With a 30% chance each of reaching 90, the couple's outcomes split into both at 9%, only one at 21% each way and neither at 49%, so at least one of them reaches 90 with a 51% chance. 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.
The relationshipp each spouse's chance of reaching 90, here 0.3 (1-p)^2 the chance neither reaches 90, assuming independence What it says in wordsThe chance at least one survives is one minus the chance that both do not.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.
Where candidates lose it
Two answers lose the room: 60%, from adding, and 9%, from multiplying. The first counts the both-survive case twice; the second answers a different question, the chance both survive.
The deeper miss is stopping at 51% without the planning point. The interviewer wants to hear that a couple's horizon is the second death, which pushes the plan longer.
What the interviewer asks next
- What is the chance that exactly one of them reaches 90?
- If the wife's chance is 40% and the husband's 25%, what is the chance at least one reaches 90?
- How would you set the planning age for this couple's retirement income?
033A retiree has Rs 1 crore earning 8% a year. He withdraws Rs 8 lakh, the full 8%, at the end of every year, while inflation runs at 6%. What happens to the real value of his income and his capital over 15 years?Indian wealth management
Try it first
After 15 years, roughly what is his Rs 1 crore worth in today's rupees?
Show the worked solution
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.
Withdrawing the full 8% keeps the retiree's capital at Rs 1 crore in rupees, but in today's rupees it falls every year to about Rs 41.7 lakh by year 15. His fixed Rs 8 lakh income shrinks to Rs 3.34 lakh of today's purchasing power. The relationship1.06^t how much prices have risen after t years of 6% inflation Rs 1 crore the nominal capital, kept flat by withdrawing the full return What it says in wordsA fixed number of rupees is worth less every year by the amount prices have risen.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.
Where candidates lose it
The trap is saying nothing happens because the capital is untouched. It is untouched in rupees, and the question is deliberately worded in real terms to see whether you notice the difference.
The second loss is saying the capital falls but not by how much. Give the {(1 + P33['inf']) ** 15:.2f} price factor, the Rs {P33['real_cap'][-1]:.0f} lakh real value and the one-line fix: spend the real return, not the nominal one.
What the interviewer asks next
- How many years until his real capital halves at 6% inflation?
- If he wants his income to keep pace with inflation for 25 years, roughly what first-year withdrawal is sustainable?
- How does a bad market in the first three years of retirement change this picture?
083A retired client has a Rs 3 crore corpus and spends Rs 12 lakh a year in today's money. With a 0% real return the money lasts 25 years. How long does it last if the corpus earns a 3% real return?Wealth management
Try it first
Your best guess for how long it lasts at 3% real.
Show the worked solution
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.
The relationship300 the corpus in Rs lakh 12 yearly spending in today's money, Rs lakh 0.03 the real return n years the corpus lasts What it says in wordsThe corpus equals the present value of the withdrawals; solve for the number of years that makes them match.At a 0% real return the Rs 3 crore corpus runs down in a straight line and is empty at year 25, while at 3% real it bends and lasts 46.9 years, because the return refills most of each year's withdrawal early on. 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.
Where candidates lose it
Most candidates add a few years, answering 28 or 30, because they apply the 3% as a small correction to 25. They miss that the return is earned on the whole corpus while spending comes out gradually.
The opposite slip is saying it lasts for ever. Check it against the perpetuity: for ever needs Rs 4 crore at 3%, and the client has Rs 3 crore.
What the interviewer asks next
- What real return makes Rs 3 crore last for ever at Rs 12 lakh a year?
- What if withdrawals are taken at the start of each year instead?
- How does a bad first five years change the answer, even with the same average?