Risk-Adjusted Return on Inventory Buys

Retailers prefer to carry products that have both high expected demand and low demand uncertainty. Unfortunately, these are competing concerns, and so it's not always obvious which product to choose when one option has both higher expected demand and higher demand uncertainty.

Here, we present a simple formula for a product's risk-adjusted expected profit, using the newsvendor model. This score can be used to rank candidate products, allowing one to rationally balance the trade-off between expected demand and demand uncertainty. We cover a simple example and also share a Google Sheet that carries out the relevant computations, useful for both candidate product ranking and profit forecasting.

Newsvendor model risk-adjusted expected return

In our prior post on the newsvendor model, we covered the well-known optimal order quantity $Q$ for a product of uncertain demand. This result says we should purchase up to the point where

$$\begin{array}{c}P(d\le Q)=\frac{c_g}{c_g+c_l}\end{array}$$

The value at left is the probability that demand $d$ comes in less than or equal to $Q$. On the right:

• $c_g$ is the gain from selling one more unit (retail price minus unit cost), and
• $c_l$ is the loss from over-ordering by one unit (unit cost minus salvage value)

Our prior post illustrates how you can use this to set buy sizes so as to optimize expected return.

But what is the risk-adjusted expected return if we order up to the optimal point (1)? In the appendix to this post we derive the following result, assuming demand follows a normal distribution with mean $\mu$ and standard deviation $\sigma$,

$$\begin{array}{c}\text{risk-adjusted expected return}=\mu \cdot c_g-\sigma \cdot c_{\sigma }\end{array}$$

Here, we've introduced a new parameter $c_{\sigma }$, defined in equation (12) of our appendix below. This is a somewhat lengthy expression so we don't quote it here. However, we point out that it is a function of the unit economic input parameters $c_g$ and $c_l$ only – in particular, it does not depend on the demand distribution.

What we learn from (2) then is that there is a clean trade-off in the net expected profit between expected demand and demand uncertainty. As $\mu$ goes up by one, we gain the profit per sale $c_g$. But as $\sigma$ goes up by one, we pay the penalty $c_{\sigma }$. The balance between these determines whether a change in demand is favorable.

Example: Ranking candidate products

We can use the risk-adjusted return formula (2) to help us decide which of a set of candidate products to carry. To help our readers carry out their own analyses, we've created a Google Sheet that evaluates all quantities. If you click this link, it will ask you to make a copy of the file, which you can then edit and use for your own applications.

To focus on the tradeoff between demand uncertainty and expected sales, we assume that $c_g=100$ and $c_l=50$ across all products we're considering. With these choices, plugging into (2) and (12) gives

$$\begin{array}{c}\text{risk-adjusted expected return}=\mu \cdot 100-\sigma \cdot 53.63\end{array}$$

From this we can see that if $\mu$ increases by $1$, the tradeoff will be favorable provided $\sigma$ does not increase by more than $100/53.63\approx 1.83$.

Consider the table below of candidate products. Notice that in the second row, the expected demand and demand uncertainty have both increased by $100$, relative to the first. In this case, total expected return has gone up. However, in the third row the demand uncertainty went up twice as much as the expected demand, relative to the first row (i.e., more than the threshold level of $1.83$). In this case, the expected return has decreased. If we can pick just one of these options, the second product is best.

Product	Expected demand $\mu$	Demand uncertainty $\sigma$	Expected Return [$]
Product $1$	$1000$	$100$	$94,546$
Product $2$	$1100$	$200$	$99,092$
Product $3$	$1100$	$300$	$93,638$

Table: Exploring the tradeoff between expected demand and demand uncertainty for a few examples. In each row, we take $c_g=100$ and $c_l=50$. In the second row, we increase both $\mu$ and $\sigma$ by $100$, relative to the first row, and expected return increases. However, in the last row, we've increased the demand uncertainty by more, and the expected risk-adjusted return is lower than in the first row. You can check these values yourself using the sheet linked above.

Discussion

Here, we've answered a very practical question: What is the risk-adjusted return of a product of uncertain demand? Our formula (2) can be used to forecast expected profit for such products, and also to rank candidate products for purchase. We've found that there is a clear tradeoff, where lifts in the expected demand lift the expected return, while lifts to the demand uncertainty reduce expected return.

In our prior post on the newsvendor model, we highlighted the unintuitive result that the optimal solution often favors buying more units of a product as its demand uncertainty rises. This can be optimal if the margin is sufficiently high, because doing so allows us to capture more upside if demand comes in hot. However, we see now that this does not always imply that we should favor products with more demand uncertainty. Rather, the natural tradeoff we've considered here determines which is the optimal choice.

Appendix: Derivations

Here, we work out the derivation of (2). To begin, we calculate the expected gain from sales. This is

$$\begin{array}{c}\frac{E(\text{gain})}{c_g}=QP(d\ge Q)+{\int }_0^{Q}q\cdot p(d=q)dq.\end{array}$$

For a normal distribution, we can integrate the term at right. Plugging in, we get

$$\begin{array}{c}\frac{E(\text{gain})}{c_g}=QP(d\ge Q)+{\int }_0^{Q}q\frac{1}{\sqrt{2\pi {\sigma }^2}}\exp \left(-\frac{(q-\mu {)}^2}{2{\sigma }^2}\right)dq\end{array}$$

To evaluate, we write $j=\frac{q-\mu }{\sigma }$, and assume that the lower limit on the integral can be approximated by minus infinity – i.e., that is very unlikely to have demand close to zero or below under the normal approximation. This gives

$$\begin{array}{c}\frac{E(\text{gain})}{c_g}=QP(d\ge Q)+{\int }_{{}^{\prime }-{\infty }^{\prime }}^J\left(\sigma j+\mu \right)\frac{1}{\sqrt{2\pi }}\exp \left(-\frac{j^2}{2}\right)dj\end{array}$$

Simplifying gives

$$\begin{array}{c}\frac{E(\text{gain})}{c_g}=Q\left[1-\Phi \right]+\mu \Phi -\sigma \varphi \end{array}$$

Here, $\Phi$ and $\varphi$ are the cumulative distribution function and probability density function for the standard normal, respectively. Both are evaluated at the upper bound, $\frac{Q-\mu }{\sigma }$.

The expected loss takes a similar form to the above,

$$\begin{array}{c}\frac{E(\text{loss})}{c_l}={\int }_0^Q\left(Q-q\right)p(d=q)dq\end{array}$$

Evaluating as above gives

$$\begin{array}{c}\frac{E(\text{loss})}{c_l}=Q\Phi -\mu \Phi +\sigma \varphi \end{array}$$

Combining (7) and (9), we get the total expected profit,

$$\begin{array}{c}E(\text{profit})=c_g\left(Q\left[1-\Phi \right]+\mu \Phi -\sigma \varphi \right)-c_l\left(Q\Phi -\mu \Phi +\sigma \varphi \right)\end{array}$$

Next we use (1) to replace $\Phi$ with $\frac{c_g}{c_g+c_l}$. Plugging this in above, a number of terms cancel and we get

$$\begin{array}{c}E(\text{profit})=\mu \cdot c_g-\sigma \cdot \left(c_g+c_l\right)\varphi \left({\Phi }^{-1}\left(\frac{c_g}{c_g+c_l}\right)\right)\end{array}$$

This is equivalent to (2). From here, we see that

$$\begin{array}{c}{c}_{\sigma }\equiv \left(c_g+c_l\right)\varphi \left({\Phi }^{-1}\left(\frac{c_g}{c_g+c_l}\right)\right),\end{array}$$

a function only of $c_g$ and $c_l$. We note that the inner quantity ${\Phi }^{-1}\left(\frac{c_g}{c_g+c_l}\right)$ can also be written as $\frac{Q-\mu }{\sigma }$.

About VarietyIQ

VarietyIQ helps retailers and brands optimize inventory through smarter forecasting and product curation. We combine advanced data science with domain expertise to improve planning accuracy and unlock growth.

Curious how this analysis might improve product selection within your assortment? Get in touch — we’d love to connect.

---

Thanks to Jaireh Tecarro for creating the header image for this post.