# Calculate a discount rate given a PV at some point in the future [closed]

Repost from CrossValidated...

I am trying to calculate a 24-month Customer Lifetime Value for a hypothetical magazine subscription service. CLV is typically calculated as the summation of the present value of future cash fows from the subscription, so if the subscription costs \$1.00 per month, that is the summation of$\frac{1}{(1+r)^n}$where$r$is the discount rate and$n$is the number of periods, taking$n$from$0 ... 23\$.

By looking at the data for my subscribers, I see that the actual survival rate is equal to 0.22 at 24 months (or, for a given 100 people who subscribe to my magazine, 22 remain after 24 months).

I want to specify the CLV equation from this data, but I'm not sure how to calculate r, given only this data point of 22% remaining after 24 months and the outline of the CLV equation above.

Just to clarify, the discount rate typically includes the cost of capital, but for this example I'm simplifying by assuming that the cost of capital is 0.

Any assistance would be greatly appreciated...

-
 This site is for professionals engaged in quantitative finance. This looks off-topic here. – Tal Fishman Apr 25 '12 at 18:19 I was referred here from CrossValidated due to the question concerning derivations of the present value equation. Is there a better forum? – user548084 Apr 25 '12 at 21:56 I'm afraid that was bad advice. I'm not sure where on SE you could get an answer to your question. Perhaps on some future marketing SE site. – Tal Fishman Apr 26 '12 at 14:38

## closed as off topic by chrisaycock♦Jan 29 at 18:23

Questions on Quantitative Finance Stack Exchange are expected to relate to quantitative finance within the scope defined in the FAQ. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about closed questions here.

## 2 Answers

If I lend money to the central bank then the right discount rate is the central bank rate. If I lend money to a company then the right discount rate is the central bank rate plus the spread for the company. In your case you are not lending any money so it's hard to see what the discount rate should be. To be honest, as it stands your question does not make much sense to me. From an economical point of view I would say the appropriate discount rate should be the cost of funding for the company that produces the magazine.

-
 True, the discount rate should include the cost of capital, but for this example I'm simplifying by assuming the cost of capital is zero, so as I understand it the subscription price (the monthly cash flow) is discounted by the "churn rate", because there are fewer and fewer subscribers from a given cohort remaining to pay the subscription price. – user548084 Apr 25 '12 at 17:12

You need at least one more piece of data, and that is how the attrition develops to get from your base of 100, to a 22% survivorship rate at the end.

The difference between the two is 78%, so I'd just pro-rate this at 3 1/4% per month. That is, your survival rate is .9675 after 1 month, .935 after two months, all the way to .22 in month 24.

Then solve from the monthly discount rate (internal rate of return) that equates the resulting cash flows to your outlays, and annualize it.

-
 I'm not sure I follow this. Can you say more about "how the attrition develops"? – user548084 Apr 25 '12 at 17:15 3.24% is the arithmetic average (.78/24), but that doesn't work as a discount rate because the "curve" you propose 0.9675 >- 0.935 >- ... >- 0.22 is linear, but a "curve" from a discount rate is not. Since CLV is a summation of the area under the curve, using a linear curve will overestimate the value. Unless I'm mistaken? – user548084 Apr 25 '12 at 17:17 @user548084: I only know the beginning and end points; 100% subscription after 0 months, 22% after 24. That's an attrition rate of 78% (100%-22%) in 24 months. 78%/24 = 3.25% a month, assuming it's "pro rata" (straight line). That's just a guess. You could argue that the "true" rate is more like 6% in months 1-13, falling to 0% in month 14 and therefter, to get to 78% That is a different profile, and a different calculation. – Tom Au Apr 25 '12 at 17:21 Got it, so the shape of the curve between the beginning and the end points looks just like if you plotted a present value on the y axis and the number of months on the x-axis. So plot y = 1 / (1+r)^x assuming some r. Not sure the technical name, sort of looks like an upside down log curve. The rate is constant (each months always has (1-r) fewer subscribers than the previous month) but the curve itself is not linear. – user548084 Apr 25 '12 at 17:30 Sorry, make that n[i] = (1-r)n[i-1] where n[i] is the number of subscribers in month i and n[i-1] is the number of subscribers in the previous month. How do you do subscripts here? – user548084 Apr 25 '12 at 17:39