Mutual Fund Mastery puzzles, solved step by step

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All topicsCompounding and time value9Statistics, correlation and diversification8Bond maths and duration10Performance measurement and returns8Costs and fee drag8Valuation riddles11Logic and numeracy brainteasers6Estimation and market sizing7Probability and expected value8NAV, units and fund mechanics7Risk, volatility and drawdown8Behavioural traps6Withdrawals and after-tax arithmetic4
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  1. 025Without writing an equation: a company has an enterprise value of Rs 1,000 crore, debt of Rs 300 crore and cash of Rs 80 crore, with 20 crore shares. What is each share worth? Explain it using a house and its mortgage.Valuation riddlesWarm upPIMCONew York · 2023

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    What is each share worth?

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    Rs 39 a share. Enterprise value is the house: Rs 1,000 crore for the business itself. The debt is the mortgage, Rs 300 crore owed to the bank, which leaves the owner Rs 700 crore of the house. The cash is money in the drawer that the owner keeps on top, Rs 80 crore. Together that is Rs 780 crore for the shareholders, which over 20 crore shares is Rs 39 each.

    What does the house stand for, and what does the mortgage stand for?

    A family owns a house worth Rs 1 crore with a Rs 30 lakh home loan, and keeps Rs 8 lakh in a drawer. If they sold everything and settled up, they would walk away with the house's value, less the loan, plus the drawer: Rs 78 lakh. Enterprise value is the value of the operating business, the house; lenders are paid first out of it, and whatever cash the company already holds belongs to the shareholders on top. Scale the same family up a thousand times and you have this company.

    Enterprise value is the house; equity is what the owner keepsMortgage: debt 300Owner's part700The house: enterprise value 1,000Cash 80Money in the drawer+What the shareholders own1,000 - 300 + 80= Rs 780 croreover 20 crore shares= Rs 39 a shareAll figures Rs crore except per share
    The business is worth Rs 1,000 crore, lenders hold Rs 300 crore of it, and the shareholders own the remaining Rs 700 crore plus Rs 80 crore of cash, Rs 780 crore in all or Rs 39 a share.

    Why is cash added back rather than subtracted?

    Because enterprise value was built to leave it out. A buyer of the whole company would pay for the business and inherit the cash, so the price of the business alone nets the cash away. Going from enterprise value back to equity, you reverse that step: take off what the lenders are owed and add back the cash the shareholders already own. Subtracting cash as well, the common slip, gives 1,000 minus 300 minus 80, Rs 620 crore, or Rs 31 a share, which hands the drawer to the bank.

    The relationship
    Equity=EV−Debt+Cash=1,000−300+80=780,78020=Rs 39\text{Equity} = EV - \text{Debt} + \text{Cash} = 1{,}000 - 300 + 80 = 780, \qquad \frac{780}{20} = \text{Rs } 39
    EVenterprise value, the operating business, Rs 1,000 crore
    Debtwhat is owed to lenders, Rs 300 crore
    Cashcash the company holds, Rs 80 crore
    What it says in wordsShareholders own the business less what lenders are owed, plus the cash in hand, shared across the shares.

    Add the refinements an interviewer will probe. Anything else with a claim ahead of the ordinary shareholders comes off too, such as preference shares or a minority partner's share of a subsidiary, the way a second loan on the house would. And the share count should include options and convertibles that are likely to become shares. The limitation of the analogy is that a house is easy to price on its own; a business's enterprise value is itself an estimate, and every rupee of error in it lands on the equity.

    Where candidates lose it

    The common slip is subtracting cash along with debt because both sound like balance sheet adjustments, giving Rs 31. It treats the company's own cash as if it belonged to someone else.

    The other slip is dividing enterprise value by the shares and answering Rs 50, forgetting that lenders are paid before shareholders. The house picture prevents both: the mortgage comes off, the drawer stays.

    What the interviewer asks next

    • How would Rs 50 crore of preference shares change the answer?
    • If the company uses Rs 80 crore of cash to repay debt, what happens to enterprise value and to the share price?
    • Why do analysts value a bank on equity rather than enterprise value?

    Asked at PIMCO, Financial Institutions Group (FIG), New York, 2023 (Wall Street Oasis): How do you get from enterprise value to equity value without an equation

  2. 071A stock trades at 20 times earnings and pays out 40% of its profit as dividends. What is its dividend yield?Valuation riddlesWarm upHoulihan LokeyChicago · 2026

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    What is the dividend yield?

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    2%. Flip the P/E to get the earnings yield: a P/E of 20 means the company earns 1/20, or 5%, of its share price each year. It pays out 40% of those earnings, so the dividend is 40% of 5%, which is 2% of the price. A Rs 400 share earning Rs 20 pays Rs 8, and 8 / 400 is 2%.

    How do you get from a P/E to a yield?

    If a flat costs Rs 20 lakh and earns Rs 1 lakh a year in rent, it costs 20 years of rent, and the rent is 5% of the price. The same flip works for shares. A P/E of 20 means the share costs 20 years of earnings, so the earnings yieldEarnings per share divided by the share price: the P/E turned upside down. A P/E of 20 is an earnings yield of 5%. is 1/20 = 5%. Only part of those earnings reach the shareholder as cash. With a payout ratioThe share of profit a company pays out as dividends. The rest is retained and reinvested in the business. of 40%, the dividend is 0.4 x 5% = 2% of the price, and the other 3% stays in the company.

    Dividend yield = payout ratio x earnings yieldEarnings yield = 1 / P/E = 1 / 20Paid out: 2%Kept: 3%= 5%40% of earnings60% of earnings0%1%2%3%4%5%One share, in rupeesPriceRs 400EarningsRs 20Dividend (40%)Rs 8Dividend yield8 / 400 = 2%P/E = 400 / 20 = 20; the kept Rs 12 is the 3% the company reinvests
    A P/E of 20 is an earnings yield of 5%, and paying out 40% of it gives a dividend yield of 2% with 3% retained, as a Rs 400 share earning Rs 20 and paying Rs 8 shows.
    The relationship
    DP=DE×EP=0.40×120=2%\frac{D}{P} = \frac{D}{E} \times \frac{E}{P} = 0.40 \times \frac{1}{20} = 2\%
    D / Pdividend yield, dividend per share over price
    D / Epayout ratio, dividend per share over earnings per share
    E / Pearnings yield, the inverse of the P/E
    What it says in wordsDividend yield is the payout ratio times the earnings yield, because the earnings cancel out.

    What does the other 3% do, and where does this identity help?

    The retained 3% is not lost; it is reinvested. If the company earns 15% on what it reinvests, keeping 60% of profit lets it grow earnings by about 0.6 x 15% = 9% a year, and the dividend yield plus that growth, 11%, is a rough estimate of the shareholder's long-run return. That is the logic of a dividend discount model in one line. The identity also works backwards in an interview: a stock yielding 2% with a 40% payout must be on a P/E of 20.

    The limits: the P/E uses one year's earnings, which may be unusually high or low, and the payout ratio can change from year to year. Buybacks return cash too, so a company paying low dividends but buying back shares can return more than its dividend yield suggests. And the growth estimate assumes the company keeps earning 15% on new money, which gets harder as it grows.

    Where candidates lose it

    The trap is dividing the wrong things: 40% by 20 gives 2 by luck, but candidates who do it cannot explain why and fall over on the follow-up. Others turn 20 into 20% or forget to flip the P/E at all.

    Say the identity out loud before the number: dividend yield equals payout ratio times earnings yield. Then the arithmetic is one line and every variation of the question is the same line.

    What the interviewer asks next

    • A stock yields 3% and pays out 60% of earnings. What is its P/E?
    • The company raises its payout to 80% with no change in price. What happens to the yield and to future growth?
    • Why might a fund manager prefer a 1% yielder that buys back shares to a 3% yielder that does not?

    Asked at Houlihan Lokey, Private Funds Advisory, Chicago, 2026 (Wall Street Oasis): Mostly technical, with standard accounting and valuation ratio questions.

  3. 075A bond with a 7% annual coupon trades at 104. Is its yield to maturity above or below 7%, and what is its current yield?Bond maths and durationWarm upVanguardMalvern · 2023

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    Which ordering is right for this bond?

    Show the worked solution

    The yield to maturity is below 7%, and the current yield is 6.73%. Current yield is the coupon over the price: 7 / 104 = 6.73%. The yield to maturity is lower still, because you pay 104 and get back only 100, so the Rs 4 premium is a loss spread over the bond's life. For an assumed five years left, the yield to maturity is about 6.05%.

    Why must the yield be below the coupon when the price is above 100?

    Pay Rs 104 for a gift voucher worth Rs 100 that also pays Rs 7 of cashback a year. The cashback is generous, but you have overpaid for the voucher by Rs 4. A bond priced above par returns less than its coupon, because the buyer pays more than the Rs 100 that comes back at maturity, and that premium is lost along the way. The coupon of 7% is set on the face value of 100 and never changes. The current yieldThe yearly coupon divided by the bond price today. It ignores any gain or loss as the price moves to face value at maturity. divides the same Rs 7 by what you actually pay: 7 / 104 = 6.73%.

    Above par, the yields sit below the coupon0%1%2%3%4%5%6%7%7.00%Coupon rate7 / 100 face6.73%Current yield7 / 104 price6.05%Yield to maturityadds the Rs 4 lossPrice pulls to parpar 100104 todayyr 0yr 1yr 2yr 3yr 4yr 5Rs 4 premium lost by maturity
    A 7% coupon bond bought at 104 has a current yield of 6.73% and, with five years left, a yield to maturity of 6.05%, because its price pulls down to 100 by maturity and the Rs 4 premium is lost.

    How far below 7% is the yield to maturity?

    That depends on how long the bond has to run, which the question does not say, so name an assumption. With five years left, the Rs 4 premium is lost at roughly Rs 0.80 a year. A quick estimate takes the coupon less that yearly loss, Rs 6.20, over the average of the purchase and redemption prices, 102: about 6.08%. The exact yield to maturityThe single discount rate that makes all remaining coupons and the final repayment worth exactly the price paid today. is 6.05%. The longer the bond, the thinner the yearly slice of the premium and the closer the yield to maturity sits to the current yield.

    The relationship
    104=∑t=157(1+y)t+100(1+y)5  ⇒  y≈6.05%104 = \sum_{t=1}^{5} \frac{7}{(1+y)^t} + \frac{100}{(1+y)^5} \;\Rightarrow\; y \approx 6.05\%
    104the price paid
    7the yearly coupon per 100 of face
    100the repayment at maturity
    ythe yield to maturity
    What it says in wordsThe yield to maturity counts the coupons and the loss of the premium, so it ends below both the coupon and the current yield.

    For a debt fund this ordering is everyday arithmetic: when rates fall, older high-coupon bonds trade above par, and the fund's quoted portfolio yield sits below the coupons it receives. The limit of yield to maturity: it assumes the bond is held to maturity, never defaults and that coupons are reinvested at the same yield, none of which is guaranteed. A bond callable before maturity at par makes the premium an even bigger risk.

    Where candidates lose it

    The trap is quoting 7% as the yield because that is the coupon. The coupon is fixed on the face value; the yield depends on the price you pay, and above par it is lower.

    The second miss is stopping at the current yield. Name the order, coupon above current yield above yield to maturity, and give the reason for the last step: the premium is lost by maturity.

    What the interviewer asks next

    • The same bond trades at 96. Put the coupon, current yield and yield to maturity in order.
    • With 20 years left instead of 5, is the yield to maturity closer to or further from the current yield?
    • Why do debt funds holding many premium bonds report a portfolio yield below their average coupon?

    Asked at Vanguard, Investments, Malvern, 2023 (Wall Street Oasis): Questions asked ranged from resume stuff, global macro/current news stuff, and simple bond math since I expressed interest in FICC.