Confused with Change in Total Return formula

CFA Level III Portfolio Management
#1

Hi guys,

Here’s the change in in TR formula i found in one video lecture, however when using it with an example it doesn’t seem to work at all.

Formula provided in the video lecture : ΔTR = ΔP * Q + ΔQ * P
With:
TR: Total revenue
P: Price
Q: Quantity output

Example:

Q1 = 10 ; Q2 = 20
P1 = 1 ; P2 = 2
TR1 = 10 ; TR2 = 40

So we agree here that ΔTR = TR2 - TR1 = 40 - 10 = 30 (Intuitive way of finding ΔTR)

However, using the formule mentioned previously, i get ΔTR = 40, instead of ΔTR = 30

ΔTR = ΔP * Q + ΔQ * P = (2 - 1) * 20 + (20 - 10) * 2 = 40

Did i miss something here? Can someone please help?

Thanks guys.

#2

This is an approximation, and it works well only when ∆P and ∆Q are small.

The proper formula is:

∆TR = \left(P + \Delta P\right)\left(Q + \Delta Q\right) - PQ
= PQ + P\Delta Q + Q\Delta P + \Delta P\Delta Q - PQ
= P\Delta Q + Q\Delta P + \Delta P\Delta Q

Here, \Delta P\Delta Q = \left(2 - 1\right)\left(20 - 10\right) = 10, which is significant.

2 Likes
#3

Dunno if this helps but if you set P and Q equal to their mean values, you get the 30 you seek for change in TR. Instead of taking the later value and ignoring the prior one as irrelevant, take the average for the 2 to get a representative P and Q value.

#4
\left[\frac{P + \left(P + \Delta P\right)}{2}\right]\Delta Q + \left[\frac{Q + \left(Q + \Delta Q\right)}{2}\right]\Delta P
= \left(P + \frac{\Delta P}{2}\right)\Delta Q + \left(Q + \frac{\Delta Q}{2}\right)\Delta P
= P\Delta Q + \frac{\Delta P\Delta Q}{2} + Q\Delta P + \frac{\Delta Q\Delta P}{2}
= P\Delta Q + Q\Delta P + \Delta P\Delta Q
= \Delta TR

More generally, if you use P^* = P + k\Delta P and Q^* = Q + \left(1 - k\right)\Delta Q, you’ll get the correct value for \Delta TR.

1 Like
#5

Hi, you did a slight mistake when you applied the formula. You need to do the following:
(Q2 - Q1) * P1 + (P2 - P1) * Q2
(20 - 10 ) * 1 + (2 - 1) * 20 = 30