Table 2-1. Interest Rate Historic Data
It can be seen from Table2-1, from the year 2009 to year 2011, the interest rates of USA and UK are constant, which are 3.25% and 0.5% respectively. The interest rate of Australia has changed a little. It decreases in the first quarter of 2009, from 3.25% to 3.00% and remains 3.00% for half a year. In the fourth quarter of 2009, it increased to 3.5% and continuously increases to 4.75% in the last quarter of 2010. It remains 4.75% in the first three quarters and decreased to 4.25% in the last quarter of 2011. When the relative interest rate is higher than other countries, just like Australia, it will attract investors from the other countries to purchase more Australian Dollars dollar. Then, the inflow of investment from other countries will increase the purchase of domestic goods. In contrast, the cash outflow of investing in other countries will decrease, which leads to an increase in the Australia Exchange Rate.
Calculation and analysis the purchasing power parity theory
Interpretation of Purchasing power parity
Purchasing power parity (PPP) theory specifies a precise relationship between relative inflation rates of two countries and their exchange rates in a specific period. It says that the real value of currencies in two countries will remain the same over time. In inexact terms, PPP theory suggests that interest rate change and exchange rate change have the same magnitude in achieving the equilibrium of purchasing power between two currencies. When one country's inflation rate rises relative to that of another, its currency depreciates as its exports increase (due to its low prices), and vice versa.
Formula derivation of Purchasing power parity
Assume that the price indexes of the domestic economy (d) and a foreign economy (f) are equal in the beginning, Pd and Pf, respectively. With time goes on, the domestic economy experiences an inflation rate of Id, while the foreign country experiences an inflation rate of If. When the domestic country experiences inflation, the price index of goods in the consumer’s home country (Pd) becomes
Pd (1+ Id)
The price index of the foreign country (Pf) will change due to inflation in the home country:
Pf (1+If)
The theory of PPP also suggests that the exchange rate will not remain constant but will adjust to maintain the parity in purchasing power, when inflation rate differentials happen. If inflation occurs and the exchange rate of the foreign currency changes, the foreign price index from the home consumer’s perspective becomes
Pf (1+If)(1+ef),
where ef represents the percentage change in the value of the foreign currency. According to PPP theory, the percentage change in the foreign currency (ef) should vary to maintain parity in the new price indexes of the two countries. We can solve for ef under conditions of PPP by setting the formula for the new price index of the foreign country equal to the formula for the new price index of the domestic country, as follows:
Pf (1+If)(1+ef) = Pd(1+Id)
Since Pd equals Pf (because price indexes were initially assumed equal in both countries), they cancel leaving, we can calculate that
ef = (1+Id)/(1+If)- 1
This formula reflects the percentage change in the value of the foreign currency due to the relative inflation rate of two countries under PPP condition.
And at this time, due to the exchange marketing effect, the percentage change in the value of the foreign currency in fact is ef’, where the so is the spot rate of the foreign currency in the previous stage, and the st is the spot rate of the foreign currency in the next stage.
We can calculate that
ef’ = (st-so)/so
And the R is the ratio of two percentage change in the value of the foreign currency, one is due to the relative inflation rate, another is due to the exchange rate. We can calculate that
R= ef/ef’
= [(1+Id) / (1+If) -1] / [(st-so)/so]
According to purchasing power parity (PPP), if the ratio (R) is not 1, there is a disparity in the purchasing power of two countries. When the ratio (R) is positive, it reflects the PPP theory exists. When the result (R) is negative, it reflects the PPP theory does not occur. And if the ratio (R) is 1, it reflects the PPP theory is absolutely effective. And if it is far away from 1, it reflects the PPP theory has less and less effect. When the ratio is greater than 1, or less than -1, it represents more favorable purchasing power on the domestic goods than on foreign goods. When the ratio is greater than -1 and less than 1, it represents more favorable purchasing power on the foreign country goods than on domestic goods.
Calculation and analysis between UK and USA
In the first quarter of 2009, the USA dollars experience a -0.0392% inflation rate. The inflation rate of UK is 3.0143%, the value of the UK pound changed due to the relative inflation rate as shown:
euk= (1+Ius)/(1+Iuk)-1
= (1-0.000392)/(1+0.030143)-1
= - 0.0296415
-
This reflects the UK should depreciate by 2.96415 percent in response to the UK higher inflation relative to the US.
In fact, the exchange rate (between the US and the UK) first quarter of 2009 is 1.4300. And the one in the second quarter of 2009 is 1.6448. The exchange rate of it was appreciated. At this time, the percentage change in the value of UK pounds euk’
euk’= (st-so)/so
= (1.4300-1.6448)/1.6448
=- 0.13059
This reflects the percentage change in the value of UK pounds in fact depreciates by 13.059 percent.
And the ratio
R=
= (- 0.0296415)/(- 0.13059)
= 0.2269755
The ratio is positive, which reflects that the PPP theory has occurred. Because the euk ratio is negative, and the euk’ ratio is also negative, it reflects that it represents more favorable purchasing power for US dollar than UK pound.
Every country comprehensive analysis
Table 2-2. USD to AUD spot and forward rate
Table 2-3. Ratio of Australia
Table 2-4. USD to GBP spot and forward rate
Table 2-5. Ratio of United Kingdom
From the data calculated above, we can see that the PPP theory sometimes can hold true and sometimes not. The ratio of Australia, in the last three quarters of 2010 and also in the first three quarters of 2009, stays within the range between -1 and 1, which shows that the purchasing power parity is right. However, in other quarters the ratio of Australia is not within that level, e.g. in 2011 this ratio exceeds ten. For UK, the ratio of all these quarters is nearly zero, which also clarifies the preciseness of PPP. Conclusively, the purchasing power theory is almost proved by the truth of these countries.
Calculate and analysis the interest rate parity theory
Interpretation of interest rate parity
Interest rate parity is a no-arbitrage condition where investors should be indifferent to the common interests of the bank deposits in the two countries. The two key assumptions of interest rate parity are capital mobility and perfect substitutability of domestic and foreign assets. Under interest rate parity condition, the expected return on domestic assets is the same as the expected return on foreign currency, due to the equal balance on the foreign exchange market. When the investor of one country receives a higher interest rate from the foreign investment, the investor is paying more per unit of foreign currency than what is received per unit when the currency is sold forward.
Formula derivation of Interest rate parity
Assume that the investor who attempts covered interest arbitrage. The amount of the domestic currency is initially invested (Qd), the spot rate (Sd) in the domestic currency when the foreign currency is purchased. The interest rate on the foreign deposit (if), and the forward rate (f) in the domestic currency are the rates that the foreign currency will be converted back to the home currency.
The amount of the domestic currency received at the end of the deposit period due to such a strategy (Qend) is:
Qend= (Qd/Sd) (1+if)f
Since f is simply the spot rate Sd times one pulls the forward premium (fp), we can rewrite this equation as
Qend= (Qd/Sd) (1+if) [Sd(1+fp)]= Qd (1+if)(1+fp)
The rate of return from this investment (R) is
R= (Qend-Qd)/Qd= (1+if)(1+fp)-1
If interest rate parity exists, then the rate of return achieved from covered interest arbitrage (R) should be equal to the rate available in the domestic country (id).
R= id= (1+if) (1+fp)-1
Therefore, it can be calculated that:
id = (1+if)(1+fp)-1
fp= [(1+id)/(1+if)]-1
In the meantime, if investors simply invest their money domestically, the forward rate (f) will let the investor receive the forward premium (fp'), which is calculated below:
fp'=(f-Sd)/Sd
And the Final Result (FR) is the ratio of two forward premiums, one is due to different interest rate between two countries, another is because that the forward rate differs from the spot rate at one time. It can be calculated as:
FR= fp'/ fp = [ (f-Sd)/Sd]/ [(1+id)/(1+if)-1]
According to the interest rate parity, when the Final Return (FR) is positive, it means the PPP theory holds. While the FR is negative, it reflects the PPP doesn’t hold. If the FR is 1, it means the PPP theory is absolutely effective. And if it is far away from 1, it reflects the PPP theory has less and less effect. When the ratio greater than 1, or less than -1, it represents the investor would achieve a lower return on foreign investment than they would on a domestic one. When the ratio greater than -1 and less than 1, it represents the investor would achieve a higher return on foreign investment than they would invest on a domestic one.
Calculation and analysis between UK and USA
In the third quarter of 2011, the USA dollars has a 3.5% interest rate. The interest rate of UK is 0.5%, the forward premium is achieved due to different interest rate between the USA and the JAPAN as shown:
fp UK= (1+iusa)/ (1+iuk) – 1= ( 1+3.25%)(1+0.5%)-1= 0.0376 or 3.76%
This implies that the UK will have a forward premium of 3.76%.
When investors invest their funds domestically, the spot rate between the USA and the UK is 0.0134. The forward rate in that time is 0.0157; they will receive a forward premium fp’uk
fp’uk= (f-Sd)/Sd = (0.0157-0.0134)/0.0134 = 0.0171
The result implies that when the USA investors invest their funds domestically, they will loss 1.71%.
And the Final Return ratio:
FR= fp'/ fp = [ (f-Sd)/Sd]/ [(1+id)/(1+if)-1]= 0.0171/0.0376= 0.45
The FR is positive, which reflects the IRP theory has been hold.
The forward rate is not an unbiased estimator of the spot rate
Statistical test of forecast bias
The forward rate is the rate that appears in a contract to exchange a currency in the future.
If the forward rate is a biased predictor of the future spot rate, this implies that there is a systematic forecast error, which could be corrected to improve forecast accuracy. If the forward rate is unbiased, it fully reflects all available information about the future spot rate. Any forecast errors would be the result of events that could not have been anticipated from existing information at the time of the forecast. A conventional method of testing for a forecast bias is to apply the following regression model to historical data.
St=ao+a1Ft-1 +μt
Where: St=Spot rate at time t
Ft-1=forward rate at time t-1
μt= error term
ao= intercept
a1= regression coefficient
If the forward rate is unbiased, the intercept should equal zero and the regression coefficient a1, should equal 1.0.The test for a1 is:
t=
t=( a1-1)/standard error of a1
If ao = 0 and a1 is significantly less than 1.0, this implies that the forward rate is systematically overestimating spot rate. Conversely if ao =0 and a1 is significantly greater than 1.0, this implies that forward rate is systematically underestimating spot rate. When a bias is detected and anticipated to persist in the future, future forecasts may incorporate the bias detected. The graph 1 shows that the a1 of Australia is 1.024052699. Graph 2 shows the a1 of UK is 0.9922228. By detecting bias, an MNC may be able to revise its forecast to adjust for the bias so it can improve its forecasting accuracy.
.
Graph 1. Linear Regression of Spot and Forward Rate of Australia
Graph 2. Linear Regression of Spot and Forward Rate in Australia
Calculate and interpret the forward premium or discount
When the forward rate exceeds the existing the spot rate, it contains a premium. This is, the forward premium occurs when the foreign currency is more expensive in the future. And if the forward rate is less than the existing the spot rate, it contains a discount.
Formula derivation of the forward premium
The forward premium or discount is always calculated as the annualized percentage difference between the spot and the forward rates a proportion of the spot rate, that is:
P= (st-so)/so××100%
Calculation and analysis with UK and USA
The spot rate in the first quarter of 2009 is UK pounds/ USA dollars = 1.4334. And the forward rate in that time is UK pounds/ USA dollars = 1.4339. It is about a quarter. If the UK investor bought the currency in the financial market, the forward premium would be obtained by comparing the rate for buying UK pounds (it is pound/ dollar 1.4334 spot and the pound/ dollar 1.4339 three-months forward.) the forward premium for buying UK pounds would be amounted to:
Puk= (st-so)/so××100%
=××100%
=0.0013953,
This reflects the UK pounds is said to be at an annualized premium of 0.13953 percentage in the three-month forward market against the US dollar based on rates for buying marks.
analysis with chart of every country
This table below illustrates the forward premium, or discount of three countries.
For UK (the relation between the USA and the UK), in the last three quarters of 2009 the ratio is negative, UK has meet the forward discount. And the first quarter of 2009 is 0. At that time, the spot rate is equal to the forward rate. In other quarters, UK meet the forward premium.
For Australia (the relation between the USA and the Australia), in the three years, the forward rate derives a lot from spot rate. In the first three quarters of 2009 and third quarter of 2011, Australia meets a forward discount and the for the rest quarters a forward premium.
Conclusion
In this report, it analyses three currencies (based on the USA dollar) quarterly with the past three years. It researches and analyses those countries by the real date. It explained the purchasing power parity theory and the interest rate parity. And it used the sophisticated quantitative techniques and the theoretical analysis to show that the PPP theory and the IRP theory do not occur perfectly; in fact, it has many barriers, which cannot be explained completely. But they have still affected the reality. Then, this report explains the forward rate is not an unbiased estimator of the spot rate. By the analysis data, it calculated and interpreted the forward premium or discount in the end.
Reference
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Hakkio, Craig S., (1986). Interest Rates and Exchange Rates-What Is the Relationship? Economic Review, Federal Reserve Bank of Kansas City, issue (November), pp. 33-43.
NIJMAN, THEOE., FRANZ C. PALM AND CHRISTIAN P. WOLFF (1993), Premia in forward foreign exchange as unobserved components: A note, Journal of Business & Economic Statistics, July 1993, 11: 361-365.
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