With fixed nominal interest rates, inflation depresses real interest rates, leading to a decreasing path of private consumption of domestic goods, so that ct becomes negative. From: Handbook of Macroeconomics, 2016
Cristina Terra, in
Principles of International Finance and Open Economy Macroeconomics, 2015 Suppose the nominal interest rate
on a 1-year US bond is 5% and the nominal interest rate in Mexico for a bond of the same maturity is 10%. The current exchange rate in the spot exchange rate market is 2.5 peso$/US$. If the uncovered interest rate parity is valid and the expected exchange rate for the next year is 2.4 peso$/US$, which of the two investments is more interesting to an American investor? If the
covered interest rate parity is valid, what should be the nominal exchange rate on the future dollar contract with a 1-year term?The Foreign Exchange Market
Exercise 1
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Testing for Causality: A Survey of the Current Literature
Nicholas Apergis, in The Economics and Econometrics of the Energy-Growth Nexus, 2018
Data
Annual time series data on nominal interest rates, inflation, and effective exchange rates for Nigeria are obtained from the Annual Report and Statements of Accounts published by the Central Bank of Nigeria, spanning the period 1970–2011. The analysis makes use of money market interest rates as nominal interests, as along with inflation can be used as proxies for expected inflation. In addition, the analysis employs the US six-month London Interbank Rate (USRATE), obtained from the World Economic Outlook Publication Report, as a proxy for the foreign interest rate. Finally, all variables are expressed in percentages.
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Financial markets: bonds
John Hill, in Environmental, Social, and Governance (ESG) Investing, 2020
Fisher’s law
The foregoing discussion is in terms of nominal interest rates. That is, there is no discussion of inflation. Fisher’s law, named after the economist Irving Fisher considers that the rate of change of prices in the real economy has an impact on required rates of return on financial instruments. This can be stated as:
i=r+p
where i is the nominal rate of interest; r is the real rate; and p is the expected rate of inflation.
Intuitively, Fisher’s law says that the nominal rate equals the real rate plus inflation.
Note also that
r=i–p
or that the real rate equals the nominal rate minus inflation. As an example, if the nominal rate on a bond is 4% and inflation is 2%, then an investor receives a real rate of return of 2%. The practical application is that if investors expect higher rates of inflation, they will demand higher rates of interest from bond issuers.
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Extracting Market-Implied Bitcoin's Risk-Free Interest Rate
Nicolas Wesner, in Handbook of Digital Currency, 2015
11.2.2 Monetary economic theory
According to Fisher (1930), nominal interest rates can be decomposed into real rate and inflation expectation: i = r + πa. Using this relation, uncovered interest rate parity can be written:
(11.3)tst+ke−st =rt−rt*+πa−π* a
Since a virtual currency as Bitcoin is decentralized, it cannot be associated with a foreign country, so that r = r* can be assumed (in the same way, equality between home and foreign national outcome y = y* can be assumed). This leads to relative purchasing power parity:
(11.4)tst+ke−st=πa−π*a
We consider then some assumptions that form the basis of the monetary approach. Some of those assumptions fit well with the current framework of a virtual currency like Bitcoin where almost all goods are priced primarily in a domestic currency like US$ and where no deep, liquid, organized bond market exists. First demand for money is stable and supply exogenous. Moreover, we consider that the money market is in continuous equilibrium and it is cleared by price movements, not interest rates.
According to this basic monetarist model, a change in the relative money supply affects the exchange rate because it affects the relative price level. A higher relative money supply implies a higher relative price level, and by relative purchasing power parity, the exchange rate changes with the relative price level. Therefore, in this framework, the movement in the spot rate is proportional to the movement in the relative money supply:
(11.5) tst+k−st=mt+k−mt –mt+k*−mt*
Here, mt* and mt represent (log) foreign money supply and domestic money supply, respectively.
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Savings
Nathan Tintle, ... Todd Swanson, in A Spiral Approach to Financial Mathematics, 2018
Conceptual questions
1.
What is the difference between effective and nominal interest rates?
2.What does the notation i(12) mean?
3.Which will result in earning more interest or are they the same: A nominal rate of 6% compounded quarterly or an effective annual rate of 6%? Why?
4.Which will result in earning more interest or are they the same: A nominal rate of 3% compounded quarterly or a nominal rate of 3% compounded daily? Why?
5.Keeping the nominal interest rate constant, as the compounding period gets shorter does the effective annual interest rate increase, decrease or stay the same? Why?
6.Which leads to more interest paid: an effective interest rate of 12%, compounded annually or a nominal interest rate of 12%, compounded monthly? Why?
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Handbook of International Economics
Charles Engel, in Handbook of International Economics, 2014
1 Exchange Rates and Interest Parity
This chapter surveys empirical and theoretical research since 1995 (the publication date of the previous volume of the Handbook of International Economics) on the determination of nominal exchange rates. This research includes innovations to modeling based on new insights about monetary policymaking and macroeconomics. While much work has been undertaken that extends the analysis of the effects of traditional macroeconomic fundamentals on exchange rates, there have also been important developments that examine the role of non-traditional determinants such as a foreign exchange risk premium or market dynamics.
This chapter follows the convention that the exchange rate of a country is the price of the foreign currency in units of the domestic currency, so an increase in the exchange rate is a depreciation in the home currency. St denotes the nominal spot exchange rate, and st≡log(St).
A useful organizing feature for this chapter is the definition of λt:
(1)λt≡it∗+ Etst+1-st-it.
In this notation, it is the nominal interest rate on a riskless deposit held in domestic currency between periods t and t+1, while it∗ is the equivalent interest rates for foreign-currency deposits.1 Etst+1 is the rational expectation of st+1: the mean of the probability distribution of st+1, conditional on all information available to the market at time t. λt is the difference between (approximately) the expected return on the foreign-currency deposit expressed in units of the domestic currency, it∗+Etst+1-st, and it. It can be called the deviation from uncovered interest parity,2 the expected excess return, or less generally, the foreign exchange risk premium. Rearrange equation (1) to get:
(2)st≡-(it-it∗)-λt+Etst+1.
An increase in the domestic to foreign short-term interest differential, it-it∗, ceteris paribus is associated with an appreciation. An increase in the expected excess return on the foreign deposit, λt, is also associated, ceteris paribus, with an appreciation. Holding interest rates and λt constant, a higher expected future exchange rate implies a depreciation.
Models of the exchange rate for the past 40 years have often focused on the case in which λt=0. Let qt be the log of the real exchange rate (the relative foreign to domestic consumer price levels, expressed in common units, where pt is the log of the domestic consumer price index (CPI), and pt∗ is the log of the foreign CPI in foreign-currency units), so qt=st+pt∗-pt, and set λt =0. Equation (2) can be written as:
(3)st≡-(it-it∗)+Etπt+1- Etπt+1∗+pt-pt∗+Etqt+1=-(r t-rt∗)+pt-p t∗+Etqt+1,
where π t+1≡pt+1-pt is approximately the domestic inflation rate and rt≡it-Etπt+1 is approximately the domestic ex ante real interest rate (with analogous definitions for the foreign variables). Iterate equation (3) forward to get:
(4)st≡-∑j=0∞Et(it+j-it+j∗)+lim j→∞Etst+j+1=-∑j=0∞Et( it+j-it+j∗)+ ∑j=0∞Et(πt+j+1 -πt+j+1∗)+pt-pt∗+limj→∞Etqt+j+1=-∑j=0∞Et(rt+j-rt+j∗)+pt-pt∗+ limj→∞Etqt+j+1.
This equation summarizes the concerns of the literature that sets λt=0, and helps to demarcate the scope of this chapter. Monetary models of exchange rates have focused on the role of monetary policy in setting interest rates and determining inflation. These models have also emphasized the macroeconomic forces that determine expected future interest rates, real and nominal, and expected future inflation. We will survey recent developments in this line of research.
The term lim j→∞Etqt+j+1 can be thought of as the long-run real exchange rate. This survey will not attempt to encompass the large literature that examines the neoclassical determinants of equilibrium real exchange rates. The chapter is about nominal exchange rates, but economists, policymakers, and individuals are concerned about nominal exchange rates mostly because they believe that their fluctuations matter for real exchange rates and other relative prices such as the terms of trade, so we focus on models in which the determination of real prices depends integrally on the nominal exchange rate level.
We are also not concerned with the pt-pt∗ term. This term is of course important in nailing down the level of nominal exchange rates—it helps to answer the question of why a dollar buys 80 yen instead of 8 yen. It also is important in high-inflation countries for understanding shocks to the nominal exchange rate, st-Et-1 (st). Here we stipulate that in these high-inflation countries, monetary growth is most influential in determining the variance of pt-Et-1(pt). There is little need to go further than that for high-inflation countries; and for low-inflation countries, the behavior of pt-Et-1(pt) contributes little to our understanding of nominal exchange rate shocks, so we set it aside.
Much recent theorizing about exchange rate determination focuses on λt. There are a number of reasons why λt may not equal zero. If agents require a higher expected return on foreign compared to domestic deposits, because of a foreign exchange risk premium or some sort of liquidity premium, then λt≠0. The definition of λt uses rational expectations, but participants in the market may form expectations using some other algorithm. There might be private information relevant for the demand for foreign and home deposits, so even if agents all form their expectations rationally on the basis of their own information, the market equilibrium condition might not aggregate to it∗+Etst+1-st=it. Individuals might have “rational inattention,” so that they do not act continuously on publicly available information. The market microstructure—how foreign-currency demand and supply gets translated into a price for foreign exchange—might affect the market equilibrium. The modern “asset-market” approach is built off the assumption that capital flows freely between markets, but capital controls or other transactions costs can upset the asset-market equilibrium, as could other limits to arbitrage such as collateral constraints. Equation (4) can be generalized to:
(5)s t≡-∑j=0∞Et(it+j-it+j∗)- ∑j=0∞Etλt+j+limj→∞Etst+j+1 =-∑j=0∞E t(it+j-it+j∗) +∑j=0∞Et(πt+j+1-πt+j+1∗)-∑j=0∞Etλ t+j+pt-pt∗+limj→∞Etqt+j+1=-∑j=0∞Et(rt+j-rt+j∗)- ∑j=0∞Etλt+j+ pt-pt∗+limj→∞ Etqt+j+1,
which demonstrates that it is not only current but also expected future values of λt that matter for the exchange rate.
Of special interest are the theories of λt that might account for the uncovered interest parity puzzle. This is the empirical puzzle that finds over many time periods for many currency pairs the slope coefficient in the regression:
(6)st+1-st=a+b(it-it∗)+ut+1
is less than one and often negative. Under the null hypothesis that λt=0 in equation (1), the regression coefficients should be a=0 and b=1. This survey will draw the link between models that are derived to explain the empirical findings concerning regression (6) and the implications of the implied behavior of λt for the exchange rate. In addition to the theoretical reasons noted in the previous paragraph for why we might have λt≠0, the literature has also raised the possibility that empirical work mismeasures Et st+1(the “peso problem”), or that econometric issues lead to spurious rejection of the null hypothesis.
The plan of the chapter is to consider first the “traditional” asset market approach, in which λ t=0. We survey how the New Keynesian literature has given theoretical and empirical insights into nominal exchange rate behavior. We consider exchange rate dynamics and volatility, and whether models are useful for forecasting exchange rate changes.
Then we consider different exchange-rate regimes and survey the large empirical literature on sterilized foreign exchange market intervention in floating-exchange rate countries.
The last part of the survey turns to models of λt. We take up the literature that has modeled foreign exchange risk premiums, and approaches that allow for violations of the representative-agent rational-expectations framework. Both have implications for the determination of exchange rates and the resolution of the uncovered interest parity puzzle.
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Cross-border mergers and acquisitions
Donald M. DePamphilis Ph.D., in Mergers, Acquisitions, and Other Restructuring Activities (Eleventh Edition), 2022
Applying the Fisher effect
The so-called Fisher effect states that nominal interest rates can be expressed as the sum of the real interest rate (i.e., interest rates excluding inflation) and the anticipated rate of inflation. The Fisher effect can be shown for the United States and Mexico as follows:
(1+ius )=(1+rus)(1+Pus) and(1+rus)=(1+ius)/(1+Pus)
(1+imex)=(1+rmex)(1+Pmex)and(1+rmex)=(1+imex)/(1+Pmex)
If real interest rates are constant among all countries, nominal interest rates among countries will vary only by the difference in the anticipated inflation rates. Therefore,
(19.6)(1+ius)/ (1+Pus)=(1+imex)/(1+Pmex)
in which
ius and imex = nominal interest rates in the United States and Mexico, respectively
rus and rmex = real interest rates in the United States and Mexico, respectively
Pus and Pmex = anticipated inflation rates in the United States and Mexico, respectively
If the analyst knows the Mexican interest rate and the anticipated inflation rates in Mexico and the United States, solving Equation 19.6 provides an estimate of the US interest rate (i.e., ius = [(1 + imex) × (1 + Pus)/(1 + Pmex)] – 1). Exhibit 19.3 illustrates how the cost of equity estimated in one currency is converted easily to the cost of equity in another using Equation 19.6. Although the historical equity premium in the United States is used in calculating the cost of equity, the historical UK or MSCI premium also could have been used.
Exhibit 19.3
Calculating the Target Firm’s Cost of Equity in Both Home and Local Currencies
Acquirer, a US multinational firm, is interested in purchasing Target, a small UK-based competitor, with a market value of ₤550 million, or about $1 billion. The current risk-free rate of return for UK 10-year government bonds is 4.2%. The anticipated inflation rates in the United States and the United Kingdom are 3% and 4%, respectively. The size premium is estimated at 1.2%. The historical equity risk premium in the United States is 5.5%.a Acquirer estimates Target’s ß to be 0.8 by regressing Target’s historical financial returns against the Standard & Poor’s 500. What is the cost of equity (ke,uk) that should be used to discount Target’s projected cash flows when they are expressed in terms of British pounds (i.e., local currency)? What is the cost of equity (ke,us) that should be used to discount Target’s projected cash flows when they are expressed in terms of US dollars (i.e., home currency)?b
ke ,uk(seeEquation19.5)=0.042+0.8×(0.055)+0.012=0 .098=9.80%
ke,us(seeEquation19.6)=[(1+0.098)]×(1undefined+ 0.03)×/(1+0.04)−1=0.0875×100=8.75%
Estimating the cost of equity in emerging (segmented) capital market countries
If capital markets are segmented, the global CAPM must reflect the tendency of investors in individual countries to hold local country rather than globally diversified equity portfolios. Thus, equity premiums differ among countries, reflecting the nondiversifiable risk associated with each country’s equity market index. What follows is a discussion of how to adjust the basic CAPM for valuing cross-border deals when the target is located in an emerging country.
Estimating the risk-free rate of return (emerging countries)
Data limitations and the absence of a legal procedure to deal with sovereign debt (i.e., government-issued debt) in default often preclude using the local country’s government bond rate as the risk-free rate. There is no court to approve a debt restructuring plan to reduce, wipe out, or convert debt to equity for commercial bankruptcies. Troubled countries negotiate directly with lenders to restructure debt by reducing the amount owed, lowering the interest rate, extending the maturity of the debt, or some combination of the three.
Countries defaulting on their sovereign debt do pay a price, but the impact is limited in duration. The increase in their borrowing costs after default tends to shrink such that it becomes negligible within 5 years after the countries return to the bond market. The default risk premium paid by these countries is higher for countries that take a long time to settle with creditors and is much higher for countries that have a history of defaulting on their bonds. These “penalties” explain why government debt defaults are infrequent, and when they occur, governments typically try to settle quickly with creditors.57
As an alternative to the local country government bond rate, the US Treasury bond rate often is used to estimate the risk-free rate if the target firm’s cash flows are in terms of local currency. To create a local nominal interest rate, the Treasury bond rate should be adjusted for the difference in the anticipated inflation rates in the two countries using Equation 19.6. Alternatively, the risk-free rate can be estimated using the buildup method as the sum of the expected inflation rate and the expected real rate. The analyst can add the expected inflation rate for the country to the US Treasury inflation-adjusted bond rate (i.e., Treasury inflation-protected securities, [TIPS]). For example, the expected inflation rate for Angola in June 2012 was 12%, and the 5-year rate on US treasury inflation-indexed securities (the real rate) was 2.38%. Therefore, the estimated risk-free rate for Angolan government bonds at that time was 14.38%.
Adjusting CAPM for risk (emerging countries)
Systematic risk for a firm operating primarily in its emerging country’s home market (whose capital market is segmented58) is measured mainly with respect to the country’s equity market index (βemfirm,country) and to a lesser extent with respect to a globally diversified equity portfolio (βcountry,global). The emerging-country firm’s global beta (βemfirm,global) can be adjusted to reflect the relationship with the global capital market as follows:
(19.7)ßemfirm,global=ßemfirm,country×ßcountry,global
The value of βemfirm,country is estimated by regressing historical returns for the local firm against returns for the country’s equity index.59 The value of βcountry,global can be estimated by regressing the financial returns for the local-country equity index (or for an index in a similar country) against the historical financial returns for a global equity index.60 Because of the absence of historical data in many emerging economies, the equity risk premium often is estimated using the “prospective method” implied in the constant-growth valuation model. As shown in Equation 7.4 in Chapter 7, this formulation provides an estimate of the present value of dividends growing at a constant rate in perpetuity. That is, dividends paid in the current period (d0) are grown at a constant rate of growth (g) such that d1 equals d0(1 + g).
Assuming the stock market values stocks correctly and we know the present value of a broadly defined index in the target firm’s country (Pcountry) or in a similar country, dividends paid annually on this index in the next period (d1), and the expected dividend growth (g), we can estimate the expected return (Rcountry) on the stock index as follows:
(19.8)Pcountry=d 1/(Rcountry−g)andRcountry=(d1/Pcountry)+g
From Equation 19.8, the equity risk premium for the local country’s equity market is Rcountry – Rf, in which Rf is the local country’s risk-free rate. Exhibit 19.4 illustrates how to calculate the cost of equity for an emerging country firm in the absence of perceived significant country or political risk not captured in the beta or equity risk premium. Note the local country’s risk-free rate is estimated using the US Treasury bond rate adjusted for expected inflation in the local country relative to the United States. This converts the US Treasury bond rate into a local-country nominal rate.
Exhibit 19.4
Calculating the Target Firm’s Cost of Equity for Firms in Emerging Countries
Assume next year’s dividend yield on an emerging country’s stock market is 5%, and earnings for the companies in the stock market index are expected to grow by 6% annually in the foreseeable future. The country’s global beta (ßcountry,global) is 1.1. The US Treasury bond rate is 4%, and the expected inflation rate in the emerging country is 4% compared with 3% in the United States. Estimate the country’s risk-free rate (Rf), the return on a diversified portfolio of equities in the emerging country (Rcountry), and the country’s equity risk premium (Rcountry – Rf). What is the cost of equity in the local currency for a local firm (ke,em) whose country beta (ßemfirm,country) is 1.3?
Rf=[(1+0.04)((1undefined+ 0.04 )/(1+0.03))−1]=0.0501×100=5.01%
Rcountry(seeEquation19.8)=5.00+6.00= 11.00%
Rcountry−Rf=11.00−5.01= 5.99%
βemfirm,global(see Equation 19.7)=1.3×1.1=1.43
ke,em=5.01+1.43( 5.99)=13.58%
Adjusting the CAPM for country or political risk (emerging countries)
A country’s equity premium may not capture all the events that could jeopardize a firm’s ability to operate, such as political instability, limits on repatriation of earnings, capital controls, and the levying discriminatory taxes. Such factors could increase the firm’s likelihood of default. Unless the analyst includes the risk of default in projecting a local firm’s cash flows, the expected cash flow stream would be overstated in that it does not reflect the costs of financial distress.
If the US Treasury bond rate is used as the risk-free rate in calculating the CAPM, adding a country risk premium (CRP) to the basic CAPM estimate is appropriate. The CRP often is measured as the difference between the yield on the country’s sovereign or government bonds and the US Treasury bond rate of the same maturity. The difference, or “spread,” is the additional risk premium that investors demand for holding the emerging country’s debt rather than US Treasury bonds.61 S&P (www.standardardandpoors.com), Moody’s Investors Service (www.moodys.com), and Fitch IBCA (www.fitchibca.com) provide sovereign bond spreads. In practice, the sovereign bond spread is computed from a bond with the same maturity as the US benchmark 10-year Treasury bond used to compute the risk-free rate for calculating the cost of equity.
Although political risk has traditionally been linked with emerging countries, instances in the recent past illustrate political risk associated with equities and sovereign bonds in developed countries. The sovereign bond crisis in Spain and Italy following the global recession in 2008 to 2009 caused government bond rates to rise and stock prices to fall until it became clear that the eurozone would remain intact. Similarly, the decision by the United Kingdom to exit the European Union in 2016 triggered concern about the long-term health of the British economy. Consequently, the adjustments for political risk suggested in this section also can apply to developed countries.
Global CAPM formulation (emerging countries)
To estimate the cost of equity for a firm in an emerging economy (ke,em), Equation 19.5 can be modified for specific country risk as follows:
(19.9)ke,em=Rf+ßemfirm,global(Rcountry −Rf)+FSP+CRP
in which
Rf = local risk-free rate or the US Treasury bond rate converted to a local nominal rate if cash flows are in the local currency (see Equation 19.6) or to the US Treasury bond rate if cash flows are in dollars
(Rcountry – Rf) = difference between expected return on a well-diversified equity index in the local country or a similar country and the risk-free rate
βemfirm,global = emerging country firm’s global beta (see Equation 19.7)
FSP = firm size premium, reflecting the additional return that smaller firms must earn relative to larger firms to attract investors
CRP = specific country risk premium, expressed as the difference between the local country’s (or a similar country’s) government bond rate and the US Treasury bond rate of the same maturity. Add to the CAPM estimate only if the US Treasury bond rate is used as a proxy for the local country’s risk-free rate.
Estimating the local firm’s cost of debt in emerging markets
The cost of debt for an emerging market firm (iemfirm) should be adjusted for default risk because of country- and firm-specific factors. When a local corporate bond rate is not available, the cost of debt for a local firm may be estimated by using an interest rate in the home country (ihome) that reflects a level of creditworthiness similar to the firm in the emerging country. The CRP is added to the appropriate home country interest rate to reflect the impact of such factors as political instability on iemfirm. Therefore the cost of debt can be expressed as follows:
(19.10)i emfirm=ihome+CRP
Most firms in emerging markets are not rated; to determine which home-country interest rate to select, it is necessary to assign a credit rating to the local firm. This “synthetic” credit rating is obtained by comparing financial ratios for the target firm with those used by US rating agencies. The estimate of the unrated firm’s credit rating may be obtained by comparing interest coverage ratios used by S&P with the firm’s interest coverage ratio to determine how S&P would rate the firm. Exhibit 19.5 illustrates how to calculate the cost of emerging-market debt.
Exhibit 19.5
Estimating the Cost of Debt in Emerging Market Countries
Assume that a firm in an emerging market has annual operating income before interest and taxes of $550 million and annual interest expenses of $18 million. This implies an interest coverage ratio of 30.6 (i.e., $550 ÷ $18). For S&P, this corresponds to an AAA rating. According to Standard & Poor’s, default spreads for AAA firms are 0.85 currently. The current interest rate on US triple A–rated bonds is 6.0 %. Assume further that the country’s government bond rate is 10.3% and that the US Treasury bond rate is 5%. Assume that the firm’s marginal tax rate is 0.4. What is the firm’s cost of debt before and after tax?
Costofdebtbeforetaxes(see Equation 19.10)=6 .0+(10.3−5.0)=11.3%
After-taxcostofdebt=11.3×(1−0.4)=6.78%
Exhibit 19.6 illustrates the calculation of weighted average cost of capital (WACC) in cross-border transactions. Note the adjustments made to the estimate of the cost of equity for firm size and country risk. Note also the adjustment made to the local borrowing cost for country risk. The risk-free rate of return is the US Treasury bond rate converted to a local nominal rate of interest.
Exhibit 19.6
Estimating the Weighted Average Cost of Capital (WACC) in Cross-Border Transactions
aNote that the expression {[(1 + 0.05) × (1 + 0.06)/(1 + 0.03)] – 1} × 100 represents the conversion of the US Treasury bond rate to a local nominal rate of interest using Equation 19.6. Also note that 1.3 × 0.7 provides the target’s global beta, as indicated in Equation 19.7.
Acquirer Inc., a US-based corporation, wants to purchase Target Inc. Acquirer’s management believes that the country in which Target is located is segmented from global capital markets because the beta estimated by regressing the financial returns on the country’s stock market with those of a global index is significantly different from 1.
Assumptions: The current US Treasury bond rate (Rus) is 5%. The expected inflation rate in the target’s country is 6% annually, as compared to 3% in the US. The country’s risk premium (CRP) provided by S&P is estimated to be 2%. Based on Target’s interest coverage ratio, its credit rating is estimated to be AA. The current interest rate on AA-rated US corporate bonds is 6.25%. Acquirer Inc. receives a tax credit for taxes paid in a foreign country. Because its marginal tax rate is higher than Target’s, Acquirer’s marginal tax rate of 0.4 is used in calculating WACC. Acquirer’s pretax cost of debt is 6%. The firm’s total capitalization consists only of common equity and debt. Acquirer’s projected debt–to–total capital ratio is 0.3.
Target’s beta and the country beta are estimated to be 1.3 and 0.7, respectively. The equity premium is estimated to be 6% based on the spread between the prospective return on the country’s equity index and the estimated risk-free rate of return. Given Target Inc.’s current market capitalization of $3 billion, the firm’s size premium (FSP) is estimated at 1.0 (see Table 7.1 in Chapter 7). What is the appropriate WACC Acquirer should use to discount Target’s projected annual cash flows, expressed in its own local currency?
ke,em(see Equation 19.9)={[(1+0.05)×(1+0.06)/(1+0.03) ]−1}×100a+1.3×0.7(6.0)+1.0+2.0=16.52%
ilocal(seeEquation19.10)=6.25+2.0=8.25%
waccem(see Equation 7.4)= 16.52×(1−0.3)+8.25× (1−0.4)×0.3=13.05%
Table 19.1 summarizes methods commonly used for valuing cross-border M&As for developed-country and emerging-country firms. The WACC calculation assumes that the firm uses only common equity and debt financing. The CRP is added to both the cost of equity and the after-tax cost of debt in calculating the WACC for a target firm in an emerging country if the US Treasury bond rate is used as the risk-free rate of return. The analyst should avoid adding the CRP to the cost of equity if the risk-free rate used to estimate the cost of equity is the local country’s government bond rate. References to home and local countries in Table 19.1 refer to the acquirer’s and the target’s countries, respectively.62
TABLE 19.1. Common methodologies for valuing cross-border transactions
| Step 1. Project and convert cash flows. a. Project target’s cash flows in local currency.b. Convert local cash flows into acquirer’s home currency using forward exchange rates projected using interest rate parity theory. | Step 1. Project and convert cash flows.a. Project target’s cash flows in local currency. b.Convert local cash flows into acquirer’s home currency using forward exchange rates. Project exchange rates using purchasing power parity theory if little reliable data on interest rates are available. |
| Step 2. Adjust discount rates. ke,dev = Rf + ßdevfirm,globala(Rm – Rf) + FSP i = cost of debtc WACC = keWe + i(1 – t) ×Wda. Rf is the long-term government bond rate in the home country. b.ßdevfirm,global is nondiversifiable risk associated with a well-diversified global, US, or local-country equity index. c.Rm is the return on a well-diversified US, local, or global equity index. d.FSP is the firm size premium. e.t is the appropriate marginal tax rate. f.We is the acquirer’s target equity–to–total capital ratio, and Wd is 1 – We. | Step 2. Adjust discount rates. ke,em = Rf + ßemfirm,globala(Rcountry – Rf)b + FSP + CRP ilocal = ihome + CRP WACC = keWe + ilocal (1 – t) × Wda. Rf is the long-term government bond rate in the local country or the US Treasury bond rate converted to a local nominal rate if cash flows in local currency; or if cash flows in dollars, the US Treasury bond rate. Note that if the local risk-free rate is used, do not add CRP. b.ßemfirm,global is nondiversifiable risk associated with target’s local-country ß and local country’s global ß. c.Rcountry is the return on a diversified local equity index or a similar country’s index. d.CRP is the country risk premium. e.ihome is the home-country cost of debt. f.ilocal is the local-country cost of debt. |
WACC, Weighted average cost of capital.
aß may be estimated directly for firms whose business is heavily dependent on exports or operating in either developing or emerging countries by regressing directly the firm’s historical financial returns against returns on a well-diversified global equity index. For firms operating primarily in their home markets, ß may be estimated indirectly by using Equation 19.7.b(Rcountry – Rf) also could be the equity premium for well-diversified US or global equity indices if the degree of local segmentation is believed to be small.cFor developed countries, either the home-country or local-country cost of debt may be used. There is no need to add a country risk premium as would be the case in estimating a local emerging country’s cost of debt.Read full chapter
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Savings Revisited
Nathan Tintle, ... Todd Swanson, in A Spiral Approach to Financial Mathematics, 2018
Summary
In this section we saw how for a given nominal interest rate, more and more frequent compounding continues to increase the corresponding effective interest rate. However, there is a limit to the increase. This limit is called the force of interest. The force of interest, essentially representing the idea of continuously compounded interest, can help to quickly approximate frequent (e.g., daily) compounding.
Notation and equation summary
Force of interest for compound interest
limm→∞(1+i(m)m)m−1=ei(∞ )−1=e∂−1
General force of interest for accumulation function a(t)
∂(t)=a′(t)a(t)
Generalized formula for future value in terms of a continuously compounded rate, r.
A(t)=Pert
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Cross-Border Mergers and Acquisitions
Donald M. DePamphilis Ph.D., in Mergers, Acquisitions, and Other Restructuring Activities (Fifth Edition), 2010
Applying the Fisher Effect
The so-called Fisher effect states that nominal interest rates can be expressed as the sum of the real interest rate (i.e., interest rates excluding inflation) and the anticipated rate of inflation. The Fisher effect can be shown for the United States and Mexico as follows:
(1+ius)=(1+rus)(1+Pus)
and
(1+rus)=(1+ius)/ (1+Pus)
(1+imex)=(1+rmex)(1+Pmex)
and
(1+rmex)=(1+imex)/(1+Pmex)
If real interest rates are constant among all countries, nominal interest rates between countries vary by only the difference in the anticipated inflation rates. Therefore,
(17-6) (1+ius)/(1+Pus)=(1+imex)/(1+Pmex)
where
ius and imex = nominal interest rates in the United States and Mexico, respectively.
rus and rmex = real interest rates in the United States and Mexico, respectively.
Pus and Pmex = anticipated inflation rates in the United States and Mexico, respectively.
If the analyst knows the Mexican interest rate and the anticipated inflation rates in Mexico and the United States, solving Equation (17-6) provides an estimate of the U.S. interest rate; that is, ius=(1+i mex)×[(1+Pus)/(1+Pmex)]−1. Exhibit 17-3 illustrates how the cost of equity estimated in one currency is converted easily to another using Equation (17-6). Although the historical equity premium in the United States is used in calculating the cost of equity, the historical U.K. or MSCI premium also could have been employed.
Exhibit 17-3
Calculating the Target Firm's Cost of Equity in Both Home and Local Currency
Acquirer, a U.S. multinational firm, is interested in purchasing Target, a small U.K.-based competitor, with a market value of £550 million or about $1 billion. The current risk-free rate of return for U.K. 10-year government bonds is 4.2 percent. The anticipated inflation rates in the United States and the United Kingdom are 3 and 4 percent, respectively. The estimated size premium for a small capitalization firm is 1.2 percent (see Chapter 7, Table 7-1). The historical equity risk premium in the United States is 5.5%.1 Acquirer estimates Target's β to be 0.8, by regressing Target's historical financial returns against the S&P 500. What cost of equity (ke,uk) should be used to discount Target's projected cash flows when they are expressed in terms of British pounds (i.e., local currency)? What cost of equity (ke,us) should be used to discount Target's projected cash flows when they are expressed in terms of U.S. dollars (i.e., home currency)?2
ke,uk,see equation(17–5),=0.042+0.8×(0.055)+0.012=0.098=9.80%
k e,us,see equation(171–6),=(1+0 .098)×[(1+0.03)/(1+0.04) ]-1=0.0875=8.75%
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Cross-Border Mergers and Acquisitions
Donald M. DePamphilis Ph.D., in Mergers, Acquisitions, and Other Restructuring Activities (Sixth Edition), 2012
Applying the fisher effect
The so-called Fisher effect states that nominal interest rates can be expressed as the sum of the real interest rate (i.e., interest rates excluding inflation) and the anticipated rate of inflation. The Fisher effect can be shown for the United States and Mexico as follows:
(1+i us)=(1+rus)(1+Pus)and(1+rus)=(1 +ius)/(1+Pus)(1 +imex)=(1+rmex)(1+ Pmex)and(1+rmex)=(1+imex)/(1+Pmex)
If real interest rates are constant among all countries, nominal interest rates among countries will vary only by the difference in the anticipated inflation rates. Therefore,
(17.6)(1+ius)/(1+Pus)=(1+imex)/(1+Pmex)
where
ius and imex = nominal interest rates in the United States and Mexico, respectively
rus and rmex = real interest rates in the United States and Mexico, respectively
Pus and Pmex = anticipated inflation rates in the United States and Mexico, respectively
If the analyst knows the Mexican interest rate and the anticipated inflation rates in Mexico and the United States, solving Eq. (17.6) provides an estimate of the U.S. interest rate (i.e., ius = [(1 + imex) × (1 + Pus)/(1 + Pmex)] – 1). Exhibit 17.3 illustrates how the cost of equity estimated in one currency is converted easily to another using Eq. (17.6). Although the historical equity premium in the United States is used in calculating the cost of equity, the historical U.K. or MSCI premium also could have been employed.
Exhibit 17.3
Calculating the Target Firm's Cost of Equity in Both Home and Local Currency
Acquirer, a U.S. multinational firm, is interested in purchasing Target, a small U.K.-based competitor, with a market value of £550 million, or about $1 billion. The current risk-free rate of return for U.K. ten-year government bonds is 4.2%. The anticipated inflation rates in the United States and the United Kingdom are 3% and 4%, respectively. The expected size premium is estimated at 1.2%. The historical equity risk premium in the United States is 5.5%.a Acquirer estimates Target's β to be 0.8, by regressing Target's historical financial returns against the S&P 500. What is the cost of equity (ke,uk) that should be used to discount Target's projected cash flows when they are expressed in terms of British pounds (i.e., local currency)? What is the cost of equity (ke,us) that should be used to discount Target's projected cash flows when they are expressed in terms of U.S. dollars (i.e., home currency)?b
ke,uk( see Eq.(17.5))=0.042+0.8×(0.055)+0.012=0.098 =9.80%ke,us(see Eq.(17.6))=[(1+0.098)×(1.+0.03)/(1+0.04)]−1=0.0875×100=8.75%
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