Ethical_and_professional_standards_quantitative_methods_and_economics

Ethical and professional standards quantitative methods and economics

Ethical and Professional Standards

The standard of professional conduct 1. Professionalism

knowledge of the law: comply with the more strict law
independence and objectivity: must not offer, solicit or accept any gift, benefit and so on
misrepresentation(曲解误报): must not knowingly make any misrepresentations
misconduct: must not engage in any ptofessional conduct involving dishonesty...

Integrity(正直) of capital markets
- material nonpublic information: must not act or cause others to act on the information
- market manipulation
Duties to Clients
- loyalty, prudence(审慎) and care: must act for the benefit of their clients and place their clients' interests before their employer's or their own interests
- fair dealing
- suitability
  - make a reasonabel inquiry prior to making any investment recommendation
  - determine that an investment is consistent with the clients' written objectives and constraints
  - judge the suitability of investments in the context of the client's total portfolio
- preformance presentation
- preservation of confidentiality: MUST keep information abount current, former and prospective clients confidential unless:
  - the information concerns illegal activities
  - disclosure is required by law, or
  - the client permits disclosure of the information
Duties to Employers
- loyalty: in matters related to employment, must act for the benefit of their employer and not cause harm to them
- additional compensation arrangements: must not accept benefits with or might reasonabley be expected to create a conflict of interest with emloyer's unless obtain written consent from all parties involved.
- responsibilities of supervisors
Investment analysis, recommendations, and actions
- diligence and reasonable basis
- communication with clients and prospective clients
  - must disclose format and principles and promptly disclose any changes
  - disclose significant limitations and risks
  - identify important factors to their investment and include factors in communication with clients and prospective clients
  - distinguish between fact and opinion
- record retention
Conflicts of interest
- disclosure of conflicts: make sure the disclosure are prominent, are delivered in plain language
- priority of transactions
- referral fees(介绍费): must disclose to emloyers, clients and prospective clients any compensation received by, or paid to, others for the recommendation of products or service
Responsibilities as a CFA Institute Menber or CFA Candidate
- conduct as participants in CFA Institute Programs: must not engage in any conduct that compromises the reputation of CFA
- reference to CFA: must not misrepresent or exaggerate the meaning

Ethical and Professional Standards: Application

Quantitative Methods for Valuation

significance level like 5%

confidence level like 95%, i.e. (1 - significance level)

correlation and regression

sample covariance $cov_{XY} = \frac{\sum_{i=1}^n(X_i-\bar{X})(Y_i-\bar{Y})}{n - 1}$

sample correlation coefficient $r_{XY} = \frac{cov_{XY}}{(s_X)(s_Y)}$

correlation coefficient

Interpretation

$r= +1$

perfect positive correlation

$0 < r < +1$

positive linear relationship

$r = 0$

no linear relationship

$-1 < r < 0$

negative linear relationship

$r= -1$

perfect negative correlation

test of hypothesis

To test whether the correlation between the population of two variable is equal to zero $H_0: \rho = 0 \ versus\ H_a:\rho \ne 0$

we can use a t-test to determine wether the null hypothesis should be rejected with $df=n-2$ :

$t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}}$

Reject $H0$ if $+t{critical} < t$, or $t < - t_{critical}$

simile linear regression model

The estimated slope coefficient for the regression line describes the change in Y for a one unit change in X. And it is calculated as:

$\hat{b_1} = \frac{cov_{XY}}{\sigma^2_X}$

the intercept term ($\hat{b_0}$) may be expressed as:

$\hat{b_0} = \bar{Y} - \hat{b_1}\bar{X}$

standard error of estimate (SEE) and coefficient of determination ($R^2$)

The confidence interval for the regression coefficient, $b1$ is calculated as: $$\hat{b_1} \pm (t_c \times s{\hat{b_1}})$$ Where, $t_c$ is the critical two-tailed t-value for the selected confidence level with the appropriate number of degrees of freedom, which is equal to the number of sample observation minus 2.

A t-test may also be used to test the hypothesis that the true slope coefficient $b_1$ is equal to some hypothesized value. Letting $\hat{b_1}$ be the point estimate for $b_1$, the appropriate test static with n-2 degrees of freedom is:

$t_{b_1} = \frac{\hat{b_1} - b_1}{s_{\hat{b_1}}}$

Reject $H0$ if $t > + t{critical}$ or $t < - t_{critical}$

confidence intervals for predicted values

The equation for the confidence interval for a predicted value of Y is:

$\hat{Y} \pm (t_c \times s_{f})$ where:

$t_c$ = two-tailed critical t-value at the desired level of significant with df = n - 2

$s_f$ = standard error of the forecast

If you do need to calculate $s_f$, it can be done with the following formula: $s_f^2 = SEE^2(1+\frac{1}{n} + \frac{(X-\bar{X})^2}{(n-1)s_x^2})$

ANOVA

Total sum of squares (SST)
$SST = \sum_{i=1}^n (Y_i - \bar{Y})^2$
where $\bar{Y}$ is the mean value of $Y$
Regression sum of squares (RSS)
$RSS = \sum_{i=1}^n (\hat{Y_i} - \bar{Y})^2$
Sum of squared errors (SSE)
$SSE = \sum_{i=1}^n (Y_i - \hat{Y_i})^2$

Thus, total variation = explained variation + unexplained variation, or:

SST = RSS + SSE

Source of variation

Degrees fo Freedom

Sum of Squares

Mean Sum of Squares

Regression (explained)

RSS

$MSR = \frac{RSS}{k} = \frac{RSS}{1} = RSS$

Error (unexplained)

n - 2

SSE

$MSE = \frac{SSE}{n-2}$

Total

n - 1

SST

k is the number of slope parameters estimated and n is the number of observation. In general, the regression $df = k$ and the error $df = (n-k-1)$. So in one independent variable linear regression, we use $k=1$ for regression and $n-1-1=n-2$ for the error.

Calculating $R^2$ and SEE

$R^2 = \frac{SST - SSE}{SST} = \frac{RSS}{SST}$

The SEE is the standard deviation of the regression error terms and is equal to the square root of the mean squared error (MSE):

$SEE = \sqrt{MSE} = \sqrt{\frac{SSE}{n-2}}$

The F-statistic

$F = \frac{MSR}{MSE} = \frac{RSS/k}{SSE/(n-k-1)}$ IMPORTANT: This is always a one-tailed test!

In fact, in simple linear regression we have $t^2 = F$, see Relationship between F and t statistics

Multiple regression and issues in regression analysis

regression model uncertainty: 遗漏变量导致
parameter estimate uncertainty: 主观影响，p值

Hypothesis testing of regression coefficients

The t-statistic used to test the significance of the individual coefficients in a multiple regression is calculated using the same formula that is used with simple linear regression. $t = \frac{\hat{b_j} - b_j}{s_{\hat{b_j}}} = \frac{estimated\ regression\ coefficient - hypothesized\ value}{cofficient\ standard\ error\ of\ b_j}$

Interpreting p-Values

The p-value or probability value or asymptotic significance is the probability for a given statistical model that, when the null hypothesis is true, the statistical summary (such as the sample mean difference between two compared groups) would be the same as or of greater magnitude than the actual observed results.:

If the p-value is less than significance level, the null hypothesis can be rejected.
If the p-value is greater than the significance level, the null hypothesis cannot be

F statistic

The F-statistic, whihc is always a one-tailed test, is calulated as:

$F = \frac{MSR}{MSE} = \frac{RSS/k}{SSE/(n-k-1)}$ where:

$RSS$ = regression sum of squares

$SSE$ = sum of squared errors

$MSR$ = mean regression sum of squares

$MSE$ = mean squared errors

Coefficient of determination, $R^2$

$R^2$ is also calculated the same way as in simple linear regression. $R^2 = \frac{SST - SSE}{SST} = \frac{RSS}{SST}$

Adjusted $R^2$

Unfortunately, $R^2$ by itself may not be a reliable measure of the explanatory pwoer of the multiple regression model. This is because $R^2$ almost always increases as variables are added to the model, even if the marginal contribution of the new variable is not statistically significant. The adjusted $R^2$ value is expressed as: $R_a^2 = 1 - [(\frac{n-1}{n-k-1})\times(1-R^2)]$

ANOVA Tables

Source

df (Degrees fo Freedom)

SS (Sum of Squares)

MS (Mean Sum of Squares)

Regression

RSS

$MSR = \frac{RSS}{k}$

Error

n - k - 1

SSE

$MSE = \frac{SSE}{n-k-1}$

Total

n - 1

SST

What is heteroskedasticity?

Heteroskedasticity occurs when the variance of the residual is not the same across all observations in the sample.

Effect of heteroskedasticity on regression analysis

There are four effects of heteroskedasticity you need to be aware of:

The stanard errors are usually unreliable estimates.
The coefficient estimates (the $\hat{b_j}$) aren't affected.
If the standard errors are too small, but the coefficient estimates themselves are not affected, the t-statistics will be too large and the null hypothesis of no statistical significance is rejected too often, The opposite will be true if the standard errors are too large.
The F-test is also unreliable.

Detecting heteroskedasticity

There are two methods to detect heteroskedasticity: examing scatter plots of the residuals and using the Breusch-Pagan chi-square ($\chi^2$) test.

The BP chi-square test, which has a chi-square ($\chi^2$) distribution, is calculated as:

BP chi-square test = $n \times R_{resid}^2$ with k degrees of freedom

where:

n = the number of observations

$R_{resid}^2$ = $R^2$ from a second regression of the squared residuals from the first regression on the independent variables

k = the number of the independent variables

Correcting heteroskedasticity

The most common remedy and the one recommended in the CFA exam is to calculate robust standard errors (also called White-corrected standard errors or heteroskedasticity-consistent standard errors). The second is to correct for heteroskedasticity is the use of generalized least squares.

$t = \frac{Coefficient}{White-corrected\ standard\ errors}$

Detecting Serial correlation

There are 2 methods used to detect the presence of serial correlation: residual plots and the Durbin-Watson statistic.

$DW = \frac{\sum_{t=2}^T(\hat{\epsilon}_t - \hat{\epsilon}_{t-1})^2}{\sum_{t=1}^T\hat{\epsilon}_t^2}$

where:

$\hat{\epsilon}_t$ = residual for period t

If the sample size is vary large:

$DW \approx 2(1-r)$ where:

r = correlation coefficient between residuals from one period and those from the previous period.

DW test statistic is approximately equal to 2 if the error terms are homoskedastic and not serially correlated(r=0). DW < 2 if the error terms are positively serially correlated (r>0), and DW > 2 if error terms are negatively serially correlated (r<0).

$H_0$: the regression has no positive serial correlation

if DW < $d_l$, the error terms are positively serially correlated (i.e. reject the null hypothesis of no positive serial correlation).
if $d_l$ < DW < $d_u$, the test is inconclusive.
if DW > $d_u$, there is no evidence that the error terms are positively correlated

Effect of multicollinearity on regression analysis

Multicollinearity makes the slope coefficients tend to be unreliable, the standard errors of the slope coefficients are artificially inflated. Hence, there is a greater probability that we will incorrectly conclude that a variable is not statistically significant.

参数误差变大。t检验显著性会改变。

异方差

序列自相关

多重共线性

参数

无影响

影响

$Sb_j$ /F-test / t-test

影响

Detecting multicollinearity

The most common way to detect multicollinearity is the situation where t-test values are significantly different than zero, while the F-test is statistically significant and the $R^2$ is high. If , in CFA exam, the absolute value of the sample correlation between any two independent variables in the regression is greater than 0.7, multicollinearity is a potential problem.

Model specification

There are three broud categories of model misspecification: 1. The function form can be misspecified.

important variables are omitted.
variables should be transformed.
data is improperly pooled.
1. Explanatory variables are correlated with the error term in time series models.
a lagged dependent variable is used as an independent variable.
a function of the dependent variable is used as an independent variable ("forecasting the past").
independent variables are measured with error.

Time-series analysis

A time series is a set of observations for a variable over successive periods of time.

Linear trend model

$y_t = b_0 + b_1(t) + \epsilon_t$

where:

$y_t$ = the value of the time series (the dependent variable) at time t

$b_0$ = intercept at the vertical axis

$b_1$ = slope coefficient

$\epsilon_t$ = error term

t = time (independent variable); t = 1, 2, 3...T

Log=linear trend model

$y_t = e^{b_0 + b_1(t)}$ When a variable grows at a constant rate, a log-linear model is most appropriate. When the variable increases over time by a constant amount, a linear trend model is most appropriate.

Autoregressive model

When the dependent variable is regressed against one or more lagged values of itself, the resultant model is called as an autoregressive model.

$x_t = b_0 + b_1x_{t-1}+...+b_px_{t-p}+\epsilon_t$

Autocorrelation & model fit

If the resisduals have significant autocorrelation, the AR model that produced the residuals is not the best model for the time series being analyzed. The procudure to test whether an AR time series model is correctly specified involves three steps:

Estimate the AR model being evaluated using linear regression: Start with a first-order AR model, i.e. AR(1)
Calculate the autocorrelations of the model's residuals (i.e., the level of correlation between the forecast errors from one period to the next).
Test whether the autocorrelations are significantly different from zero: If the model is correct specified, none of the autocorrelations will be statistically significant. To test for significance, a t-test is used to test the hypothesis that the residuals are zero. The t-statistic is the estimated autocorrelation divided by the standard error ($\hat{xt} = 1/\sqrt{T}$), so the test statistic for each autocorrelation is $t = \frac{\rho{\epsilont,\epsilon{t-k}}}{1/\sqrt{T}}$ with (T-2) degrees of freedom.

t-test is difference of the test index divided by the standard error.

A time series must have a finite mean-reverting level (i.e., $\frac{b_0}{1-b_1}$) to be covariance stationary. Thus, a random walk, with or without a drift, is not covariance stationary, and exhibits what is known as a unit root ($b_1$=1).

A sequence of random variables is covariance stationary if all the terms of the sequence have the same mean, and if the covariance between any two terms of the sequence depends only on the relative positions of the two terms, that is, on how far apart they are located from each other, and not on their absolute position, that is, on where they are located in the sequence.

Def. A sequence of random variables ${X_n}$ is covariance stationary if and only if

$\exists \mu \in \R : E[X_n] = \mu, \forall n > 0$ $\forall j \ge 0, \exists \gamma_j \in \R : Cov[X_n, X_{n-j}] = \gamma_j, \forall n > j$

Unit Root testing for nonstationary

Dick and Fuller created a rather ingenius test for a unit root. Instead of testing whether the original coefficient ($b_1 - 1$) is different from zero using a modified t-test. If ($b_1 - 1$) is not significantly different from zero, they say that $b_1$ must be equal to 1.0 and, therefore, the series must have a unit root and is nonstationary (i.e., it is not covariance stationary).

seasonality

To adjust for seasonality in an AR model, an additional lag of the dependent variable is added to the orignal model as another independent variable.

using ARCH models

An ARCH model is used to test for autoregressive conditional heteroskedasticity. With in the ARCH framework, an ARCH(1) time series is one for which the variance of the residuals in one period is dependent on (i.e., a function of ) the variance of the residuals in the preceding period. To test whether a time series is ARCH(1), the squared residuals from an estimated time-series model, $\hat{\epsilon}t^2$, are regressed on the first lag of the squared residuals $\hat{\epsilon}{t-1}^2$. The ARCH(1) regression model is expressed as: $\hat{\epsilon}_t^2 = a_0 + a_1 \hat{\epsilon}_{t-1}^2+ \mu_t$

Cointegration

Cointegration means that two time series are economically linked (realted to the same macro varibales) or follow the same trend and that relationship is not expected to change.

can linear regression be used to model the relationship between two time series?

Independent variable time series

Is covariance stationary

Is NOT covariance stationary

Depandent variable time series

Is covariance stationary

Yes

Is NOT covariance stationary

Yes, IF the two time series are cointegrated

To determin what typr of model is best suited to meet your needs, follow these guidelines: 1. Determin your goal.

Are you attempting to model the relationship of a variable to other variables (e.g., cointegrated time series, cross-sectional multiple regression)?
Are you trying to model the variabel over time (e.g., trend model)
1. Plot the data to see the trend. A structural change is indicated by a significant shift in the plotted data at a point in time that seems to divide the data into two or more distinct patterns. You have to run different models for the structural change.
2. If there is no seasonality or structural shift, use a trand model.
If the data plot on a straight line with an upward or downward slope, use a linear trend model.
If the data plot in a curve, use a log-linear trend model.
1. Run the trend analysis, compute the residuals, and test for serial correlation using the Durbin Watson test.
If you detect no serial correlation, you can use the model.
If you detect serial correlation, you must use another model (e.g., AR)
1. If the data has serial correlation, reexamine the data for stationarity before running an AR model. If it is not stationary, treat the data for use in an AR model as follows:
If the data has linear trend, first-difference the data.
If the data has an exponential trend, first-difference the natural log of the data.
If there is a structural shift in the data, run two separate models as discussed above.
If the data has seasonal component, incorprate the seasonality in the AR model as discussed below.
1. After first-differencing in 5 above, if the series is covariance stationay, run an AR(1) model and test for serial corrlation and seasonality.
If there is no remaining serial correlation, you can use the model.
If you still detect serial correlation, incorporate lagged values of the variable (possibily including one for seasonality--e.g., for monthly data, add the 12th lag of the time series) into the AR model until you hvae removed any serial correlation.
1. Test for ARCH. Regress the square of the residuals on squares of lagged values of the residuals and test whether the resulting coefficient is significantly different from zero.
2. If the coefficient is not significantly different from zero, you can use the model.
3. If the coefficient is significantly different from zero, ARCH is present. Correct using generalized least squares.
4. If you have developed two statistically reliable models and want to determin which is better at forecasting, calculate their out-of-sample RMSE.

Probabilistic approaches: scenario analysis, decision trees, and simulations

simulation

Steps in simluation

Determin the probabilistic variables.
Define probability distributions for these variables.
There are three approaches to specifying a distribution:
- Historical data
- croess-sectional data
- Pick a distribution and estimate the parameters
Check for corrlations among variables
Run the simulation.
The number of simulations needed for a good output is driven by:
- The number of uncertain variables
- The types of distributions
- The range of outcomes. The wider, the more.
  Advantages if simulations
  There are two advantages of a carefully crafted simulation:
Better input quality.
Provides a distribution of expected value rather than a point estimate.
Constraints
There are threee types of constraints:
Book value constraints
- regulatory capital requirements
- negative equity
Earings and cash flow constraints
- imposed internally ot meet the analyst expectation
Market value constraints

Limitation

There are several limitations of using simulations as a risk assessment tool: 1. Input quality. 2. Inappropriate statistical distributions. 3. Non-stationary distributions. 4. Dynamic correltions.

compare scenario analysis, decision tree, and simulation.

Scenario analysis and decision trees are used to analyze discrete risk while simulation are used to analyze continuous risk.
Scenario analysis computes finite cases and the combined probability of the outcome is less than 1.
Desicion trees is suitable for the condition when risk is both discrete and sequential.

Appropriate method

Distribution of risk

Sequential?

Accommodates correlated variables?

Simulations

Continuous

Does not matter

Yes

Senario analysis

Discrete

Yes

Decision trees

Discrete

Yes

Economics for Valuation

Currency exchange rates: determination and forecasting

Exchange rates

For example, 1.4126 USD/EUR means that each euro costs $1.4126. In this example, euro is called the base currency and USD the price currency.

A spot exchange rate is the currency exchange rate for immediate delivery, which for most currencies means the exchange of currencies takes place two days after the trade.

A forward exchange rate is a currency exchange rate for an exchange to be done in the future (e.g., 30 days, 60 days, 90 days or one year).

Dealer quotes often include both bid (买入价) and offer （卖出价）(ask) rates. For example, the euro could be quoted as $1.4124 - 1.4128. The bid price ($1.4124) is the price at which the dealer will buy euros, and the offer price ($1.4128) is the price at which the dealer will sell euros. So the rule is buy the base currency at ask and sell the base currency at bid for investors.

up-the-bid-and-mutiply, down-the-ask-and-divide
Foreign exchange spread
The difference between the offer and bid price is called the spread. Spreads are often stated as 'pips' and one pip is 1/10000.

The spread quoted by the dealer depends on:

The spread in the interbank market for the same currency pair.
The size of the transaction.
The relationship between the dealer and client.

The interbank spread on a currency pair depends on:

currencies involved. High-volume currency pairs command lower spreads than do lower-volume currency pairs.
Time of day. The time overlap during the trading day when both the New York and London currency markets are open is considered the most liquid time window.
Market volatility.

Cross rate

The cross rate is the exchange rate between two currencies implied by their exchange rates with a common third currency, usually the USD or EUR.

For example, we have the following quotes: USD/AUD=0.60 and MXN/USD=10.70. The cross rate between Australian dollars and pesos (MXN/AUD): $\frac{MXN}{AUD} = \frac{USD}{AUD}\times\frac{MXN}{USD} = 0.60 \times 10.70 = 6.42$

Cross rates with bid-ask spreads

Rules: $(\frac{A}{C})_{bid} = (\frac{A}{B})_{bid} \times (\frac{B}{C})_{bid}$ $(\frac{A}{C})_{offer} = (\frac{A}{B})_{offer} \times (\frac{B}{C})_{offer}$ $(\frac{A}{C})_{offer} = \frac{1}{(\frac{C}{A})_{bid} }$ $(\frac{A}{C})_{bid} = \frac{1}{(\frac{C}{A})_{offer} }$

相除对角，相乘同边

A currency is quoted at a forward premium relative to a second currency if the forward price is greater than the spot price and forward discount if less than the spot price.

The value of a forward contract (to the party buying the base currency) at maturity (time T) is: $V_T = (FP_T - FP)(contract size)$ where:

$V_T$ = value of the forward contract at time T, denominated in price currency

$T$ = maturity of the forward contract

$FP$ = forward price locked in at inception to buy the base currency (and with a maturity of T)

$FP_T$ = forward price to sell the base currency at time T

Value prior to expiration

$V_t = \frac{(FP_T - FP)(contract\ size)}{[1 + R(\frac{days}{360})]}$ where:

$V_t$ = value of the forward contract at time t, (t < T) denominated in price currency

$FP_t$ = forward price (to sell base currency) at time t in the market for a new contract maturing at time T

$days$ = number of days remaining to maturity of the forward contract (T - t)

$R$ = interest rate of price currency

International parity conditions

International parity conditions form the building blocks of most models of exchange rate determination. The key international parity conditions are as follows: 1. convered interest rate parity 2. uncovered interest rate parity 3. forward rate parity 4. purchasing power parity 5. the implementation Fisher effect

The exception to the rule that parity conditions do not hold in the short term is covered interest rate parity, which is the only parity condition that is enforced by arbitrage.

1. Covered interest rate parity (平价)

Given the spot exchange rate and domestic and foreign yields, the forward exchange rate must equal the rate that gives these alternative investment strategies -- exactly the same holding period return. Covered interest rate parity is thus said to be a no-arbitrage condition.

For covered interest rate parity to hold exactly, it must be assumed that there are zero transaction costs and that the underlying domestic and foreign money market instrements being compared are identical in terms of liquidity, maturity and default risk. The freer the markets are the differentials are found to be more close to zero and covered interest rate parity holds.

2. Uncovered interest rate parity

According to the uncovered interest rate parity condition, the expected return on an unvovered foreign currency investment should equal the return on a comparable domestic currency investment. Uncovered interest rate parity states that the change in spot rate over the investment horizon should, on average, equal the differential in interest rates between the two countries. That is, the expected appreciation/depreciation of the exchange rate will just offset the yield differential.

In domestic currency terms. the investment return on an uncovered foreign-cuurency-denominated investment is equal to $(1 + i_f)(1 - \%\Delta S_{f/d}) - 1$ This return can be approximated using the following equation: $\cong i_f - \%\Delta S_{f/d}$

Using the example, suppose the return on the one-year foreign money market instrument 10% while the return on the domestic money market instrument is 4%.

The $S{f/d}$ rate is expected to remain unchanged: that is, $\%\Delta S{f/d}$ = 0. So the investor would prefer the foreign-currency-denominated money market investment.
The domestic currency is expected to appreciate by 10%. So the investor would prefer domestic because the foreign invesmtne is 0% (=10% - 10%)
The domestic currency is expected to appreciate by 6%. In this case, the risk-neutral investor is assumed to be indifferential between the alternatives since 4% = 10% - 6%.

Uncovered interest rate parity says the expected change in the spot exchange rate over the investment horizon should be relfected in the iterest rate differential:

$\%\Delta S_{f/d}^e = i_f - i_d$

where $\Delta S^e$ indicates the change in the spot rate expected for future periods.

3. Forward rate parity

Forward rate parity states that the forward exchange rate will be an unbiased predictor of the future spot exchange rate. Formally, covered interest rate parity requires that (given A/B quote structure): $F = \frac{1+R_f(\frac{days}{360})}{1+R_d(\frac{days}{360})} S_0$ $F- S_0= S_0 (R_f - R_d)\frac{\frac{days}{360}}{1+R_d(\frac{days}{360})}$

where:

$F$ = forward rate (quoted as f/d)

$S_0$ = spot rate (quoted as f/d)

$days$ = number of days in the underlying forward contract

$R_f$ = intereest rate for foreign currency

$R_d$ = intereest rate for domestic currency

For the sake of simplicity, we assume that the investment horizon is one year, so that

$F - S_0 = S_0 \frac{(R_f - R_d)}{1+R_d} \approx S_0 (R_f - R_d)$

So we have

$\% \Delta S_{f/d} = \frac{F - S_0}{S_0} = R_f - R_d$

Covered interest rate parity (平价)

Given a quote structure of A/B, the base currency (i.e., currency) is expected to appreciate by approximately $RA - R_B$. (When $R_A - R_B$ is negative, currency B is expected to depreciate). $$E(\%\Delta S){(A/B)} = R_A - R_B$$ If the forward rate is equal to the expected future spot rate, we say that the forward rate is an unbiased predictor of the future spot rate. In such an instance, $F = E(S_1)$

International Fisher Relation

We can write this relation (known as the Fisher relation) as: $R_{nominal} = R_{real} + E(inflation)$ Taking the Fisher relation and real interest rate parity together gives us the international Fisher effect: $R_{nominal\ A} - R_{nominal\ B} = E(inflation_A) - E(inflation_B)$ This tells us that the difference between two countries' nominal interest rates shoud be equal to the difference between their expected inflation rates.

In an ideal world:

relative expected inflation rates should determine relative nominal interest rates;
relative interest rates should determine forward exchanges; and
forward exchange rates should correctly anticipate the path of the future spot exchange rate.

Purchasing power parity

The law of one price.

Instead of focusing on individual products, absolute purchsing power parity (absolute PPP) compares the average price of a representative basket of consumption goods bwtween countries. $S(A/B) = CPI(A) / CPI(B)$ $P^x_f = S_{f/d} \times P^x_d$

Relative PPP states that changes in exchange rates should exactly offset the price effects of any inflation differential between the two countries.

$\%\Delta S(A/B) = Inflation_{(A)} - Inflation_{(B)}$ where:

$\%\Delta S(A/B)$ = change in spot price (A/B)

Real exchange rate

$real\ exchange\ rate = S_t \frac{CPI_B}{CPI_A}$

where:

$CPI$ = consumer price index at time t $S_t$ = spot rate at time t (given as A/B)

Balance of payments

Balance-of-payments (BOP) accounting is method used to keep track of transactions bwtween a country and its international trading partners. The BOP equation is: $current\ account + financial\ account + official\ reserve\ account = 0$

current account measures the exchange of goods, services, investment income and unilateral transfers (gifts).
financial account measures the flow of funds for debt and equity investment into and out of the country.
offical reserve account transactions are those made from the reserves held by the official monetary authorities of the country. Normally the official reserve account balance does not change significantly from year to year, and hence, economists focus onthe first two parts of the BOP equation.

Influence of BOP on exchange rates

current account influences

flow mechanism. Current account deficits in a country increase the supply of that currency in the markets. The decrease in the value of the currency may restore the current account deficit to balance -- depending on the following factors:
- The initial deficit. The larger the initial deficit, the larger the depreciation of domestic currency needed to restore current account balance.
- The influence of exchange rates on domestic import and export prices. However, some of the increase in cost may not be passed on to consumers.
- Price elasticity of demand of the traded goods. If the most important imports are relatively price inelastic, the quantity imported will not change.
Portfolio composition mechanism. When investor countries decide to rebalance their investment portfolios, it can have a significant negative impact on the value of those investee country currencies.
Debt sustainablility mechanism. When the level of debt gets too high relative to GDP, investors ma question the sustainability of this level of debt, leading to a rapid depreciation of the borrower's currency.

capital account influences

As capital flows into a coutry, demand for the country's currency increase, resulting in appreciation. Excessive capital inflows into emerging markets create problems for those countries such as:

excessive real apprecaition of the domestic currency.
financial asset and/or real estate bubbles.
increase in external debt by businesses or government.
excessive consumption in the domestic market fueled by credit.

Emerging market governments often counteract excessive capital inflows by inposing capital controls or by dirext intervention in the foreign exchange markets. $real\ echange\ rate(A/B) = equilibrium\ real\ exchange\ rate(A/B) + (real\ interest\ rate_B - real\ interest\ rate_A) - (risk\ premium_B - risk\ premium_A)$

The real value of a currency is positive related ot its real interest rate and negatively related to the risk premium investors demand for investing in assets denominated in the currency.

Correlations between equity returns and exchange rates are unstable in the short term and tend toward zero in the long run.

Taylor Rule

Central banks are usually charged with setting policy interest rates so as to 1) maintain price stability (inflation target) and 2) achieve the maximum sustainable level of the emloyment. $R = r_n + \pi + \alpha (\pi - \pi^*) + \beta (y - y^*)$

where:

$R$ = Central bank policy rate implied by the Taylor rule

$r_n$ = Neutral real policy interest rate

$\pi$ = current inflation rate

$\pi^*$ = Central bank's target inflation rate

$y$ = log of current level of output

$y^*$ = log of central bank's target(sustainable) output

$\alpha, \beta$ = policy response coefficients (>0, Taylor suggested a value of 0.5 for both)

Subtracting inflation from both sides of the equation above, we get:

$Real\ interest\ rate = r = (R - \pi) = r_n + \alpha(\pi - \pi^*) + \beta (y - y^*)$

Further we get:

$Real\ exchange\ rate(A/B) = equilibrium\ real\ exchange\ rate (A/B) + difference\ in\ neutral\ real\ policy\ interest\ rate\ (B - A) + \alpha [difference\ in\ inflation\ gap (B - A)] + \beta [difference\ in\ output\ gap (B - A)] - (risk\ premium_B - risk\ premium_A)$

The IMF assesses long-term equilibrium real exchange rate based on three complementary approaches:

Macroeconomic balance approach
External sustainablility approach
Reduced-form econometric model approach(简化形式的计量经济模型方法)

Reduced-form 的翻译有待考证。

FX carry trade

Aninvestment strategy that involves taking long positions in high-yield currencies and short positions in low-yield currencies. The FX carry trade attempts to capture an interest rate differential and is a bet against uncovered interest rate parity. Carry trades typically perform well during low-volatility periods.

$return = interest\ earned\ on\ investment - funding\ cost - currency\ depreciation$

Assume a trader can borrow Canadian dollars at 1% and earn 9% on an investment in Brazilian reals for one year. To execute the trade to earn 8% from the interest rate differential, the trader will do the following:

Borrow Canadian dollars at t=0
Sell the dollars and buy Brazilian reals at the spot rate at t=0
Invest in a real-denominated investment art t=0
Liquidate the Brazilian investment at t=1
Sell the reals and buy dollars at the spot rate at t=1
Pay back the dollar loan
If the real appreciates, the trader’s profits will be greater than 8% because the stronger real will buy more dollars in one year. If the real depreciates, the trader’s profits will be less than 8% because the weaker real will buy fewer dollars in the future. If the real falls in value by more than 8%, the trader will experience losses.
risk of the carry trade
The carry trade is profitable only if uncovered interest rate parity does not hold over the investment horizon.

Risk management in carry trades

There are two approaches to managing crash risk in a carry trade: 1. Volatility filter: Whenever implied volatility increases above a certain threshold, the carry trade positions are closed. 2. Valuation filter: A valuation band is established for each currency based on PPP or other models. If the value of a currency falls below the band, the trader will overweight that currency in the trader's carry trade protfolio.

MUNDELL-FLEMING Model

Mundell-Fleming model evaluates the impact of monetary and fiscal policies on interest rates -- and consequently on exchange rates. Changes in inflation rates due to changes in monetary or fiscal policy are not explicityly modeled by the Mundell-Fleming model.

High capital mobility

Expansionary monetary policy -> reduce interest rate -> reduce the inflow of capital investment in physical and financial assets(不缺钱，不需引入投资) -> reduce the demand for the domestic currency -> depreciation of the domestic currency

Expansionary fiscal policy (an increased deficit from lower taxes or higher government spending) -> increase government borrowing -> increase real interest rates -> attract foreign investment -> improve the financial account -> increase demand for the domestic currency

However, in the long run, an excessive buildup in debt may eventually casue downward pressure on the domestic currency.

Low capital mobility

In emerging markets, capital flows may be restricted. In that case, the impact of trade imbalance on exchange rates (goods flow effect) is greater than the impact of interest rates (financial flows effect). Expansionary fiscal/ monetary policy leads to increases in net imports, leading to depreciation of the domestic currency. Similarly, restrictive monetary/ fiscal policy leads to an appreciation of domestic currency.

When capital mobility is low, the effects of monetary and fiscal policy on exchange rates will operate primarily through trade flows rather than capital flows. The combination of expansionary monetary and fiscal policy will be bearish for a currency. Earlier we said that expansionary fiscal policy will increase imports and hence the trade deficit, creating downward pressure on the currency. Layering on an expansive monetary policy will further boost spending and imports, worsening the trade balance and exacerbating the downward pressure on the currency.

The combination of restrictive monetary and fiscal policy will be bullish for a currency. This policy mix will tend to reduce imports, leading to an improvement in the trade balance.

The impact of expansionary monetary and restrictive fiscal policies (or restrictive monetary and expansionary fiscal policies) on aggregate demand and the trade balance, and hence on the exchange rate, is indeterminate under conditions of low capital mobility. Exhibit 6 summarizes these results.

贸易顺差（所谓贸易顺差是指在特定年度一国出口贸易总额大于进口贸易总额，又称 “出超”。表示该国当年对外贸易处于有利地位）使得国内外汇市场上的外币供给大于外币需求，必然产生外币贬值的预期和人民币升值的预期。因此，贸易顺差产生人民币升值的压力：贸易顺差越大，人民币升值的压力越大；人民币升值的预期，又加大了外资流入和贸易顺差的扩大，进一步增强了人民币升值的压力。
资本项目的顺差或逆差直接影响着货币汇率的涨跌，当一国资本项目有大量逆差，国际收支的其他项目又不足以弥补时，该国国际收支会出现逆差，从而引起本国货币对外汇率下跌。反之，则会引起本国货币汇率的上升。
简单说，外国人买本国东西要先买本国货币，造成本币升值。另，商品货币除外，贸易对汇率的影响微乎其微。且相当一部分贸易是以美元结算的一国贸易顺差说明这个国家宏观形势好，外国人就会来这个国家投资，那外汇流入就增加了，外汇供大于求，那外汇就贬值了，相对本国货币就升值了。
Monetary and Fiscal Policy and Exchange Rates

Monetary Policy/ Fiscal Policy

Capital Mobility

High

Low

Expansionary/Expansionary

Uncertain

Depreciation

Expansionary/Restrictive

Depreciation

Uncertain

Restrictive/Expansionary

Appreciation

Uncertain

Restrictive/Restrictive

Uncertain

Appreciation

Fixed exchange rate regimes

Under a fixed exchange rate regime, an expansionary monetary policy would lead to depreciation of domestic currency as stated above. The government would then have to purchase (sell) its own currency in the foreign exchange market. This action essentially reverses the expansionary (restrictive) stance.

Expansionary (restrictive) fiscal policy leads to appreciation (depreciation) of domestic currency. The government would then sell (buy) its own domestic currency to keep exchange rates stable -- reinforcing the impact of its fiscal policy on aggregate demand.

扩张性的财政政策，主要是指政府减税，增加支出，从而增加社会总需求，进一步增加对货币的总需求。而在货币政策保持不变的情况下，货币总需求增加，导致利率上升。 Accommodative: 宽松的

The Portfolio Balance Approach

portfolio balance approach: A theory of exchange rate determination that emphaizes the portfolio investment decisions of global investors and he requirement that global investors willingly hold all outstanding securities denominated in each currency at prevailing prices and exchange rates. In this framework, a growing government budget deficit leads to a steady increase in the supply of domestic bonds outstanding. These bonds will be willingly held only if investors are compensated in the form of higher expected return. Such return could come from 1) higher interest rate and/or a higher risk premium, 2) immediate depreciation of the currency to a level sufficient to generate anticipation of gains from subsequent currency appreciation, or 3) some combination of these two factors.

Expansive fiscal policy

short-run response -> increase in real interest rate differential -> currency appreciation
long-run response -> governent debt buildup -> 1) central bank monetizes debt; 2) fiscal stance turns restrictive -> currency depreciates

exchange rate management: intervention and controls

Examples of better economic management by a government include:

a decrease in inflation and inflation volatility
more-flexible exchange rate regimes
improved fiscal position
privatization of state-owned entities
liberalization of financial markets
lifting of foreign exchange regulations and controls

Some studies find, however, that EM policymakers might have greater success in controlling the exchane rates than their inductrial country counterparts because the ratio of EM central bank FX reserve holdings to average daily FX turnover in their domestic currencies is actually quite sizable.

warning signs of a currency crisis

If capital inflows come to a sudden stop, the result may be a financial crisis, in which the economy contracts, asset values plummet, and the currency sharply depreciates. 1. Prior to a currency crisis, the capital markets have been liberalized(自由) to allow the free flow of capital 2. There are large inflows of foreign capital in the period leading up to a crisis, with short-term funding denominated in a foreign currency being particularly problematic 3. Currency crises are often preceded by (and often coincide with) banking crises 4. Countries with fixed or partially fixed exchange rates are more susceptible to currency crises than countries with floating exchange rateds 5. Foreign exchange reserves tend to decline↓ precipitously as a crisis approuches 6. In the period leading up to a crisis, the currency has risen↑ substantially relative to its historical mean. （均值复归，大涨大跌） 7. The ratio of exports to imports (known as "the terms of trade") often deteriorates↓ before a crisis 8. Broad money growth and the ratio of M2 (a measure of money supply) to bank reserves tend to rise↑ prior to a crisis 9. Inflation tends to be significantly higher↑ in pre-crisis periods compared with tranquil periods.

A high level of foreign exchange reserves held by a country typically decreases the likelihood of a currency crisis.

Economic growth and the investment decision

Cross-country comparisons of GDP should be based on purchasing power parity rather than current market exchange rates. PPP, purchaing power parity: The idea that exchange rates move to equalize the purchaing power of different currencies.

preconditions for growth

Savings and investment
Financail markets and intermediaries
1. screening those who seek funding and monitoring those who obtain funding, the financail sector channels financial capital to projects that are likely to generate the highest risk-adjustment returns.
2. the financial sector may encourage savings and assumption of risk by creating attractive investment instruments that facilitate risk transfer and diversification and enhance liquidity
3. the existence of well-developed financial markets and intermediaries can mitigate the credit constraints that companies might otherwise face in financing capital investments.
The political stability, rule of law, and property rights
Investment in human capital: investing in education and health care
Tax and regulatory systems: All else equal, the lower the tax and regulatory burdens, the higher the rate of economic growth
Free trade and unrestricted capital flows: Unrestricated capital flows mitigate the problem of insufficient domestic savings as foreign capital can increase a country's capital, allowing for greater growth.

summary of factors limiting growth in developing countries

low rates of saving and investment
poorly developed financial markets
weak, or even corrupt, legal systems and failure to enforce laws
lack of property rights and political instability
poor public education and health services
tax and regulatory polices discourageing entrepreneurship
restrictions on international trade and flows of capital

why potential growth matters to investors

potential GDP, the maximum amount of output an economy can sustainably produce without inducing an increase in the inflation rate. The output level that corresponds to full employment with consistent wage and price expectations.

Letting P represent the aggregate value (price) of equities and E represent aggregate earnings, we can write: $P = GDP (\frac{E}{GDP}) (\frac{P}{E})$

This equation can be expressed in terms of logarithmic rates of change over a time horizon T:

$(1/T)\%\Delta P = (1/T)\%\Delta GDP + (1/T)\%\Delta (E/GDP) + (1/T)\%\Delta (P/E)$

Over the long-term, we have to recognize that growth in earnings relative to GDP is zero; labor will be unwilling to accept an ever decreasing share of GDP. Similarly, growth in the P/E ratio will also be zero over the long term; investors will not continue to pay an ever increasing price for the same level of earnings forever. Hence over a sufficiently long time horizon, the potential GDP growth rate equals the growth rate of aggregate equity valuation.

determinants of economic growth

production function

$Y = A F(K, L)$

where Y denotes the level of aggregate output in the economy, L is the quantity of labor or number of workers or hours worked in the economy, and K is an estimate of the capital services provided by the stock of equipment and structures used to produce goods and services. A is a multiplicative scale factor referred to as total factor productivity (TFP). The Cobb-Douglas production function, given by $F(K, L) = K^\alpha L^{1 - \alpha}$

The parameter $\alpha$ determines the shares of ouput (factor shares) paid by companies to capital and labor and is assumed to have a value between 0 and 1. Profit maximization requires that the marginal product of capital equal the rental price of capital. The marginal product of capital (MPK) for the Cobb-Douglas production function is

$MPK = \frac{\partial Y}{\partial K} = \alpha A K^{\alpha - 1} L^{1 - \alpha} = \alpha Y/K = r$

Defining y = Y/L as the ouput per worker or (average labor productivity) and k = K/L as the capital-to-labor ratio, we get

$y = Y/L = AK^\alpha L^{1 - \alpha}/L = A(K/L)^\alpha(L/L)^{1 - \alpha} = Ak^\alpha$

A value of α close to zero means diminishing marginal returns to capital are very significant and the extra output made possible by additional capital declines quickly as capital increases. In contrast, a value of α close to one means that the next unit of capital increases output almost as much as the previous unit of capital.

Capital deepening vs. technological progress

Capital deepening, an increase in the capital-to-labor ratio, is reflected in the exhibit by a move along the production function line. Different $(K/L)$, different $(Y/L)$. Further additions to capital have relatively liitle impact on per capita ouput because the marginal product of capital declines as more capital is added to labor input. TFP makes the increase move to another line. Same $(K/L)$, different $(Y/L)$.

Growth accounting

The growth accounting equation states that the growth rate of output equals the rate of technological change plus α times the growth rate of capital plus (1 – α) times the growth rate of labor. $\Delta Y/ Y = \Delta A / A + \alpha \Delta K / K + (1 - \alpha) \Delta L / L$

An alternative method of measuring potential GDP is the labor productivity growth accounting equation.

Growth rate in potential GDP = Long-term growth rate of labor force + Long-term growth rate in labor productivity

Extending the production function

Raw material: natural resources such as oil, lumber, and available land (N)
Quantity of labor (L)
Human capital: eduction and skill level of these workers (H)
Information, computer, and telecommunications (ICT) capital: computer hardware, software, and communication equipment ($K_{IT}$)
Non-ICT capital: transport equipment, metal products and plant machinery other than computer hardware and communications equipment, and non-residential buildings and other structures ($K_{NT}$)
Public capital ($K_P$)
Technological knowledge: the production methods used to convert inputs into final products, reflected by total factor productivity (A)

The expanded production function is expressed mathematically as: $Y = AF(N, L, H, K_{IT}, K_{NT}, K_P)$

Natural resources

renewable resources
non-renewable resources

Even though access natural resources is important, ownership and production of natural resourcese is not necessary for a country to achieve a high level of income. Dutch disease: currency appreciation driven by strong export demand for resources makes other segments of the economy, in particular manufacturing, globally uncompetitive.

Labor supply

The labor force is defined as the working age population (ages 16 to 64) that is either employed or available for work but not working (i.e., unemployed). Thus growth in the labor input depends on four factors: population growth, labor force participation, net migration, average hours worked.

population growth
Note that although population growth may increase the growth rate of the overall economy, it has no impact on the rate of increase in per capita GDP.
labor force participation
labor force pariticipation is defined as the population of working age population in the labor force
labor force participation = labor force / working age population
net migration
Immigration poses a potentail solution to a declining labor force. Immigration represents a potential source of continued economic grwoth in developed countries.
average hours worked
For most countries, the general trend in average hours worked is downward. Possible explanations include legislation limiting the number of hours worked, the "wealth effect" which induces individuals to take more leisure time, high tax rates on labor income, and an increase in part-time and temporary workers.

human capital

An economy’s human capital is increased through investment in education and on-the-job training. In addition, education may also have a spillover or externality impact. Increasing the educational level of one person raises not only the output of that person but also the output of those around that person.

capital: ICT and non-ICT

The physical capital stock increases from year to year as long as net investment (gross investment less the depreciation of the capital) is positive. The correlation between economic growth and investment is high.

A second, and closely related, explanation is that the impact of investment spending on available capital depends on the existing physical capital stock.For the developed countries, a sustained high level of investment over many years is required to have a meaningful relative impact on the physical capital stock even though the absolute size of the increase in any given year is still larger than in the developing countries.

Third, because physical capital is not really homogeneous, the composition of investment spending and the stock of physical capital matters for growth and productivity.

Network externalities: the more people in the network, the greater the potential producitivity gains.

High levels of capital spending for the non-ICT capital should eventually result in capital deepening and thus have less impact on potential GDP growth. In contrast, a growing share of ICT investments in the economy, through their externality impacts, may actually boost the growth rate of potential GDP.

Nanotechnology could well become the next “super GPT,” at which point investing in ICT may begin to look like mere capital deepening.

Technology

The most important factor affecting growth of per capita GDP is technology, especially in developed countries.

public infrastructure

As with R&D spending, the full impact of government infrastructure investment may extend well beyond the direct benefits of the projects because improvements in the economy’s infrastructure generally boost the productivity of private investments.

Theories of growth

Classical model

The key assumption underlying the classical mdoel is that population growth accelerates when the level of per capita income rises above the subsistence income, which is the minimum income needed to maintain life.

The classical model predicts that in the long run, the adoption of new technology results in a larger but not richer population.

The prediction from Malthusian model failed for two reasons: 1. the link between per capita income and population broke down. In fact, as the growth of per capita income increased, population growth slowed rather than acceleration as predicted by the classical growth model. 2. Growth in per capita income has been possible becuase technological progress has been rapid enough to more than offset the impact of diminishing marginal returns.

Neoclassical model

Its primary focus is on estimating the economy's long-term steady state growth rate. When the output-to-capital ratio is constant, the labor-to-capital ratio and output per capita also grow at the equilibrium growth rate, $g^*$. Letting $\theta$ denote the growth rate of TFP (i.e., $\Delta A/A$), $\alpha$ the elasticity of output with repect of captial, $n$ the growth rate of labor.

$g^* = growth\ rate\ of\ output\ per\ capita = \Delta y/y = \frac{\theta}{1 - \alpha}$ $growth\ rate\ of\ output = \Delta Y/Y = \Delta y/ y + n = \frac{\theta}{1 - \alpha} + n$

Considering the equation $\Delta k/k = sY - \delta K$. Where $s$ is the fraction of incoming that is saved and $\delta$ is the change in the physical capital stock. We can get the constant, output-to-capital ratio $\Psi$

$\frac{Y}{K} = (\frac{1}{s})[(\frac{\theta}{1 - \alpha}) + \delta + n] \equiv \Psi$

Note that even though the capital-to-labor ratio (k = K/L) is rising at rate $[\theta/(1 - \alpha)]$ in the steady state, the increase in the capital-to-labor ratio (k) has no impact on the marginal product of capital, which is not changing. Captial deepening is occurring, but it has no effect on the growth rate of the economy or on the marginal product of capital once the steady state is reached.

Steady State in the Neoclassical Model ($[\frac{\theta}{1 - \alpha}) + \delta + n] = sf(k)$)

Saving rate(s): higher saving rate, higher capital-to-labor ratio
Labor force growth (n): higher labor force growth, lower capital-to-labor ratio
Depreciation rate($\delta$): higher depreciation rate, lower capital-to-labor ratio
Growth in TFP ($\theta$): higher growth in TFP, lower capital-to-labor ratio

Implications of the Neoclassical Model

Capital accumulation
1. capital accumulation affects the level of output but not the growth rate in the long run
2. regardless of its initial capital-to-labor ratio or initial level of productivity, a growing economy will move to a point of steady state growth.
3. in a steady state, the growth rate of output equals the rate of labor force growth plus the rate of growth in TFP scaled by labor's share of income. The growth rate of output does not depend on the accumulation of capital or the rate of business investment.
Capital deepening vs. technology
1. rapid growth that is above the steady state rate of growth occurs when countries first begin to accumulate capital; but growth will slow as the process of accumulation conitnues
2. long-term sustainable growth cannot rely solely on capital deepening investment.
3. more generally, increasing the supply of some input too rapidly relative to other inputs will lead to diminishing marginal returns and cannot be the basis for sustainable growth.
4. in the absense of improvements in TFP, the growth of labor productivity and per capita output would eventually slow.
5. because of diminishing marginal returns to capital, the only way to sustain growth in potential GDP per capita is through technological change or growth in total factor productivity.
Convergence
1. Neoclassical growth theory predicts two types of convergence: absolute convergence (developing countries, regardless of their particular characteristics, will eventually catch up with the developed countries and match them in per captia output) and conditional convergence (convergence is conditional on the coutries having the same saving rate, population growth rate, and production function). In addtion to the first two convergence concepts, we have the notion of club convergence, where only rich and middle-income countries that are members of the club are converging to the income level of the richest countries in the world.
2. given the relative scarcity and hence high marginal productivity of capital and potentially higher saving rates in developing countries, the growth rates of developing countries should exceed those of developed countries.
3. as a result, there should be a convergence of per capita incomes between developed and developing countries over time.
Effect of savings on growth
1. the initial impact of a higher saving rate is to temporarily raise the rate of growth in the economy. However, the economy returns to the balanced growth path after the transition period.
2. during the transition period, the economy moves to a higher level of per capita output and productivity.
3. once an economy achieves steady state growth, the growth rate does not depend on the percentage of income saved or invested. Higher savings cannot permanently raise the growth rate of output.
4. however, coutries with higher saving rates will have a higher level of per capita output, a higher capital-to-labor ratio, and a higher level of labor productivity.

If the economy has not yet reached the steady state, the growth rates of output per capita and the capital-to-labor ratio as, respectively,

$\frac{\Delta y}{y} = \frac{\theta}{1 - \alpha} + \alpha s (\frac{Y}{K} - \Psi) = \frac{\theta}{1 - \alpha} + \alpha s (y/k - \Psi)$

and

$\frac{\Delta k}{k} = \frac{\theta}{1 - \alpha} + s (\frac{Y}{K} - \Psi) = \frac{\theta}{1 - \alpha} + s (y/k - \Psi)$

Where $\Psi$ is the new steady state output-to-capital ratio, y/k is the actual output-to-capital ratio which doesn't change immediately.

Endogenous growth theory

$\Delta y_e / y_e = \Delta k_e / k_e = sc - \delta - n$ Examination of the equation shows that a higher saving rate (s) implies a permanently higher growth rate. This is the key result of the endogenous growth model.

Growth in an open economy

According to the neoclassical model, convergence should occur more quickly if economies are open and there is free trade and international borrowing and lending. Opening up the economy should increase the rate at which countries’ capital-to-labor ratios converge.

Economics of Regulation

Regulatory interdependencies

Regulatory capture theory
Regulatory competition
Regulatory arbitrage

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