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  1. 028Rs 10 a year forever, starting next year, is worth Rs 100 at a 10% discount rate. What is the same stream worth today if the first payment arrives only in year 4?Valuation riddlesWarm upAsset managementMutual funds

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    About Rs 75.1. The formula C over r gives the value one year before the first payment. With the first payment in year 4, the stream is worth 10 over 0.10, or Rs 100, at year 3. Discount that back three years: 100 over 1.1 cubed is Rs 75.1. As a check, Rs 100 less the present value of the three missing payments, Rs 24.9, gives the same answer.

    Where does the perpetuity formula put its answer in time?

    Think of a pension that starts paying at retirement. Its value on the day before the first cheque is one number; its value to a 30-year-old is that number shrunk by three decades of waiting. C over r values a level perpetuity exactly one period before its first payment, so a delayed stream is the ordinary perpetuity discounted from that point. Here the first payment is in year 4, so C over r lands at year 3, where it reads Rs 100. One more discounting step gets it to today.

    Value the stream where it starts, then bring that one number homeTodayYear 1Year 2Year 3Year 4Year 5Year 6Year 7Year 8...nonenonenone1010101010Rs 10 a year, forever, from year 4At year 3: 10 / 0.10= Rs 100discount 3 yearsToday75.1Check another way: an immediate perpetuity is worth Rs 100 today.Take away the three missing payments, worth 9.09 + 8.26 + 7.51 = 24.87, and Rs 75.13 is left.
    The stream of Rs 10 a year from year 4 is worth Rs 100 at year 3, and three years of discounting at 10% brings that to Rs 75.1 today. Subtracting the three missing payments, worth Rs 24.87 today, from an immediate perpetuity of Rs 100 gives the same Rs 75.1.

    How do you check it without the formula?

    Start from what you know: Rs 10 a year from year 1 is worth Rs 100. The delayed stream is that same stream with the first three payments cut out. Their present values are 9.09, 8.26 and 7.51, which add to 24.87, and Rs 100 less 24.87 is 75.13. Two methods agreeing to the paisa is what makes the answer safe to say out loud.

    The relationship
    PV=C/r(1+r)3=1001.13=1001.331≈75.13PV = \frac{C/r}{(1+r)^{3}} = \frac{100}{1.1^{3}} = \frac{100}{1.331} \approx 75.13
    Cthe yearly payment, Rs 10
    rthe discount rate, 10%
    3the years between today and one period before the first payment
    What it says in wordsValue the perpetuity where it begins, then discount that single sum back to today.

    This is the same step as a terminal value in a discounted cash flow. The Gordon growth value at the end of year 5 is a year 5 number and must be discounted five years, not six. Counting the periods wrong by one is the most frequent error in valuation models, and this puzzle is the cleanest place to catch it.

    Where candidates lose it

    The two wrong answers come from the timeline. Discounting by 1.1 to the fourth assumes the formula lands at the first payment date rather than one year before it, which gives about Rs 68.3. Subtracting Rs 30 at face value ignores that early money is worth more than later money.

    Draw the timeline, even in the air with your finger. Say where C over r lands before you discount, and offer the subtraction check.

    What the interviewer asks next

    • What is the stream worth if the payments grow at 3% a year from year 4?
    • How much is the delay costing, as a share of the undelayed value?
    • Where does the same off-by-one error show up in a DCF terminal value?