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  1. 070One lender quotes 12% a year compounded monthly; another quotes 12.5% a year compounded annually. Which loan is cheaper, and what is the effective annual rate of each?Compounding and time valueWarm upCarlyle GroupNew York · 2015

    Try it first

    Which loan is cheaper?

    Show the worked solution

    The 12.5% annual loan is cheaper. Twelve per cent compounded monthly is an effective 12.68% a year, against 12.50%. A 12% nominal rate charged monthly means 1% a month, and interest earns interest eleven more times within the year: 1.01 to the power 12, less 1, is 12.68%. Convert every quote to an effective annual rate before comparing.

    Why is 12% a year compounded monthly more than 12%?

    Leave a credit card balance unpaid and the interest charged in January is itself charged interest in February. The bank's 'monthly rate' grows faster than the same rate charged once a year. A nominal rate tells you how interest is quoted; the effective annual rate tells you what it costs, and only effective rates can be compared. Here 12% a year becomes 1% a month, and after twelve months of compounding Rs 100 has grown to Rs 112.68.

    Same 12% headline, five different costs: convert before you compare12.0%12.4%12.8%Lender B: 12.50% compounded annually12.00%Annual12.36%Half-yearly12.55%Quarterly12.68%Monthly12.75%DailyLender A: 12% monthlyCompounding frequency of a 12% nominal rate
    The same 12% nominal rate costs anything from 12.00% to about 12.75% a year depending on how often it compounds, and at monthly compounding its 12.68% sits above the other lender's 12.50%, so the annual quote is cheaper.
    The relationship
    EAR=(1+0.1212)12−1=1.0112−1≈12.68%  >  12.50%\text{EAR} = \left(1 + \frac{0.12}{12}\right)^{12} - 1 = 1.01^{12} - 1 \approx 12.68\% \;>\; 12.50\%
    0.12the nominal annual rate
    12compounding periods a year
    EAReffective annual rate, the true yearly cost
    What it says in wordsDivide the nominal rate by the number of periods, compound it over a year, and subtract one to get the rate you can compare.
    Compounding of 12% nominalEffective annual rate
    Annual12.00%
    Half-yearly12.36%
    Quarterly12.55%
    Monthly12.68%
    Daily12.75%
    Continuous12.75%
    More frequent compounding raises the effective rate, but with sharply diminishing steps: going from monthly to continuous adds less than a tenth of a point.

    What would the monthly lender need to quote to match?

    Run the conversion backwards: the monthly rate that compounds to 12.5% is 1.125 to the power 1/12, less 1, about 0.98% a month, or a nominal 11.84% a year. Any monthly quote above 11.84% is dearer than 12.5% annual. For a quick mental check, the extra from monthly compounding at these rates is roughly half the rate squared: 0.5 x 0.12 x 0.12 is 0.72 points, close to the true 0.68.

    What else decides which loan is really cheaper?

    The effective rate prices the money, not the whole deal. Processing fees, prepayment penalties and insurance bundled into the loan all add to the cost and must be folded in. Watch for flat-rate quotes too: a 7% flat rate on a three-year loan charges interest on the original amount even as it is repaid, which works out to an effective rate of about 13.6% a year. Confirm how any lender computes its rate before comparing; the limit of the EAR is that it compares like with like only once every cost is in it.

    Where candidates lose it

    The fast wrong answer picks 12% because it is the smaller number. It compares quotes built on different compounding, which is comparing prices in two currencies without converting.

    The other slip is the reverse: knowing that compounding adds cost but guessing the size. Monthly compounding adds about 0.68 points at 12%, not one or two points, so a 12.5% annual quote only just wins. Compute it rather than estimate it.

    What the interviewer asks next

    • What is the effective annual rate of 1.5% a month on a credit card?
    • A deposit pays 7% compounded quarterly. What is its effective annual yield?
    • Why does continuous compounding at 12% give about 12.75%, and what function produces it?

    Asked at Carlyle Group, Generalist, New York, 2015 (Wall Street Oasis): Some math brainteasers and accounting questions ranging from compounding rates to how an inventory purchase would flow