CHAPTER 5

Cross-Border Valuation

Typical textbook advice is that the choice between the home currency and foreign currency approaches is irrelevant in the valuation of an overseas project, and to accept/reject decisions. You will see in this chapter that this advice is based on the implicit assumption that FX rates are correctly valued. That is, the home and foreign currency approaches to cross-border valuation yield consistent results only if the current and forecasted spot FX rates are intrinsic FX rates (consistent with equilibrium).

This chapter also shows that if the time-0 spot FX rate is misvalued, or if FX rate forecasts differ from expected intrinsic spot FX rates, the home and foreign currency approaches result in different valuations for the same asset. Thus, NPV analyses in the different currency perspectives may imply different decision outcomes.¹

For example, if the home currency is overvalued at time 0 versus the foreign currency, a foreign project with a negative NPV in the foreign currency may have a positive NPV from the home currency perspective, because of the windfall FX gain of using an overvalued currency to undertake the investment. Or, if the time-0 spot FX rate is correctly valued but the home currency is forecasted to be overvalued versus the foreign currency in the future, a foreign project with a positive NPV in the foreign currency may have a negative NPV from the home currency perspective, because of the expected windfall FX loss in converting future cash flows from an overvalued foreign currency to the home currency.

Framework for Cross-Border Valuation

Reviewing notation, refers to the forecasted actual time-N spot FX rate, and is the expected intrinsic time-N spot FX rate. Similarly, denotes the actual time-0 spot FX rate, and denotes the intrinsic time-0 spot FX rate. Also, refers to the forecasted rate of actual FX change in the euro versus the US dollar, and represents the expected rate of intrinsic FX change in the euro versus the US dollar.

The expected spot FX rate for time N is related to the time-0 spot FX rate and the expected rate of FX change per the relationship shown in equation (5.1).

Expected Spot FX Rate at Time N

For example, assume the time-0 spot FX rate, , is 1.21 $/€, and the euro is expected to depreciate by 0.81% versus the US dollar over the next year: = −0.81%. Then the expected spot FX rate at time 1, , is (1.21 $/€)[1 − 0.0081] = 1.20 $/€.

And, because of something called Siegel’s paradox, we cannot simply assume that E(x^$/C) = −E(x^C/$), or even that 1 + E(x^$/C) = 1/[1 + −E(x^C/$)]. Instead, we need to use the approximation in equation (5.2):²

Expected FX Change and Currency Direction

For example, assume that the US dollar is expected to appreciate by 1.87% versus the euro, E(x^€/$) = 1.87%, and the volatility of x^$/€, , is 0.103. (The volatility of x^€/$ and x^$/€ are approximately equal.) Equation (5.2) says that the euro’s expected rate of change versus the dollar is E(x^$/€) ≈ −0.0187 + (0.103)² = −0.0081, or −0.81%.

Exhibit 5.1 Currency Volatility Estimates (versus US Dollar)

Euro	10.3%
Japanese Yen	9.5%
Chinese Yuan	1.7%
British Pound	8.5%
Canadian Dollar	9.1%
Australian Dollar	12.7%
Taiwan Dollar	4.9%
Swiss Franc	10.8%
Indian Rupee	7.3%
Korean Won	11.1%
Brazilian Real	22.6%
Mexican Peso	9.6%
Swedish Krona	11.3%
Hong Kong Dollar	0.4%
Norwegian Krone	11.6%
Denmark Krone	10.2%
New Zealand Dollar	13.5%
Singapore Dollar	5.5%
South African Rand	16.0%
Thai Baht	6.7%

Source: Author’s computations using month-end data, 1999 to 2016, from St. Louis Federal Reserve.

Currency volatility estimates are available in many places. Example estimates for some currencies versus the US dollar are shown in Exhibit 5.1.

Cross-Border Valuation and Intrinsic FX Rates

Now let’s return to the Aerotech-Vienna Gear example from Chapter 4 and show that the home currency and foreign currency approaches to valuation are equivalent if the FX rates are the intrinsic FX rates. For now, to emphasize the main issues, we assume that Vienna Gear will only be in business for one year, and will generate only next year’s operating cash flow, in euros. Vienna Gear’s uncertain time-1 cash flow in euros, , has an expectation, , of €10,000.

Aerotech’s managers have estimated Vienna Gear’s cost of capital is 8% in US dollars. That is, . Here is a common scenario where corporate managers have estimated an overseas project’s cost of capital in the home currency but have measured the expected operating cash flow stream in the foreign currency. Aerotech has also estimated that in US dollars, Vienna Gear’s FX business exposure to the euro, , is 0.40.

Let’s first apply the foreign currency valuation approach and find Vienna’s time-0 intrinsic business value in euros. To do this, we first need to convert Vienna’s from US dollars to euros, using equation (4.1): . Assume that the risk-free rates are 3% in US dollars and 2.39% in euros, the volatility of the euro versus the US dollar, , is 0.103, and E^*(x^€/$) = 1.87% (based on equation (4.2) and the ICAPM currency risk premium estimate for the US dollar versus the euro in Exhibit 4.1, −0.20%.) Given that Vienna Gear’s is 8%, equation (4.1) says that the estimated is (1 + 0.08) [1 + 0.0187] − 0.40(0.103)² − 1 = 0.096, or 9.60%.

Next, we find Vienna Gear’s time-0 intrinsic business value in euros as the present value of the expected future cash flow in euros: = €10,000/1.096 = €9,124. [In general, denotes asset i’s intrinsic value in euros. In this case, we know that asset i is the Vienna Gear business operation, so we suppress the i subscript, but use the B subscript to denote the focus on business value.]

We now want to find Vienna Gear’s business value from the US dollar (home currency) perspective. For now, assume that Aerotech’s managers forecast that the future spot FX rate will be equal to the expected intrinsic spot FX rate. Given this assumption, we’ll show that if the parameters of the valuation analysis in each currency are consistent, the present value of the project’s expected cash flow in US dollars is equivalent to the present value in euros, given the time-0 intrinsic spot FX rate.

Assume that the time-0 intrinsic spot FX rate, . Next, we want to use equation (5.1) to estimate the expected time-1 intrinsic spot FX rate, . To do this, we need the expected rate of intrinsic FX change of the euro versus the US dollar, E^*(x^$/€). Given that E^*(x^€/$) = 0.0187 and = 0.103, equation (5.2) says that E^*(x^$/€) is approximately −0.0187 + (0.103)² = −0.0081, or -0.81%. Then equation (5.1) says that the estimated time-1 intrinsic FX rate is = (1.21 $/€)[1 − 0.0081] = 1.20 $/€. Since we assume (for now) that the forecasted time-1 actual spot FX rate, , is equal to the expected time-1 intrinsic spot FX rate, the forecasted time-1 actual spot FX rate is also 1.20 $/€.

We are ready to use equation (4.3) to estimate Vienna Gear’s expected time-1 operating cash flow in US dollars: is (approximately) €10,000(1.20 $/€)[1 − 0.60(0.103)²] = $11,924. Note that we have applied equation (4.4) to convert the FX exposure to the euro, 0.40, to the FX exposure to the US dollar, 0.60.

Since Vienna Gear’s cost of capital in US dollars is 8%, the operation’s intrinsic business value in US dollars is: = $11,924/1.08 = $11,041. This result is the same (except for rounding) as converting the operation’s intrinsic business value in euros, €9,124, to US dollars at the time-0 intrinsic spot FX rate of 1.21 $/€: $11,040.

Now let’s bring the project’s investment outlay into the analysis. Assume that Vienna Gear’s owners are asking €9,500 for the business. Thus, €9,500 would be Aerotech’s investment outlay, viewed in euros, I^€. Looking at the proposal from the euro point of view, the NPV is = €9,124 − 9,500 = −€376. That is, NPV^€ = −€376. From the euro perspective, the acquisition has a negative NPV, implying rejection of the investment.

Next, we find the NPV in US dollars, if the time-0 actual spot FX rate is 1.21 $/€ and is thus correctly valued. Aerotech’s investment outlay from the US dollar perspective, I^$, would thus be 1.21 $/€(€9,500) = $11,495. Therefore, the NPV in US dollars is $11,040 − 11,495 = −$455. That is, NPV^$ = −$455. The NPV^$ is equivalent to NPV^€ at the time-0 spot FX rate, because −€376(1.21 $/€) = −$455.

In this example with correctly-valued (intrinsic) FX rates, the two currency perspectives lead to equivalent NPV amounts, given that the inputs to the analysis are consistent across the currencies. Thus, the two currency perspectives imply the same decision outcome. In principle, it does not matter which currency is chosen for the NPV analysis, because you reach the same accept/reject decision by conducting the NPV analysis in euros, with a euro cost of capital and an expected euro cash flow, or in US dollars, with a US dollar cost of capital and an expected cash flow converted from euros into US dollars.

You will sometimes see this irrelevance conclusion stated as a general proposition with the implication that it always holds. As you will see next, however, that generalization is incorrect. The home and foreign currency approaches to international investments give equivalent NPVs only if the FX rates are the intrinsic FX rates.

Misvalued Time-0 Spot FX Rate

Next, we look at a scenario where the euro is undervalued versus the US dollar at time 0, but the FX rate is expected to be correctly valued in the future.³ For a given time-0 investment outlay in euros, the undervaluation of the euro at time 0 implies that the project’s NPV^$ will be higher than the US dollar equivalent of NPV^€, given the time-0 actual spot FX rate.

Assume that the time-0 actual spot FX rate is , whereas the time-0 intrinsic spot FX rate is 1.21 $/€. Aerotech’s investment outlay from the US dollar perspective is 1 $/€(€9,500) = $9,500. That is, I^$ = $9,500. Vienna Gear’s is not affected by the time-0 spot FX misvaluation, given that the FX forecast is still the expected time-1 intrinsic spot FX rate, 1.20 $/€. That is, is still $11,924/1.08 = $11,040, regardless of the time-0 spot FX rate. So NPV^$ = $11,040 − 9,500 = $1,540. That is, NPV^$ = $1,540.

Note that even though the NPV^$ is affected by whether the time-0 actual spot FX rate is correctly valued, the NPV^€ is not affected. Regardless of the time-0 actual spot FX rate, NPV^€ = €9,124 − 9,500 = −€376. So, we see that the two NPVs are not necessarily equivalent to each other when the time-0 spot FX rate is misvalued. Indeed, in this example, it matters for the accept/reject decision which currency is chosen for the NPV analysis, because the project’s NPV in euros is negative and the NPV in US dollars is positive. A reject decision is implied by NPV^€, but an accept decision is implied by NPV^$.

In this example, the FX market conditions allow Aerotech to convert US dollars into euros for the investment outlay at time 0 at a “bargain” spot FX rate, yet expect to convert the acquisition’s future cash flow at a correctly valued future spot FX rate. The time-0 spot FX misvaluation is temporary and forecasted to be corrected by time 1. The box on Toronto-Dominion Bank, October 2007 is a real-world example where the home currency is overvalued versus the foreign currency at time 0.

It is relatively easy to think of an example of the opposite situation, where the foreign currency (euro) is overvalued versus the home currency (US dollar) at time 0, and the acquisition’s NPV^€ is positive, but the NPV^$ is negative. For example, assume that Vienna Gear’s owners are only asking €8,500 for the business and that the time-0 actual spot FX rate is 1.40 $/€. The acquisition has a positive NPV^€: €9,124 − 8,500 = €624. But in US dollars, the outlay for the acquisition is 1.40 $/€(€8,500) = $11,900. So, the NPV^$ is negative, $11,040 − 11,900 = -$860.

Forecasts of Misvalued Future FX Rates

This section extends the analysis to the case where the managers’ actual forecast of a future FX rate differs from the expected intrinsic FX rate. As you might suspect, the two present values of expected cash flows will not be equivalent across the currency perspectives, even converting with the time-0 intrinsic spot FX rate.⁴

Let’s pick up the Aerotech/Vienna Gear example and assume again that the expected time-1 intrinsic spot FX rate is 1.20 $/€, but that Aerotech’s managers’ forecast for the actual time-1 spot FX rate is 1.30 $/€. That is, = 1.20 $/€ and = 1.30 $/€. In this scenario, the euro is expected to be overvalued versus the US dollar when the project’s cash flow is received at time 1. If the time-0 spot FX rate is correctly valued, the forecast for the euro to be overvalued at time 1 implies that the project’s NPV^$ will be higher than the US dollar equivalent of NPV^€.

Vienna Gear’s actual expected cash flow, measured in US dollars, is = €10,000(1.30 $/€)[1 − 0.60(0.103)²] = $12,917. To correctly value the operation in US dollars, we need to break down into two components. The first component is the expected cash flow given the expected time-1 intrinsic spot FX rate (1.20 $/€), which we already know is $11,924. We’ll call this component the intrinsic expected cash flow, . The second component is simply the difference between the actual expected cash flow and the intrinsic expected cash flow, − , which is $12,917 − 11,924 = $993.

The reason for breaking into two components is that we need to discount the components at different rates, because of the different systematic risks. We need to discount the intrinsic expected cash flow component at Vienna Gear’s cost of capital in US dollars, 8%, which is based on the operation’s systematic business risk. We already know that the present value of this component is $11,924/1.08 = $11,040.

Since the expected cash flow’s second component, the FX windfall of $993, is due solely to FX market conditions and not business risk, we need to discount the second component at a required rate of return that reflects the systematic FX risk of the euro versus US dollar FX rate. Therefore, the correct discount rate for the FX windfall component is the required rate of return in US dollars on a one-year risk-free euro deposit. The deposit pays the risk-free rate in euros, , and in equilibrium, the investor also gets the risk compensation of . Therefore, a US dollar investor’s required rate of return on a euro deposit, , is the correct discount rate for expected FX windfall cash-flow components, as shown in equation (5.3).

Discount Rate for FX Windfall Component

As we assumed earlier that = 2.39% and = −0.81%, the required rate of return in US dollars on a one-year risk-free euro deposit is 2.39% − 0.81% = 1.58%. Therefore, the present value of the expected FX windfall component equals $993/1.0158 = $978. Adding the component present values, the value of the $12,917 actual expected time-1 cash flow is $11,040 + 978 = $12,018. That is, Vienna Gear’s intrinsic business value in US dollars is $12,018.

So, you see that if one forecasts a future spot FX rate that is NOT the expected intrinsic spot FX rate, the present value of the expected cash flows is NOT equivalent for the two currency choices, even with conversion at the time-0 intrinsic spot FX rate. If you convert the US dollar cost of capital to euros, find the present value in euros, and then convert the present value to US dollars using the time-0 intrinsic spot FX rate, you get $11,040. If you convert the expected euro cash flow to US dollars at the forecasted actual time-1 spot FX rate, the intrinsic business value is $12,018. If the outlay for the project were $11,500, the project would have a negative NPV from the euro perspective but a positive NPV from the US dollar perspective because of the forecasted time-1 overvaluation of the euro versus the US dollar.

Multiperiod Cash Flows

This section tackles the more common scenario of multiperiod cash flows. We examine the case where the analyst forecasts the foreign currency to be misvalued for multiple periods. For this scenario, we use a “short-cut” that involves converting the present value of the expected operating cash flows in euros to US dollars using the time-0 intrinsic spot FX rate.

To establish the “short-cut” method, note first that if forecasted future spot FX rates are equal to the expected intrinsic spot FX rates, the home currency present value of a foreign project’s future expected cash flows is equal to foreign currency present value of the project’s future expected cash flows times the time-0 intrinsic spot FX rate, .

Each period’s expected FX windfall may be found using the difference between the actual forecasted FX rate and the expected intrinsic FX rate: . Let denote the present value of the expected FX windfall components. Then a foreign operation’s intrinsic business value in the home currency is given in equation (5.4).

Intrinsic Business Value in US Dollars

For example, say Vienna Gear will generate five years of euro cash flows, with the expected cash flow being €10,000 each year. Still assume that the operation’s cost of capital in US dollars is 8%, the FX operating exposure to the euro is 0.40, the volatility of the euro versus the US dollar is 0.103, = −0.81%, = 1.58%, and the time-0 intrinsic spot FX rate, , is 1.21 $/€. So, the operation’s cost of capital in euros is still 9.60%. The intrinsic business value in euros is = €10,000/1.096 + 10,000/1.096² + 10,000/1.096³ + 10,000/1.096⁴ + 10,000/1.096⁵ = €38,298.

Exhibit 5.2 shows the expected intrinsic future FX rates, based on = −0.81% and = 1.21 $/€. Assume that Aerotech’s managers forecast that the time-1 FX rate will be 1.30 $/€, and that the euro’s forecasted overvaluation will gradually dissipate to the correct FX valuation by year 5, 1.16 $/€, per Exhibit 5.2. Exhibit 5.2 shows the operation’s expected FX windfall cash flows in US dollars, found using the short-cut described previously.

Exhibit 5.2 5-Year FX Windfall Cash Flows

N
1	1.30 $/€	1.20 $/€	$994
2	1.26 $/€	1.19 $/€	$691
3	1.22 $/€	1.18 $/€	$392
4	1.19 $/€	1.17 $/€	$195
5	1.16 $/€	1.16 $/€	0

The present value of the expected FX windfall cash flows, , is $994/1.0158 + 691/1.0158² + 392/1.0158³ + 195/1.0158⁴ = $2,205. Per equation (5.4), Aerotech’s estimate of the investment’s intrinsic business value in US dollars is (1.21 $/€)€38,298 + $2,205 = $48,546.

Debate: Foreign Versus Home Currency Valuation Approaches

We have seen that the home and foreign currency approaches give equivalent NPVs with intrinsic FX rates, but not with any time-0 or fore-casted FX misvaluation. The home currency approach incorporates the FX misvaluation, whereas the foreign currency approach does not. Which approach should you use?

Some think that an investment decision should not consider expected windfall FX gains and losses and thus that the foreign currency NPV indicates the correct investment decision. Proponents of this approach make two arguments. The first is that managers do not know enough about FX rates to have informed FX valuation opinions and forecasts, and so the managers should stick to managing their business instead of trying to estimate intrinsic FX values and forecast FX rates.

Second, even if managers could reliably judge intrinsic FX values and forecast FX rates, they could better exploit their judgments by using a financial market transaction instead of a business investment. To see this point, consider the Vienna Gear scenario where the euro is undervalued versus the US dollar at time 0, but the FX rate is forecasted to be correctly valued by time 1. Instead of acquiring Vienna Gear, Aerotech would be better off to invest the $9,500 in a risk-free euro deposit, paying $9,500 for an asset that is intrinsically worth $11,495, for an NPV of $1,995. Acquiring Vienna Gear, with a US dollar NPV of $1,540, should be rejected in favor of the euro deposit with an NPV of $1,995. In other words, why package the business investment together with an FX speculation when the FX speculation by itself is better?

A different school of thought favors the home currency approach because the foreign currency NPV ignores the windfall FX gains/losses. Proponents of this approach argue that if managers have an informed view on FX rates, why should they ignore this information when making investment decisions? Maybe the managers’ FX information comes from FX research experts of a top-notch FX forecasting service.

Second, a financial market transaction like a euro deposit is usually not a realistic alternative to a business investment. For one thing, acquiring Vienna Gear is consistent with Aerotech’s business, but speculating in a euro deposit is not. That is, FX speculation may be frowned-upon on its own, but may be “OK” if embedded in a business decision. For another thing, managers are likely to want to avoid the volatility that the mark-to-market (MTM) changes in the euro deposit would create in reported earnings. From Aerotech’s home currency perspective, acquiring Vienna Gear adds less value than investing in a euro deposit, but acquiring Vienna Gear may still be a more practical way for Aerotech’s managers to add value, given a view that the euro is currently undervalued versus the US dollar and their forecast that the current FX misvaluation will be corrected by time 1.

On one point, there is agreement: If a company wants to manage an overseas investment’s FX operating exposure with financial hedging, say with foreign currency debt, this tactic has its own windfall FX effects that will tend to offset the FX windfall effects of the investment. It seems plausible that an optimal level of financial hedging would trade off the hedging benefit of lower financial distress costs with the nonhedging benefit of capturing windfall FX gains. A model of this type is beyond our scope. Another consideration is if the company already has some natural short exposure to the foreign currency that will serve as an operational hedge. If so, then financial hedging of the FX operating exposure of a proposed investment is less needed, and a new project with a predicted windfall FX gain might be acceptable.

The debate is difficult to resolve, because both sides make reasonable points. Assessing whether a spot FX rate is misvalued is difficult even for economists, let alone managers. Yet we often see opinions and forecasts from major global banks and other financial services about FX misvaluation. And both sides make good points about the “bundling” of a corporate investment with FX speculation.

Academic researchers have empirically studied whether foreign direct investment (FDI) relates to spot FX rates, but the evidence is inconclusive, particularly on how FDI relates to FX misvaluations.⁵

So, the choice is yours: use the home currency approach if you want to incorporate a view on FX misvaluation in the NPV analysis, or use the foreign currency approach if you want to avoid doing so.

Summary Action Points

Glossary

Foreign Currency Approach (to Cross-Border Valuation): The analyst discounts the expected cash flows denominated in the foreign currency using a cost of capital denominated in the foreign currency.

Home Currency Approach (to Cross-Border Valuation): The analyst converts the expected foreign currency cash flows into home currency equivalents using forecasted FX rates, and then discounts them using a cost of capital denominated in the home currency.

Problems

For 1 to 8: The U.S. multinational Southwest Materials Co. is evaluating a Swedish acquisition target that will generate a single operating cash flow, one year from now, expected to be Sk10m. From the US dollar perspective, the target’s FX operating exposure to the Swedish krona is 1.50. The target’s private equity owner is asking Sk9m for the operation. Southwest has estimated the target’s cost of capital (and hurdle rate) is 10% in US dollars. Southwest forecasts that the future spot FX rate will be the intrinsic spot FX rate. The volatility of the Swedish krona versus the US dollar = 0.113; the time-0 intrinsic spot FX rate = 6.25 Sk/$; the US dollar risk-free rate is equal to the krona risk-free rate, 3%; and the equilibrium expected rate of intrinsic FX change in the US dollar versus the krona is −0.82%.

For 9 to 11: The British firm Bristol Tools Ltd. will generate one uncertain British pound operating cash flow, expected to be £1m at time 1. In US dollars, Bristol’s FX operating exposure to the British pound is 0.70 and cost of capital is 9%. Deck and Blacker Company, a U.S. multinational, owns Bristol Tool. The time-0 actual spot FX rate is 1.60 $/£, but Deck and Blacker’s managers believe that the time-0 intrinsic spot FX rate is 1.50 $/£. The managers also believe that the British pound’s time-0 overvaluation versus the US dollar will be corrected by time 1 and thus that the time-1 spot FX rate will be the intrinsic spot FX rate. The volatility of the British pound versus the US dollar is 0.085, and the equilibrium expected rate of intrinsic FX change in the US dollar versus the pound is 1.32%. Deck and Blacker’s managers want to indicate an asking price in British pounds that would give the buyer a positive NPV in pounds and Black and Decker a positive NPV in US dollars.

For 12 to 14: A U.S. multinational expects its Swiss operation to generate Sf 2,000 per year in operating cash flow perpetually. The multinational estimates that in US dollars, the operation’s FX operating exposure to the Swiss franc is −0.50 and the cost of capital is 10%. Assume that the US dollar and Swiss franc risk-free rates are 3% and 3.10%, respectively; and E^*(x^Sf/$) = 0.58% and E^*(x^$/Sf) = 0.59%. The volatility of the $/Sf FX rate changes is 0.108. The time-0 actual and intrinsic spot FX rates are 1.25 $/Sf and 1 $/Sf. Managers forecast that the actual spot FX rate will gradually converge to the intrinsic spot FX rate by year 5, as follows: = 1.20 $/Sf; = 1.15 $/Sf; = 1.10 $/Sf; = 1.05 $/Sf.

Answers to Problems

Discussion Questions