Nrr formula
Net Revenue Retention (NRR) Formula:
$$\text{NRR} = \frac{\text{Starting MRR} + \text{Expansion MRR} - \text{Contraction MRR} - \text{Churned MRR}}{\text{Starting MRR}} \times 100$$
Where:
- Starting MRR — recurring revenue from existing customers at the beginning of the period
- Expansion MRR — additional revenue from upsells, cross-sells, or seat additions
- Contraction MRR — revenue lost from downgrades
- Churned MRR — revenue lost from cancellations
Example:
| Component | Value |
|---|---|
| Starting MRR | $100,000 |
| Expansion MRR | +$15,000 |
| Contraction MRR | −$5,000 |
| Churned MRR | −$8,000 |
| NRR | 102% |
→ ($100k + $15k − $5k − $8k) / $100k × 100 = 102%
Benchmarks:
- >120% — World-class (typical of top SaaS companies like Snowflake, Datadog)
- 100–120% — Strong; growth from existing customers outpaces churn
- ~100% — Break-even; expansion offsets losses
- <100% — Revenue base is shrinking from existing customers
An NRR above 100% means the business grows revenue even with zero new customer acquisition.
Net reproduction rate
Net Reproduction Rate (NRR) Formula:
$$\text{NRR} = \sum_{x} \left( f_x \cdot L_x \right)$$
Or more explicitly:
$$\text{NRR} = \sum_{x=\alpha}^{\beta} \frac{l(x)}{l(0)} \cdot f_x^F$$
Where:
- l(x) / l(0) — probability of surviving from birth to age x (from a life table)
- fₓᶠ — age-specific fertility rate for female births only at age x
- α, β — lower and upper bounds of reproductive age (typically 15–49)
Simplified form:
$$\text{NRR} = \text{GRR} \times \text{(mean probability of surviving to reproductive age)}$$
Where GRR (Gross Reproduction Rate) = TFR × proportion of female births (~0.4878)
$$\text{NRR} \approx \text{GRR} \times p(\text{surviving to mean age of childbearing})$$
Interpretation:
| NRR | Meaning |
|---|---|
| = 1.0 | Exact replacement; population stable long-term |
| > 1.0 | Population will grow (each woman replaces herself with >1 daughter) |
| < 1.0 | Population will decline (sub-replacement fertility) |
Key distinctions:
| Measure | Accounts for mortality? | Sex-specific? |
|---|---|---|
| TFR | ✗ | All births |
| GRR | ✗ | Female births only |
| NRR | ✓ | Female births only |
NRR is considered the most complete single-number summary of whether a population is replacing itself, because it accounts for both fertility and the survival of daughters to reproductive age. Most high-income countries currently have NRR < 1.
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Park's textbook preventive social medicine NRR net reproduction rate formula
Here is the Net Reproduction Rate (NRR) as defined in Park's Textbook of Preventive and Social Medicine (Demography and Family Planning chapter):
Net Reproduction Rate (NRR) — Park's Definition
"NRR is defined as the number of daughters a newborn girl will bear during her lifetime, assuming fixed age-specific fertility and mortality rates."
Formula
$$\text{NRR} = \sum_{x=15}^{49} \left( {}_nL_x \cdot f_x^F \right)$$
Where:
- ₙLₓ — person-years lived between age x and x+n from the life table (reflects survival probability)
- fₓᶠ — age-specific fertility rate for female births only at age x
- Summation is over the reproductive age group: 15–49 years
Relationship to GRR (as per Park)
| Measure | Definition |
|---|---|
| GRR (Gross Reproduction Rate) | Average number of daughters a woman would bear over her lifetime assuming no mortality |
| NRR | Same as GRR, but adjusted for mortality (survival of daughters to childbearing age) |
$$\text{NRR} = \text{GRR} \times \frac{L_x}{l_0}$$
Since NRR accounts for the probability of dying before completing reproduction, NRR ≤ GRR always.
Interpretation (Park)
| NRR Value | Meaning |
|---|---|
| = 1 | Exact replacement level — population will stabilize |
| > 1 | Population will increase in the long run |
| < 1 | Population will decline (sub-replacement) |
- NRR = 1 corresponds approximately to a TFR of ~2.1 in low-mortality populations
- Park notes NRR is the best single indicator of whether a population is replacing itself because it incorporates both fertility and female survival
Reference: Park's Textbook of Preventive and Social Medicine — Demography and Family Planning chapter
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