Question 1: Covered Interest Arbitrage

Given Information:

Spot Rate  (S) (¥/$) = 115.47

180-day Forward Rate (F) (¥/$) = 120.70

180-day U.S. interest rate  = 4.00% (foreign interest r_f)

180-day yen interest rate = 2.00% (home interest r_h)

Interest rate Parity Theorem:

As per Interest rate parity theorem,

1 + r_h = F/S * (1 + r_f )   (Sridhar, 2013)

Where –

F is the forward rate

S is the spot rate

r_h = interest rate of home currency

r_f = interest rate of foreign currency

Now let us consider earnings in each country:

In Japan the effective earnings (1 + r_h ) => [(1 + 0.02) – 1] = 0.02 or 2%

In USA, the effective earnings F/S (1 + r_f) => [(¥120.70 / ¥115.40) * (1.04)] – 1 = 1.08776 – 1 = 0.8776 (or) 8%

Since in the given situation, the earnings from investment in both the countries are not equal, Interest rate parity theorem doesn’t hold good and hence there is scope for arbitrage. We will obviously invest in that country which provides more return while borrowing  in that country which provides less return thereby end up getting more return and paying less interest. 

• Borrow in ¥
• Invest in $ (enter into 6 forward contract for sale of $ proceeds redeemable at the expiry of 180 days @ ¥/$ 120.70 on the same day)
• Realise proceeds after 1 year in $
• Repay the ¥ loan

Step 1: Borrow $1,000,000 equivalent of ¥ at spot

¥ borrowed => $1,000,000 x (¥/$) 115.40 = ¥115,400,000

Step 2: Convert ¥115,400,000 into $1,000,000 and invest in USA

Step 3: @ expiry of 180 days period, $ proceeds received from deposit in USA => $1,000,000 x (1 + 0.04) = $1,040,000

Step 4: Convert $ proceeds into ¥ @ forward rate (¥/$) 120.70

¥ Proceeds received => $1,040,000 x (¥/$)120.70 = ¥125,528,000

¥ Payable to bank at expiry of ¥ loan is ¥115,400,000 x (1 + 0.02) = ¥117,708,800

Therefore gain from arbitrage in terms of –

¥ => ¥125,528,000 – ¥117,708,800 = ¥7,820,000

$ => ¥7,820,000 ÷ (¥/$) 120.70 = $64,788.73

As per Interest Rate Parity theorem:

1 + r_h = F/S * (1 + r_f)

• F = [(1 + r_h) * S] / (1 + r_f)

We have –

• S = ($/£) 1.5890
• r_h = 4% or 0.04
• r_f = 6.25% or 0.0625
• F = [(1 + 0.04) * $1.5890] / (1 + 0.0625)
• F = ($/£) 1.5583

1. Dustin Green Model:

Subsitituting the values of obtained by Mr. Green in interest rate parity theorem, we get

Returns from investment in home currency => 1 + r_h = 1 + 0.04 = 1.04

Returns from investment in foreign currency => F/S * (1 + r_f) = ($1.5473 / $1.5890) * (1 + 0.0625) = 1.03125

Therefore as per his model, Dustin Green should borrow in foreign currency and invest in home currency. After realizing the proceeds from deposit at the end of 1 year he should reconvert the proceeds in £ and repay the loan along with interest in £. Therefore he should purchase a forward contract for £ currency

Question 2: Forecasting Exchange Rates

If everyone happens to imitate the transaction adopted by Mr Green, the following are the consequences and ultimately the end result:

• Every one starts borrowing £ and invest in $;
• As a result demand for £ increases and results in increase of £ value in terms of $ thereby increasing the exchange rate, making returns from investment in £ a bit higher. Further shortage of funds in UK market will also lead to raise of UK interest rates making the return even higher;
• The inflow of more funds in USA means increase in availability of funds more than demand and hence it decreases the interest rate and this will result in reduction of  return from investment in $; and
• This automatic adjustment mechanism created by arbitraguers will continue till the returns from both the currency becomes equal and no more arbitrage is possible

Thus if everyone follows the transaction adopted by Mr. Green, the interest rate in USA decreases and the interest rates in UK increases.  

Given information –

Spot Exchange Rate = ($/€) 1.124

Expected Exchange Rate in 1 year = ($/€)1.000

Therefore Appreciation / (Depreciation) of Currency => [(F – S) / S] x 100 = [(1.000 – 1.124)/1.124] x 100 = (11.032%)

I.e., Euro is expected to depreciate by 11.032% over next 1 year

Expected $ return and standard deviation of € risk-free bonds:

Let us assume $100 is invested in € risk-free Bonds

Amount invested in € Bonds => $100 / ($/€)1.124 = €88.9679

By investing €88.9679 in € bonds for 1 year, we realize €88.9679 * (1.01) = €89.8576

By converting the amount into $ we get €89.8576 * ($/€)1.00 = $89.8576

Hence Return = [($89.8576 – $100) / $100] = -10.14% (approx.)

Alternatively, return can also be computed as Return on foreign currency asset + / (-) appreciation / (depreciation) of foreign currency in terms of home currency.

• Return = 1% – 11.032% = -10.032% (this is an approximation method. Hence answer varies from the original return)

We know that variance of stock is computed using following equation:

(R_DC) = ?² (R_FC) + ?² (R_FX) + 2 ?(R_FC) ?(R_FX)ρ(R_FC, R_FX)

Where –

(R_DC) is the Standard Deviation of foreign currency asset in terms of domestic currency

 (R_FC) is the variance of foreign currency = 24%

 (R_FX) is the variance of foreign asset = 0% (since the asset is risk-free and hence variance is always 0)

(R_FC, R_FX) is the correlation between the foreign currency and foreign asset = 0, since the foreign asset is uncorrelated to the foreign currency (as given in the question)

By substituting the above values in the equation, we get:

(R_DC) = 24%² + 0² + 0*24*0

(R_DC) = 24%²

(R_DC) = 24%

1. Expected $ return and standard deviation of € stock index:

Let us assume $100 is invested in € Index

Amount invested in € Index => $100 / ($/€)1.124 = €88.9679

By investing €88.9679 in € index for 1 year, we realize €88.9679 * (1.10) = €97.8648

By converting the amount into $ we get €97.8648 * ($/€)1.00 = $97.8458

Hence Return = [($97.8648 – $100) / $100] = -2.135% (approx.)

Standard Deviation:

(R_DC) = ?² (R_FC) + ?² (R_FX) + 2 ?(R_FC) ?(R_FX)ρ(R_FC, R_FX)

 (R_FC) = 24%

 (R_FX) = 35%

ρ(R_FC, R_FX) = 0

By substituting the values, we get

(R_DC) = 24%² + 35%² + 0*24%*35%

• (R_DC) = 576 + 1225
• (R_DC) = 1801
• (R_DC) = 1801^1/2 = 43.48%
  1. Sub-parts:

1. Standard Deviation if correlation is -0.20

Interest Rate Parity Theorem

(R_DC) = 24%² + 35%² – 0.20*24%*35%

• (R_DC) = 1633
• (R_DC) = 1633^1/2
• (R_DC) = 40.41%

1. Currency Exposure:

The correlation between the exchange rate and price of index is -0.20 which implies that if the stock price increases by 1% then the exchange rate decreases by 1% x 0.20 = 0.20%

Since the price of the index raised by 10%, the exchange rate decreases by 2%

Therefore the overall currency exposure to a US investor for investing in € => 1-0.20 = 0.8

1. As per ICAPM:

E(R) = R_f,DC + β_WM * WMRP + γ_DC * FCRP (“ICAPM (International CAPM)”, 2011)

Where –

γ_DC = sensitivity of the asset’s domestic currency return to a change in the local (foreign) currency

FCRP: foreign currency risk premium

γ_DC = γ_FC + 1

• γ_DC= 0.80 + 1 = 1.80

and

FCRP = (E(S₁) – S₀)/S₀ – (r_DC – r_FC)          (“Foreign Currency Risk Premium (FCRP)”, 2011)

Where –

(E(S₁) – S₀)/S₀ => Expected exchange rate movement = [(1.00 – 1.124) / 1.124] = -11.032%

(r_DC – r_FC) => Interest Rate differential = 4% – 1% = 3%

By substituting the values, we get

FCRP = -11.032% + 3% = -8.032%

Therefore γ_DC * FCRP => 1.80 * -8.032% = -14.4576% (approx.)

Due to depreciation in currency, the expected loss will be -14.4576% of expected returnc

Since the investment was made in €, the investment has a risk of € falling. To mitigate the same, the € forwards are to be sold. Once the hedging position is established, there will be no impact of either currency appreciating or depreciating on the value of portfolio and hence falling or rasining of currency exchange can neither contribute to nor decrease the return. Therefore return on portfolio will be the  naggregate of expected return of the stock and bond as multiplied by their respective weights.

weight of eurpoean stock – 50%

weight of risk-free bond – 50%

Expected return of the portfolio E(R ) =

Where –

W_i is the weight of stock i and

R_i is the Expected return of Stock i

Therefore expected return of given portfolio = (0.5 * 0.01) + (0.5 * 0.1)

= 5.5%

Since the risk of foreign currency exposure is eliminated with hedging, the only risk is risk of the portfolio consisting of risk-free € bond and € index.

We know that variance of portfolio:

?²p=
Therefore ?_p = [w_i² ?²ⁱ + w_j ²?_j² + w_i w_j ?_i ?_jij]^1/2

We know that

_i = 35%

_j = 0% (being risk-free stock)

ij = 0 (since covariance of any asset with risk-free asset is 0

By substituting, we get

p = [0.50²*35² + 0.50²*0² + 0.50*0.50*35*0]^1/2

• p = 306.25^1/2
• p = 17.5%  

  Current Exchange Rate between R$ and $:

Arbitrage Strategy

As per Absolute Purchasing Power Parity Model:

Spot Exchange rate is arrived by dividing the price of a commodity of 2 countries

Spot Exchange Rate => Price of commodity in Brazil / Price of commodity in USA = R$600 / $200

Therefore Spot Rate = (R$/$) 3.00

Theoretical Exchange Rate [(R$/$) 3.00] is less than the actual market exchange rate [(R$/$) 2.58].

i.e., A person will end up paying lesser amount in R$ per $ purchased than he should have paid considering theoretical exchange price

Therefore R$ is overvalued. It is overvalued by (R$/$ 3.00) – (R$/$ 2.58) = R$0.42

Since the R$ is overvalued, the exporter will get more amount than he is expected to get theoretically and hence the overvaluation of R$ will help a Brazilian exporter in next year

As per Relative form of Purchasing Power Parity Theorem,

(P_t / P₀) = (1 + i_h )^t / (1 + i_f )^t

Where –

P_t = Price in terms of foreign currency in period t;

P₀ => Price in terms of foreign currency in period 0 = (R$/$ 2.58)

I_h => Domestic Inflation Rate = 24% ; and

I_f => foreign inflation rate = 3.0%

By rearranging the above equation, we get:

P_t = [(1 + i_h )^t x P₀] / (1 + i_f )^t

By substituting the given values in the equation, we get:

P_t = [(R$/$) 2.58 x (1 + 0.24)]¹ / (1 + 0.03)¹

• P_t = (R$/$) 3.1062

Expected return of Brazilian Bond in terms of $:

If we invest $1.00 in Brazilian bond, we will end up investing R$1.00 * (R$/$) 2.58 = R$ 2.58

At maturity, we will realize R$ proceeds of R$2.50 * (1 + 0.165) = R$2.9125

Now this will give us $ proceeds of R$2.9125 / [(R$/$)3.1062 = $0.9376

Therefore return in terms of $ => [($0.9376 – $1.00) / $1.00 = -6.23% (approx.)

1. As per Internation Fischer Effect:

(1 + r_h) / (1 + r_f) = (1 + i_h ) / (1 + i_f )

By rearranging we get:

(1 + r_h) * (1 + i_f ) = (1 + r_f) * (1 + i_h )

This is a no arbitrage condition and the investor will be indifferent among investing in the currencies.

However if the above equation is not satisfied, then parity does not exist and hence arbitrage is possible

Let us compute LHS and RHS separately

(1 + r_h) * (1 + i_f ) => (1.05) * (1.24) = 1.302 [Since it is given that the rate of interest in US is 5%]

(1 + r_f) * (1 + i_h ) => (1.03) * (1.165) = 1.19995

From the above results, we can see that the return of investing in $ adjusted to the inflation of brazil is more than the return of investing in R$ adjusted to the inflation of USA. Therefore in short term perspective it is not suggested to hold a Brazilian Stock in portfolio. However from a long-term view, the differences in earnings will waive off owing to operation of International Fischer effect and hence the investor would be indifferent in having a Brazilian stock in portfolio. Hence he can hold the same. (“An introduction to the international fisher effect”, n.d.)

The Pension manager expects a cash flow of $250 million. This implies that he didn’t have money in hand now. He expects that the prices of equity share rises and do not want to miss the encash the rise in prices. Therefore he should undertake a Long hedge wherein he purchases futures of the equity contracts at predetermined prices. Here the ultimate motive of the investor is not making profit but to lock the price to be paid in future upfront.

Contracts that should be purchased is computed as follows:

Total Value of Portfolio = $250 million;

Current Price of Index = $1125;

No. of indices per a single forward contract = 250

Therefore Price per contract = $1125 * 250 indices per contract = $281,250

Beta of the portfolio = 1.1

Therefore, no. of contracts that should be purchased => ($250 million / $281,250)*1.1 = 978 contracts (approx.)

Hedge Ratio => β times the value of portfolio = 1.1 times [Since the value of portfolio is to be hedged completely] 

Sridhar, A. (2013). Strategic Financial Management (9th ed., pp. 542-552). Mumbai: Shroff Publishers & Distributors Pvt Ltd.

An introduction to the international fisher effect. Retrieved from https://www.investopedia.com/articles/economics/10/international-fisher-effect.asp

ICAPM (International CAPM). (2011). Retrieved from https://cfaglossary.blogspot.com/2011/04/icapm-international-capm.html

Foreign Currency Risk Premium (FCRP). (2011). Retrieved from https://cfaglossary.blogspot.com/2011/04/foreign-currency-risk-premium-fcrp.html