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math exercise

时间:2015-09-26 22:19来源:未知 作者:admin 点击:
1.
Purchasing Power Parity (PPP)
PPP predicts that national price levels will be equal when converted to a common currency
Purchasing power should be equalized across countries
PPP is built upon the international law of one price (LOOP), which applies to internationally traded goods
Goods market arbitrage is crucial for LOOP
 
The law of one price (LOOP)
LOOP is basically PPP at the level of individual goods
It states that the domestic and foreign prices of an identical good
should be equal when expressed in terms of a common currency
i.e., for any good j:

where:
Pj,t= domestic price of good j at time t in domestic currency
= foreign price of good j at time t in foreign currency
St=bilateral nominal exchange rate at time t (S units of domestic
currency against 1 unit of foreign currency)
LOOP and PPP: a simple 2-good example
With 2 goods the home and foreign Consumer Prices Index (CPI)
at home and abroad is defined as, respectively
      and    
LOOP implies that and
Therefore,
  and
Add these equations together

In other words,
 
LOOP and PPP: general case with n-goods
Suppose there are N goods and  is the weight given to good j in the CPI
CPIs at home and abroad are defined as
 and
Multiplying LOOP by  on both sides gives
Summing on both sides implies PPP

Absolute PPP
The equation is known as Absolute PPP
It states that national price levels are equal when expressed in
terms of a common currency
APPP requires several assumptions:
Perfect competition
Identical goods and tastes ( ) across countries
All goods must be traded
No transport costs
No barriers to trade (e.g. tariffs or quotas)
 
APPP and the real exchange rate
We define the real exchange rate as
The log real exchange rate is  , or more compactly

where lowercase letters are natural logarithms (e.g. )
If APPP is correct, then  and 
Test
We can test whether these predictions hold true using data
LOOP requires that

LOOP therefore predicts

The actual exchange rate on January 14, 2015 was 0.6573 GBP
per USD, therefore:

UK price should be pushed down by arbitrage – cheap US
iPhones should be imported
 
Relative PPP
It is clear from data that APPP does not hold as an exact relationship
This is hardly surprising given that trade costs are not zero in reality
Even if they were, we would get deviations from APPP since different countries have different price index weights()
Goods are not exactly identical across countries()
Therefore, baskets of goods differ across countries
Relative PPP makes allowance for these differences, but it does require that they be stable over time.
 
Suppose that , where = deviation from APPP
The real exchange rate is then
But because is constant by assumption, the change in st does not depend on  :
RPPP:
Where ,  is domestic inflation, and  is foreign inflation
Implication: if  then  domestic exchange rate depreciates.
Test
We can test whether this more realistic PPP theory holds in the data of two countries’ exchange rate.
Empirical evidence on APPP
APP does not hold perfectly and continuously
Deviations from APPP often last several years
But overall there is a clear relationship between the two series, so APPP may hold in the long run
APPP predicts that real exchange rates are constant, but they are actually very volatile
For example, real (and nominal) exchange rates are far more volatile than interest rates
We can also see how real exchange rates tend to revert back to mean, as suggested by the half-life results we discussed above
 
Empirical evidence on RPPP
There are substantial deviations from RPPP at a yearly horizon, but when we look at a 29-year average it holds almost exactly
Taylor and Taylor (2004) report that the results for a 10-year average are very similar
So, although RPPP clearly does not hold at a short horizon, it does appear to hold (at least approximately) at longer horizons such as 10+ years
This conclusion is not just true for the US but it holds for other industrialized economies as well
 
2.
The two versions of the monetary approach to exchange rate determination are the flex-price monetary model of the exchange rate and the Dornbusch (1976) sticky price monetary model of the exchange rate.

The flex-price monetary model of the exchange rate has perfectly flexible prices. This model has four assumptions including the domestic and foreign demand for real money balances, money market equilibrium, PPP and UIP. The model attempts to explain nominal exchange rate movements by relating the exchange rate to money supply. For example, it can help to explain why countries with high money supply growth tend to have depreciating currencies. And FPMM predicts that the nominal exchange rate is a function of relative money supply, relative levels of real GDP and nominal interest rate differential.

The Dornbusch (1976) sticky price monetary model of the exchange rate has sticky prices. It assumes that prices are fixed for only a short period of time and then gradually adjust as time goes on. It also assumes that financial prices are flexible, so UIP holds. According to the Dornbusch Overshooting Model,in the short run, because of the stickiness of prices of goods, when money supply increases, the price level increases by less, and the new short run equilibrium level will first be achieved through shifts in financial market prices. The excess money supply will lower the interest rate up to the point where equilibrium is restored. As the UIP holds, markets expect home currency to appreciate in the future, so the foreign exchange market will initially overreact to a monetary change. Then, gradually, as prices of goods increase over time, the interest rate increases, pushing the real money supply back. And the foreign exchange market continuously approaches its new long-term equilibrium level that is the same as the flex-price model.

Mark confirms that the monetary model does significantly better than a random walk out-of-sample at longer horizons. Mark’s results suggest that monetary models are useful predictors of future exchange rates,but only at relatively long horizons. At horizons less than one year, exchange rate movements are driven also by transitory factors which are not related to economic fundamentals, and these transitory factors that affect exchange rates but which are unrelated to fundamentals are called noise.

First-generation tests of monetary models were strongly supportive of both the two models – FPMM and Dornbusch. Second-generation tests found far less favorable results, but small modifications of the monetary model can account for some of the discrepancy in performance. Overall, early empirical tests mostly supportive of monetary models of exchange rate.MR reasoned that if monetary models are effective models, they should outperform simple models in out-of-sample forecast tests. MT use their cointegrating relationship and the model of short-run dynamics to predict out-of-sample, and these findings overturn the MR result. MM forecasted out- of-sample when RHS variables must also be forecasted, and showed at longer horizons the model significantly outperform a random walk. Mark confirms that the monetary model does significantly better than a random walk out-of-sample at longer horizons and suggests that monetary models are useful predictors of future exchange rates, but only at relatively long horizons.
  
3.
When modeling equilibrium exchange rates, we combine PPP and UIP. We should considerate all the possible factors that affect the equilibrium exchange rates and choose an appropriate indicator to measure the inflation rate. We calculate the long term equilibrium exchange rates and short term equilibrium exchange rates separately.
Behavioural Equilibrium Exchange Rates (BEER)
The BEER model builds on the equation:
If we assume  depends on long-run economic fundamentals, then real exchange rates are related to
①A long-run equilibrium component  ;
②A short-run deviation due to real interest rates and the risk premium.
Once we have chosen some long run economic fundamentals and collected the data, we can estimate the model.
A standard approach is to use the Johansen cointegration method.
Clark and MacDonald (1998)
CM assume that the risk premium depends on government debt ratios.
The variables they choose as long-run economic fundamentals are: Net foreign assets; Terms of trade; Relative price of traded to non-traded goods.
So the model they estimate is of the form:

4.
Causes of the Financial Crisis
Several different explanations have been put forward for the Crisis. Economists disagree about what single factor was most important, but there is some consensus on the factors that were the major drivers
The main explanations are as follows:
1. Global imbalances
2. Monetary policy
3. Ineffective regulation
4. Mispricing of risk and poor incentives.
Reason 1: Global Imbalances
The US current account deficit rose from around 1.5% of GDP in1995 to around 6.5% in 2005.
The increase in the current account deficit was driven by a dramatic fall in the US savings rate.
This appears to have been driven by households’ desire for high consumption today, but it also reflects rising government spending at a time when taxes were cut.
The US current account deficit was made possible by increasing current account surpluses in oil-exporting and emerging Asian economies, especially China.
Savings rates were high in emerging economies as they amassed precautionary reserves to deal with ‘sudden stops’.
Because sudden stops are associated with sharp decreases in GDP and credit, policymakers and international institutions encouraged this.
The apparent shortage of reliable and tradable assets made the US the obvious destination for excess savings. High demand for US assets pushed down longer-term interest rates, leading to a ‘search for yield’.
These encouraged investors to move into riskier assets with higher expected returns, but risk ratings on many assets were not reliable

Reason 2: Monetary Policy
It is argued that monetary policy contributed to the ‘search for yield’, because central bank rates were low in the decade before the Crisis.
In fact, short-term real interest rates appear to have been negative or near zero in Japan, U.S. and Euro aera.
Central bank mandates say they should set interest rates based on inflation and output gap.
Inflation expectations and output were consistent with low interest rates.

Reason 3: Ineffective Regulation

The IMF argues that deficient regulation was the main culprit.
The charge is that regulators did nothing to prevent the creation of new financial instruments which were riskier than they appeared.
Moreover, widespread use of off-balance-sheet vehicles hid the maturity mismatch of the banking sector from investors, so even the well-informed could not assess the risks.
The question now is how regulators should deal with these risks and how much power they should be given.

Reason 4: Mispricing of Risk and Poor Incentives

Banks paid mortgage brokers an upfront fee for each mortgage they arranged, with no penalties if the loan later went into default.
Under this model, brokers had no incentive to refuse mortgage loans likely to default. Many brokers would also have received large bonuses for meeting short run sales targets.

The bonuses paid to investment bankers in the City and Wall St rewarded short-term revenues rather than long-term performance.
There were also issues with pricing of risky securities.

The risk ratings attached to securities were far too low, because they assumed that house prices would continue to rise.
The downside risk if house prices fell was extremely large, but this was not priced-in appropriately by risk-rating agencies.
This led to much higher demand for risky securities than would otherwise have been the case.
 
Policy Implications of the Financial Crisis
There are at least two lessons to be learnt from the Crisis:
1 Monetary policy mandates should include financial stability as well as price stability;
Many central banks are introducing ‘macro prudential’ regimes which give them direct powers to remove or reduce risks that threaten financial system stability .
Two main features of these regimes are banking sector capital requirements and a countercyclical capital buffer
Aim of the latter is to make banks more able to cope with unexpected losses in a downturn, so that lending conditions are more favorable
Relaxing banks’ capital requirements in an upturn is sensible, but it is important that they remain stringent enough to alter behavior.
2 Fiscal policies should be more precautionary in booms to leave fiscal space to fight crises.
After the Crisis had hit, many central banks soon reduced their policy rates to almost zero, leaving no room for further cuts.
In theory, we still had fiscal policy to stimulate the economy.
But in practice for countries whose government debt levels were already high there has been little space to maneuver.
 
Long-term Policy Implications
Global rebalancing
Financial development in emerging economies would reduce fears of sudden stops.
Better measurement of financial sector activity is needed – and more transparency.
 

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