1. A case study
Groupe Ariel S.A.: Parity conditions and cross-border valuation

2. The problem
The setting is a 2008 proposal from Groupe Ariel’s Mexican subsidiary to purchase and install cost-saving equipment in its plant in Monterrey.
Ariel requires a DCF analysis and an estimate of NPV for capital expenditures of this size in its newer foreign subsidiaries.
One of the questions confronting the analyst at headquarters in Mulhouse, France, is whether to perform this analysis in Euros or pesos.

3. Two approaches to estimating Euro NPV
Starting point for either approach: the estimated future peso cash flows (exhibit 2).
First approach (#1): to discount these peso cash flows at a peso discount rate to arrive at a peso NPV, and then, translate the peso NPV into Euros.
Need to generate a peso discount rate.
Second approach (#2): to translate the annual peso cash flows into Euros at expected future exchange rates, and then discount these Euro cash flows at a Euro discount rate to get an NPV in Euros.
Need to generate future peso/Euro exchange rates.
When parity conditions hold, approaches #1 and #2 give the same Euro NPV. When parity does not hold, the two approaches can give very different answers and may even cause managers at headquarters and in the subsidiary to disagree about whether the project should be undertaken.

4. Purchasing power parity
Relative Purchasing power parity (PPP): It states that the cost of living in different countries is equal and exchange rates adjust to offset inflation differentials across countries.
If PPP holds,
If purchasing power parity is expected to hold, then the best prediction for the one-period spot rate should be

5. Generalized version of Fisher effect
It says that real interest rates in all countries should be equal
Since we know that(1+a)(1+i)=1+r, where i is the inflation rate and r is the nominal interest rate, so

6. Interest rate parity f1 /e0 =(1+ rh)/(1+rf).
Or, (f1 - e0 )/e0 =rh - rf.

7. Peso cash flows The initial outlay: 3.5 million pesos (€220,000).
Depreciation tax shields = 3500,000/10 *35%=122,500 pesos/year. This is because the new equipment would have a useful life of 10 years.
The old equipment is sold in year 0 for 175,000 pesos, but the book value is 250,000 pesos.
This generates a loss of 75,000 pesos.
The resulting tax shield is 75,000 *35% = 26,250 pesos.
The after-tax salvage value is 175, ,250 = 201,250 pesos.
Lost depreciation tax shields =250,000/3 *35% = 29,167

8. Peso cash flows
After-tax cost savings = (old operating costs-new operating costs)*(1-35%).
Year 2009: (3,360,000-2,632,571)*0.65=
Year 2010: (3,774,960-2,930,951)*0.65=
2011 …
Sum the preceding five items for each year: initial outlay + after-tax salvage value + after-tax cost savings – lost old depreciation tax shields + new depreciation tax shields

9. Peso discount rate Various options:
8%, the same as Ariel would use for Euro flows;
8.1% to 9.2%, the current cost in Mexico of short- and long-term cost of peso debt;
8% +(Mexican inflation – French inflation), or roughly 12%
Various other numbers from case Exhibit 3.
Fisher effect: the peso equivalent of the 8% Euro discount rate is (1.08X1.07)/1.03-1=12.19%

10. NPV: approach #1 Discount all the peso cash flows at 12.19%
NPV = 1,497,678 pesos.
Translate the peso NPV at the spot exchange rate (MXN15.99/EUR):
NPV=1,497,678/15.99=€92,495

11. Approach #2
Expected future exchange rates: PPP implies that XR1=XR0(1+Mexican inflation)/(1+French inflation).
XR1=15.99X(1.07/1.03)=MXN16.61/EUR
XR2=16.61X(1.07/1.03)=17.26
XR3=17.26X(1.07/1.03)=17.93 ….
XR10=XR9X1.07/1.03=23.41
Dividing the peso CFs by these rates gives the projected CFs in Euros, conditioned on PPP holding.
NPV (r=8%)=€92,459

12. A summary of both approaches
Both approaches incorporated identical assumptions about interest rates and inflation rates. The Euro discount rate is 8%, and French and Mexican inflation rates are 3% and 7%, respectively.
They will give the same answer if the basic parity conditions are expected to hold. These include Fisher effect, PPP, Interest rate parity, and forward parity.

13. Modifications to the initial assumptions
We consider the effect of a lower Mexican inflation rate. Assume that this rate is 3%, rather than 7%.
After-tax cost savings are affected (TN exhibit 4).
Total peso cash flows (TN exhibit 5): lower
Approach #1: NPV = 1,770,540 pesos=€110,728 (peso discount rate =8%). Higher
Approach #2: same. The row of future exchange rates shows no change in currency values because inflation is the same in both countries.

14. A projected violation of PPP
Assume Mexican and French inflation rates are both 3% and that peso and Euro discount rates remain at 8%.
Suppose Ariel expects a real depreciation of the peso in the near term --- in years 1 and followed by parity during years 3-10.
XR1=22
XR2=25
XR3=XR4=…=XR10=25
Approach #1: NPV=1,770,540 pesos
Approach #2: NPV=-€670= pesos.