Mutual Fund Mastery puzzles, solved step by step

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  1. 072A toll-road InvIT unit pays Rs 12 in a good year (40% chance), Rs 8 in a normal year (45%) and Rs 2 in a bad year (15%). What is the expected payout, and what is the unit worth at a 10% required return if this pattern continues forever?Probability and expected valueCoreNuveenChicago · 2023

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    What is the unit worth at a 10% required return?

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    The expected payout is Rs 8.70 and the unit is worth about Rs 87. Weight each year by its chance: 0.40 x 12 = 4.80, 0.45 x 8 = 3.60, 0.15 x 2 = 0.30, which sum to Rs 8.70. A payment expected every year forever is worth that payment divided by the required return, so 8.70 / 0.10 = Rs 87. Valuing the likeliest year alone would give Rs 80.

    How do you value income that changes every year?

    A farmer whose crop is good four years in ten, ordinary in about half, and poor in the rest does not plan around a normal year. He plans around what the land produces on average over many years. When income is uncertain, you value the expected valueThe probability-weighted average of all possible outcomes: each outcome times its chance, added up. of the income, not the likeliest outcome and not the plain average of the outcomes. For this unit: 0.40 x Rs 12 = 4.80, 0.45 x Rs 8 = 3.60, 0.15 x Rs 2 = 0.30, a total of Rs 8.70 a year.

    Weight every outcome, then capitalise the averageGood yearRs 12 x 40%Rs 12= 4.80Normal yearRs 8 x 45%Rs 8= 3.60Bad yearRs 2 x 15%Rs 2= 0.30ExpectedRs 8.70Bars to scale: 14 px per rupeePerpetuity at 10%8.70 / 0.10Rs 87Using only the likeliest year8 / 0.10 = Rs 80Rs 7 too low: ignores the good years
    Weighting Rs 12, Rs 8 and Rs 2 by their chances gives an expected payout of Rs 8.70, which at a 10% required return forever values the unit at Rs 87, while valuing only the likeliest year would give Rs 80.
    The relationship
    V=E[D]r=0.40(12)+0.45(8)+0.15(2)0.10=8.700.10=87V = \frac{E[D]}{r} = \frac{0.40(12) + 0.45(8) + 0.15(2)}{0.10} = \frac{8.70}{0.10} = 87
    E[D]the expected yearly distribution
    rthe required return, 10%
    Vthe value of the unit, assuming the pattern repeats forever with no growth
    What it says in wordsAverage the payouts by their chances, then treat the average as a level payment forever.

    Where do the two common wrong answers come from?

    The likeliest year pays Rs 8, so some candidates value the unit at Rs 80. That throws away the 40% chance of Rs 12, which is worth Rs 7 of value here. Others average 12, 8 and 2 to get Rs 7.33, as if each year were equally likely, which gives Rs 73 and punishes the unit for a bad year that happens only 15% of the time. The weighting is the whole answer.

    Now the honest limits, because the follow-up is usually how you would assess such an asset in practice. The expected value hides the spread: the payout's standard deviation here is about Rs 3.36, large against Rs 8.70, and an investor needing steady income cares about that. Uncertainty usually shows up in a higher required return, not in a lower expected payout, so the 10% must be chosen with the spread in mind. A toll road's traffic also trends and its concession ends, so the forever assumption is a simplification to state openly.

    Where candidates lose it

    The trap is valuing the most likely year, Rs 8, which gives Rs 80. It feels prudent but ignores that good years happen 40% of the time.

    The second miss is double counting risk: weighting the payouts down for the bad year and then also using a high required return for the same risk. Say that probabilities go in the cash flow and the price of uncertainty goes in the rate, once each.

    What the interviewer asks next

    • The bad-year chance rises to 30%, taken from the good years. What is the unit worth now?
    • Why might two investors pay different prices for this unit with the same expected payout?
    • The concession ends after 20 years with nothing left. Roughly what is the unit worth then?

    Asked at Nuveen, Multifamily, Chicago, 2023 (Wall Street Oasis): Walk me through how you would assess the value of a property if the income stream is unpredictable?