Portfolio Management puzzles, solved step by step

About Rs 80.5 crore. Vacancy of 8% takes Rs 10 crore of gross potential rent down to Rs 9.2 crore collected. Operating costs of 30% of that are Rs 2.76 crore, leaving net operating income of Rs 6.44 crore. Divide by the 8% cap rate: 6.44 over 0.08 is Rs 80.5 crore, or 12.5 times the income.

Why does the cap rate price net operating income and not the rent?

Think of buying a shop that a friend runs. You would not pay for the sales the shop could make if every shelf sold out; you pay for what is left after empty days and the electricity bill. A cap rate is net operating income divided by value, so the value is only ever as good as the income that actually reaches the owner after vacancy and running costs. Gross potential rent is the ceiling, not the income.

What are the lines between gross rent and value, in order?

Say them as a ladder. Gross potential rent is what a full building earns at the rent roll. Take off vacancy and bad debt to get effective gross incomeThe rent a building actually collects after vacancy and unpaid rent, before any running costs.. Take off operating expenses, such as maintenance, insurance, property tax and management, to get net operating income. Every line above NOI moves the price by 12.5 times its size at an 8% cap rate, so a small leak near the top is a large leak in value. One extra point of vacancy costs Rs 0.1 crore of rent, Rs 0.07 crore of NOI, and Rs 0.875 crore of value.

For an exit value, the same arithmetic runs on the NOI expected in the sale year and an exit cap rate, which analysts often set a little above the entry cap rate to allow for an older building. Say which cap rate you are using and why, and say that capital spending such as a new roof sits below NOI and is not captured by this shortcut.

Only 20%. Take 1,000 applicants. 40 will default and the model flags 36 of them. 960 are good and the model wrongly flags 15% of them, 144 people. The flagged pile holds 180, and 36 of those default: one in five. The model is good at catching defaulters, but defaulters are so rare that false alarms outnumber them four to one.

Why is 90% the wrong answer when the model catches 90% of defaulters?

Picture a smoke alarm that always sounds when there is a fire and also sounds now and then for burnt toast. Because toast burns far more often than houses do, most alarms in a year are toast. A flag's meaning depends on how common the thing it looks for is: when defaulters are 4% of applicants, even a modest false alarm rate on the other 96% produces more wrong flags than right ones. The 90% is the chance a defaulter gets flagged; the question asks the reverse, the chance a flag is a defaulter.

Why count people instead of using the formula?

Bayes' rule gives the same answer, but natural frequencies are faster to say and harder to get wrong under pressure. Turn every percentage into a count of people out of a round number, and the answer is simply the flagged defaulters over everyone flagged. The formula version is 0.04 x 0.9 over (0.04 x 0.9 plus 0.96 x 0.15), which is 0.036 over 0.18, or 20%. Offer it as the check after the counts.

Then say what a lender does with it. A flag at 20% is a reason for a closer look, not a rejection. The cleared pile, by contrast, holds only 4 defaulters in 820, about 0.5%, against 4% before the model, so the model is most useful for waving through the safe majority. Cutting the false alarm rate from 15% to 5% would lift the flagged default share to about 43%.

About Rs 19.5 lakh less, and about 62% of that gap opens in the last five years. Rs 48 lakh paid in grows to about Rs 1.73 crore at 11.5% net in the direct plan and Rs 1.54 crore at 10.5% in the regular plan. The gap is Rs 7.4 lakh after 15 years, then more than doubles in five. The fee is a percentage of a balance that is largest at the end.

Why does a one-point fee gap cost so little early and so much late?

Think of a small leak in a water tank that is being filled from a tap. When the tank is nearly empty, the leak loses a trickle; when it is almost full, the same size of hole loses a lot more, because more water is pressing on it. A fund's costs are a percentage of the balance, and a monthly plan's balance is tiny in the early years and large in the last few, so most of the rupee cost falls at the end. On top of that, every rupee lost early would itself have compounded, so the gap grows faster than the balance.

How do you estimate the gap without a spreadsheet?

Use the future value of a monthly annuity at each net rate and subtract. The gap is about 13% of the regular plan's final value, which is far more than the one-point fee suggests, because the fee compounds for the whole life of every instalment. A quick mental check: the average rupee is invested for about ten years, and 1% a year for ten years is roughly 10% of the pot. The exact figure is a little higher because the early instalments sit in the fund for up to twenty years.

State the assumptions: a steady 12% gross return every year, costs that never change, and no tax. Real returns arrive unevenly, and expense ratios differ by scheme and change over time, so confirm the current figures for any real scheme. The limitation does not change the shape: whatever the numbers, the drag on a monthly plan is back-loaded.

About 8.2%. The bond pays 7 a year and also rises from 95 to 100 by maturity, which is about 1 point a year over five years. That is 8 a year of return on money that averages about 97.5 invested, and 8 over 97.5 is 8.21%. The exact yield with annual coupons is 8.26%, so the shortcut is within a few hundredths of a point.

Where does the return on a discount bond come from?

Buy a Rs 100 gift voucher for Rs 95 that also pays you Rs 7 each year until it can be cashed at full value in five years. You collect the Rs 7 every year, and you also pocket the Rs 5 discount at the end. A discount bond's yield is its coupon plus the discount it recovers as the price is pulled to par, spread across the years to maturity. Spread evenly, that is 1 point a year, so the bond earns roughly 8 a year.

Why divide by the average price rather than 95?

The amount you have invested is not fixed at 95: in this rough picture the bond's value drifts up towards 100 over the five years, so the capital at work averages about 97.5. Dividing by the average price corrects most of the error from spreading the discount in a straight line. Dividing 8 by 95 instead gives 8.42%, too high. As a check, a bond priced at 8.21% comes out at 95.21, within a fraction of 95.

Know when the shortcut drifts. It is close for short bonds near par and gets worse for long maturities or deep discounts, where the true discounting curve bends away from a straight line. For a quick answer in the room, 8.2% with the one-line derivation is what the interviewer is after; then say that the exact figure is a touch higher.

Roughly 12 crore 5G phones a year, on stated assumptions. Take about 60 crore smartphone users out of 140 crore people. If they replace their phone every 4 years, that is 15 crore replacements, plus about 1 crore first-time buyers, so 16 crore phones. If three quarters of new phones sold are 5G, the answer is about 12 crore. A household check, one phone every two years across 30 crore households, gives 15 crore, close enough.

Why start from users and replacement, not from the population?

Think of how many school shoes a town buys in a year. The number of children matters, but the answer is set by how often each child outgrows a pair. In a market where most people already own the product, yearly sales are the user base divided by the replacement cycle, plus a smaller flow of first-time buyers. Population only tells you the ceiling; the cycle turns the stock of users into a yearly flow.

Which assumptions should you say out loud, and how do you check them?

Every number here is an assumption to be stated, not a fact to be quoted: the user base, the cycle, the first-time flow and the 5G share. The replacement cycle deserves the most care because a one-year change in it swings the answer by between a fifth and a third; the population barely matters by comparison. A three-year cycle gives 15.75 crore and a five-year cycle 9.75 crore. The 5G share is the other moving part, because it changes fast from one year to the next; ask which year the interviewer means.

Then check from a different direction. With about 30 crore households, one new phone per household every two years gives 15 crore phones, close to the 16 crore from the user build. Two methods that land near each other are more convincing than one precise-looking number. For a real estimate, replace every assumption with published industry shipment data and confirm the current figures.

If the interviewer wants value rather than units, multiply by an assumed average selling price and say that 5G phones are skewed to the middle and upper price bands, so the price assumption needs as much care as the cycle.

About 47%, close to a coin toss. One manager beats the benchmark five years running with probability one half to the fifth, or 1 in 32. The chance that none of the 20 does it is 31/32 to the 20th power, about 53%. So the chance that at least one does is about 47%. In a crowd of unskilled managers, a five-year streak is almost as likely as not.

Why is a five-year streak weak evidence of skill?

Put 20 people in a room and ask each to call five coin tosses. Somebody calling all five correctly would look gifted, but with that many people, somebody usually does. The chance of a streak for one named person is small, but the chance that someone in a crowd produces one is large, and fund selection looks at the crowd. An investor who screens a category for managers with five straight winning years is sampling exactly the lucky tail of this distribution.

How do you compute at least one without adding up cases?

Use the complement. The chance that at least one of many independent trials succeeds is one minus the chance that every trial fails. Each manager fails the streak with probability 31/32. All 20 fail with probability (31/32) to the 20th, which is about 0.53. One minus that is 0.47. The expected number of streaks is 20 over 32, or 0.625, which is a separate and also useful number: less than one streak on average, yet close to even odds of seeing at least one.

The limitation runs in two directions. Real managers are not independent, since many hold similar stocks, which makes streaks cluster and the calculation rougher. And some managers do have skill; the point is only that a streak on its own cannot tell the skilled from the lucky. That takes longer records, consistent process and a view on why the edge should persist.

About 112 rupees to the dollar. Relative PPP says the higher-inflation currency loses value at the inflation gap. Each year the rate rises by 1.05 over 1.02, about 2.94%. Over ten years that compounds to a factor of 1.336, and 84 times that is about 112.2. The simple 3% shortcut gives 112.9, close enough in the room.

Why should higher inflation weaken a currency?

If a cup of tea costs Rs 20 this year and Rs 21 next year while the same cup abroad goes from 1 dollar to 1.02, the rupee has lost more buying power than the dollar. If money is to buy roughly the same basket in both countries over time, the exchange rate has to move by the gap between the two inflation rates. That is relative purchasing power parity: it does not say what the rate should be today, only how it should drift.

What does this mean for an investor holding assets abroad?

For a rupee-based investor, a US asset earns its local return plus the rupee's depreciation. A US bond yielding 3 points less than an Indian bond is not worse on this reasoning, because the expected currency drift gives back roughly the same gap. Measured the other way, the rupee loses about 25% of its dollar value over the ten years, which is what a dollar-based investor in Indian assets must earn back through higher local returns.

Say the limitation plainly. PPP is a weak guide over one or two years: capital flows, interest rate moves and risk appetite can push the rate far from this path and hold it there. Over a decade the evidence for the drift is better, but it is a tendency, not a rule. The inflation rates themselves are assumptions; for a real view, confirm current figures and consider what each central bank is targeting.

About 0.6 points a year here. Cash drag is the cash weight times the gap between what equities and cash earn: 0.08 x (14 minus 6.5) is 0.6. The fund returns 13.4% instead of 14%. The drag is largest in strong markets and turns into a cushion when equities fall: in a year equities lose 10%, the same cash saves about 1.3 points.

Why is the cost the gap, not the whole equity return?

Suppose you keep a little of your salary in a savings account instead of a higher-paying deposit. You have not lost the whole deposit rate on that money, only the difference between the two rates. Cash in an equity fund still earns something, so the return given up is the cash weight times the gap between the equity return and the cash return. Here that is 8% of 7.5 points, which is 0.6 of a point for the whole fund.

Why does it matter most in strong markets?

The drag scales with the equity return. In a 30% year the same 8% of cash costs about 1.9 points, in a 14% year 0.6, and in a year equities lose 10% it adds about 1.32 points. That is why cash drag shows up most in the years a fund is judged hardest: a fund that trails its benchmark in a strong rally often simply held more cash than the index, which holds none.

Funds hold cash for real reasons: redemptions, new money not yet invested, and dry powder for opportunities. The implementation answer is not zero cash but equitising it, for example by holding index futures against the cash balance so that the fund keeps market exposure while holding liquidity. Say that as the practical close, and note that futures carry their own costs and margin needs.

Volatility is about 11.3%, and equities contribute about 93% of the risk. The variance is 0.36 x 324 plus 0.16 x 36 plus 2 x 0.6 x 0.4 x 0.1 x 18 x 6, which is 116.64 plus 5.76 plus 5.18, or 127.58. The square root is 11.3%. Equities' own term plus half the cross term is 119.2, which is 93% of the total.

Why does 60% of the money carry more than 90% of the risk?

Picture a household with two earners, one a salaried clerk and one a commission-only salesperson. The salesperson may bring in 60% of the money but almost all of the month-to-month swings. Risk contribution depends on weight times volatility, and variance squares it, so a sleeve three times as volatile with more capital swamps the other. Equities' variance term is 0.36 times 324, or 116.6; bonds' is 0.16 times 36, or 5.76. Twenty to one before the cross term.

How do you split the risk between the two sleeves?

Give each asset its own variance term plus half of the cross term. An asset's risk contribution is its weight times its covariance with the whole portfolio, and the contributions add to the total variance. For equities that is 0.6 x (0.6 x 324 + 0.4 x 10.8), which is 0.6 x 198.72, or 119.23. Divided by 127.58, that is 93.5%. Bonds take the remaining 6.5%.

This is the arithmetic behind risk parity, which sizes each sleeve so that the risk contributions are equal rather than the capital. To give bonds half the risk here, the portfolio would hold roughly three times as much bond capital as equity, and often borrow to lift the return. The limitation: correlation is not stable. In some years stocks and bonds fall together, and the 7% can grow quickly.

Stake 10% of capital each time; staking 20% drives long-run growth to about zero. For an even-money bet the growth-maximising, or Kelly, stake is p minus q, 0.55 minus 0.45, which is 10%. That earns about 0.50% a bet in log terms. At 20%, 0.55 x log 1.2 plus 0.45 x log 0.8 is about zero, so twice the stake gives no growth at all despite double the expected profit per bet.

Why does betting more of a winning edge ever make you poorer?

A shopkeeper who wins more often than she loses still goes under if one bad month can wipe out half the shop, because rebuilding from half takes a doubling. Wealth compounds, so what matters over many bets is the average log return, and a loss hurts the log more than an equal-sized gain helps it. With a small stake, the edge dominates. With a large stake, the losses' extra damage grows faster than the edge, and past a point it wins.

How do you find the 10% and check the 20%?

Write growth per bet as g(f) = p log(1 + f) + q log(1 minus f), set its slope to zero, and the answer for even money is f = p minus q. The Kelly fraction for an even-money bet is simply the edge: 55% minus 45% is 10%. At 10%, g is 0.55 x 0.0953 minus 0.45 x 0.1054, about 0.50%. At 20% it is 0.55 x 0.1823 minus 0.45 x 0.2231, about -0.014%: essentially zero, and in fact a touch below it. Over 100 bets at the Kelly stake, capital grows by a factor of about 1.65 on the typical path; at 20% it goes nowhere.

Practitioners usually bet a fraction of Kelly. Half Kelly, 5%, keeps about 75% of the peak growth with half the volatility, and it protects against the real problem: in markets you never know p exactly. If the true edge were 52.5% rather than 55%, the 10% stake would already sit near zero growth.