Private Equity puzzles, solved step by step

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  1. 076An investment is funded half with debt at 7% and half with equity. The asset earns 12% a year before financing, and tax is ignored. What return does the equity earn? What if the asset earns only 5%?Leverage and capital structureCoreHPS Investment Partnersnew york · 2024

    Try it first

    Before you calculate: with the asset at 12%, what does the equity earn?

    Show the worked solution

    Equity earns 17% when the asset earns 12%, and only 3% when it earns 5%. Per Rs 100 of asset, lenders take a fixed 3.5. At 12% equity keeps 8.5 on its 50; at 5% it keeps 1.5 on its 50. Leverage adds the spread between the asset return and the cost of debt, times debt over equity, in both directions.

    Why does borrowing push the equity return above the asset return?

    Think of buying a flat for Rs 100 lakh with Rs 50 lakh of your own money and a Rs 50 lakh loan at 7%. If the flat earns 12 lakh a year in rent and value, the bank takes 3.5 lakh and everything else is yours. Lenders get a fixed amount whatever the asset earns, so any return above the cost of debt on the borrowed half lands on the equity. The borrowed 50 earns 12% and costs 7%; that 5 point spread on 50 is 2.5, added to the 6 the equity's own half earns.

    Lenders take a fixed 3.5; equity keeps whatever is leftAsset earns 12%12Lenders: 7% x 50 = 3.5Equity keeps 12 - 3.5 = 8.58.5 on equity of 50= 17% return on equityUnlevered 12%, levered 17%: +5 pointsAsset earns 5%5Lenders: 7% x 50 = 3.5Equity keeps 5 - 3.5 = 1.51.5 on equity of 50= 3% return on equityUnlevered 5%, levered 3%: -2 pointsEach bar is Rs 100 of asset: Rs 50 of debt at 7%, Rs 50 of equity
    On Rs 100 of asset funded half with debt at 7%, lenders take 3.5 whatever happens: at a 12% asset return equity keeps 8.5 on 50 for 17%, and at 5% it keeps only 1.5 on 50 for 3%.
    The relationship
    rE=rA+(rA−rD)DE=12%+(12%−7%)×1=17%r_E = r_A + (r_A - r_D)\frac{D}{E} = 12\% + (12\% - 7\%)\times 1 = 17\%
    r_Ereturn on equity
    r_Areturn the asset earns before financing
    r_Dinterest rate on the debt
    D/Edebt over equity, here 50 over 50, which is 1
    What it says in wordsEquity earns the asset return plus the spread over the cost of debt, scaled up by how much debt there is for each rupee of equity.

    What happens when the asset earns less than the debt costs?

    The same formula runs backwards. At 5%, the spread is minus 2 points, and with one rupee of debt per rupee of equity the equity return falls to 5 minus 2, which is 3%. Leverage is not a return booster; it is a multiplier on the spread, and the spread can be negative. Below a 3.5% asset return equity earns nothing at all, and below zero it loses money faster than the asset does.

    In the room, give both numbers and then the rule in one sentence. A private credit interviewer is checking that you see the downside as clearly as the upside, because their job is to sit on the other side of that fixed 3.5.

    Where candidates lose it

    The fast wrong answer is 24%, doubling the asset return because equity is half the funding. That forgets the lenders are paid first. The other slip is subtracting the 7% rate from 12% and calling 5% the equity return, which treats the whole asset as if it were borrowed.

    Work in rupees on Rs 100 of asset: asset income, less interest, over equity. Then give the 5% case without being asked; that is the half of the answer that shows judgement.

    What the interviewer asks next

    • What equity return do you get at 70% debt and the same 12% asset return?
    • At what asset return does the equity earn exactly the same as the asset, and why?
    • Now add tax at 25%. What happens to the 17%?

    Asked at HPS Investment Partners, Financial Sponsors, new york, 2024 (Wall Street Oasis): I was asked to calculate the returns of an investment that consisted of 50% debt with an interest rate of 7%