Private Wealth Management puzzles, solved step by step

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  1. 050A Rs 20 crore client relationship earns the bank 0.8% a year in revenue. The assets grow 8% a year, there is a 10% chance each year that the client leaves, and the bank discounts at 12%. Roughly what is the relationship worth to the bank today?Wealth business economicsHardPrivate banking

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    Which denominator turns the Rs 16 lakh of first-year revenue into a lifetime value?

    Show the worked solution

    About Rs 1.1 crore. First-year revenue is 0.8% of Rs 20 crore, Rs 16 lakh. Attrition works like extra discounting and asset growth offsets it, so the shortcut divides by 12% + 10% - 8% = 14%: Rs 16 lakh / 0.14 is about Rs 1.14 crore. Summing year by year, where growth and survival multiply rather than add, gives Rs 1.08 crore. Both round to Rs 1.1 crore.

    Why does attrition belong in the discount rate?

    A shopkeeper values a regular customer by the purchases he expects, but also by how likely the customer is to keep coming. A 10% chance of losing the client each year shrinks every future year's expected revenue by a further 10%, exactly as if the discount rate were 10 points higher; asset growth pushes the other way. So a growing, leaky revenue stream is valued like a perpetuity at the discount rate plus attrition minus growth.

    The relationship
    V≈R1r+a−g=160.12+0.10−0.08=114.3 lakhVexact=R1/(1+r)1−(1+g)(1−a)1+r=108.1 lakhV \approx \frac{R_1}{r + a - g} = \frac{16}{0.12 + 0.10 - 0.08} = 114.3 \text{ lakh} \qquad V_{\text{exact}} = \frac{R_1/(1+r)}{1 - \frac{(1+g)(1-a)}{1+r}} = 108.1 \text{ lakh}
    R_1first-year revenue, Rs 16 lakh
    rthe bank's discount rate, 12%
    athe yearly attrition rate, 10%
    gthe yearly growth of the client's assets, 8%
    What it says in wordsA relationship is worth its first-year revenue divided by the discount rate plus attrition less growth; the exact annual sum is a little lower.
    What a relationship is worth: revenue, thinned by attrition, discounted15101520Year of the relationship, revenue in Rs lakhif he stays: Rs 69.1 lakhyear 1: Rs 16 lakhrevenue if the client staysafter 10% attrition a yeardiscounted at 12%Sum of dark barsRs 1.08 croreShortcut R / (r + a - g)16 / (0.12 + 0.10 - 0.08)Rs 1.14 croreFirst 10 years: 76%of the whole value
    Revenue of Rs 16 lakh growing 8% a year is thinned by 10% annual attrition and discounted at 12%, so each year's present value is smaller than the last. The discounted bars add to about Rs 1.08 crore against Rs 1.14 crore from the shortcut, and the first ten years carry 76% of the value.

    Why do the shortcut and the annual sum differ, and what moves the answer most?

    The shortcut adds the rates, which is exact only for continuous compounding; with annual steps, growth and survival multiply, 1.08 x 0.90 = 0.972 rather than 0.98, so the yearly sum is about 5% lower. Neither is wrong; say which you used. The lever is attrition: cutting it from 10% to 5% lifts the shortcut value from Rs 1.14 crore to Rs 1.78 crore, because the denominator falls from 14% to 9%. That is the arithmetic behind a private bank's spending on service and retention.

    The limits: this values revenue, not profit, so the relationship manager's cost and the platform's cost must come off before anyone calls it value. Growth above the discount rate less attrition would make the formula break down, which is a warning that the assumptions, not the client, have become unrealistic.

    Where candidates lose it

    The first trap is dividing Rs 16 lakh by 12% and calling the relationship worth Rs 1.33 crore, forgetting that clients leave and assets grow. The second is subtracting attrition instead of adding it, which inflates the answer several times.

    Say the denominator in words, discount plus attrition minus growth, before any number. Then note that the annual sum is slightly lower and that attrition is the lever the business can pull.

    What the interviewer asks next

    • What is the relationship worth if attrition falls to 5%?
    • How much would the bank rationally spend to win this client?
    • Why should the calculation use contribution after the relationship manager's cost, not revenue?