Case 099Products and fund selectionHard

Price a 5-year capital-protected note on Rs 1 crore: a 7.5% zero-coupon yield sets the bond floor, 2% goes to distribution, and the rest buys at-the-money index calls costing 30% of notional at 18% volatility. What participation results, and how would you explain the Black-Scholes inputs to the client?

Goldman SachsZurich · 2025

The situation

A private bank's structuring desk is building a five-year capital-protected note for a client's Rs 1 crore. The issuer's five-year zero-coupon yield is 7.5% a year. The distribution fee is 2% of notional, paid upfront. The desk prices five-year at-the-money calls on a broad equity index, the index's price return only, at 30% of notional, using the Black-Scholes model with 18% volatility, the 7.5% rate and an index dividend yield of about 0.9%.

The note repays 100% of capital at maturity plus a share of any rise in the index. A five-year bank deposit at 7.5% is the client's alternative. Figures are illustrative.

Your task

Work out the bond floor, the option budget and the participation rate, show when the note beats the deposit, and explain the model's inputs in words a client would follow.

Quick check

Roughly what participation in the index rise does the client get?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

Participation is about 94.5%: Rs 69.66 per Rs 100 buys the bond floor, Rs 2 goes in fees, and Rs 28.34 buys calls priced at Rs 30. The client gets his capital back plus 94.5% of any index rise, but he gives up the dividends and the deposit's certain Rs 43.6 of interest; the note beats a 7.5% deposit only if the index rises more than about 46% in five years. The protection is only as good as the issuer.

Step 1How is the note built from two pieces?

A safe piece that guarantees the capital and a risky piece that buys the upside. Think of a family setting aside enough in a deposit today to be sure of Rs 1 lakh in five years, then spending what is left on a lottery ticket with a good chance of paying. The bond floorThe part of a protected note invested in a zero-coupon bond so that it grows back to the full capital by maturity. costs 100 divided by 1.075 to the fifth power, Rs 69.66 per Rs 100; after the Rs 2 fee, Rs 28.34 is left for options. At Rs 30 per Rs 100 of index exposure, that budget buys 94.5% of the index's rise.

Of each Rs 100, Rs 69.66 buys a five-year zero-coupon floor at 7.5%, Rs 2 pays distribution, and the Rs 28.34 left buys calls costing Rs 30, so the client participates in 94.5% of the index rise.

Step 2How would you explain Black-Scholes to the client?

As a price for a right, set by five things he can picture. The call is the right to the index's rise over five years; Black-Scholes prices it from the index level, the strike, the time left, interest rates net of dividends, and volatility, how much the index typically swings. More swing and more time make the right more valuable, because there is more chance of a big rise and the downside is capped at zero. Higher rates also raise a call's price. Here the model gives d1 of about 0.99 and d2 of about 0.59, and a price of 30% of notional.

The relationship

C = S e^{-qT} N(d_1) - K e^{-rT} N(d_2), \quad d_{1,2} = \frac{\ln(S/K) + (r - q \pm \sigma^2/2)T}{\sigma \sqrt{T}}

S, K	index level and strike, equal for an at-the-money call
T	five years
r, q	rate 7.23% continuous and dividend yield about 0.9%
sigma	volatility, 18% a year
N()	the normal distribution's cumulative probability

What it says in wordsThe call is worth the expected index level above the strike, weighted by the chance of finishing there, less the discounted strike weighted by the chance of paying it.

Step 3What moves participation, and when does the note beat a deposit?

Two inputs move it most. If volatility rises to 22%, the calls cost about 32.2% and participation falls to 88%; if rates fall to 6.5%, the floor costs Rs 72.99 and participation falls to about 90%. Against a deposit paying Rs 143.6 per Rs 100 in five years, the note wins only if the index rises more than about 46%, roughly 7.9% a year in price, before dividends the note does not pay. Below that, the client would have done better in the deposit.

At maturity the note pays 100 whatever the index does below zero and 100 plus 94.5% of any rise, so it beats a 7.5% deposit's 143.6 only when the index rises more than about 46% over five years.

Close with what the client must hear. The protection is a promise by the issuer, not a guarantee from the market, so it is worth only as much as the issuer's credit. The note locks the money for five years, and selling early means selling at the desk's bid. A capital-protected note is a deposit's certainty exchanged for a share of equity upside, and the client should know exactly what he gives up to get it.

Where candidates lose it

Candidates say capital protected, so the client gets 100% of the upside for free. Participation is set by the option budget; here the fee and the floor leave room for about 94.5%, and the client also gives up dividends and the deposit's interest.

The second miss is explaining Black-Scholes as a formula. The interviewer asked for the client version: what a call is, what the inputs mean, and which way each one pushes the price.

What the interviewer asks next

The desk offers 100% participation with a cap at 40% index return. How would that be built?
How would a 1% distribution fee change the participation?
What is the client's exposure if the issuer is downgraded in year two?

Asked at Goldman Sachs, Wealth Management, Zurich, 2025 (Wall Street Oasis): even though the job was in English, and to explain Black-Scholes

Company names and figures are illustrative.