Category: Learning

Club de Paris

The Paris Club (Club de Paris, 巴黎俱乐部) has reached 478 agreements with 102 different debtor countries. Since 1956, the debt treated in the framework of Paris Club agreements amounts to $ 614 billion.

Low-income countries generally do not have access to these markets. The assistance from bilateral and multilateral donors remains vital for them. Non-Paris Club creditors are becoming an increasingly important source of financing for these countries. Yet despite the fact that Paris Club creditors now have to deal with far more complex and diverse debt situations than in 1956, their original principles still stand.

Members

Duty of Members

Ad Hoc Participants & 6 principles

Permanent Members

The 22 Paris Club permanent members are countries with large exposure to other States woldwide and that agree on the main principles and rules of the Paris Club. The claims may be held directly by the government or through its appropriate institutions, especially Export credit agencies. These creditor countries have constantly applied the terms defined in the Paris Club Agreed Minutes to their bilateral claims and have settled any bilateral disputes or arrears with Paris Club countries, if any. The following countries are permanent Paris Club members:

AUSTRALIA
AUSTRIA
BELGIUM
BRAZIL
CANADA
DENMARK
FINLAND
FRANCE
GERMANY
IRELAND
ISRAEL
ITALY
JAPAN
KOREA
NETHERLANDS
NORWAY
RUSSIAN FEDERATION
SPAIN
SWEDEN
SWITZERLAND
UNITED KINGDOM
UNITED STATES OF AMERICA

Ad Hoc Members

Other official creditors can also actively participate in negotiation sessions or in monthly “Tours d’Horizon” discussions, subject to the agreement of permanent members and of the debtor country. When participating in Paris Club discussions, invited creditors act in good faith and abide by the practices described in the table below. The following creditors have participated as creditors in some Paris Club agreements or Tours d’Horizon in an ad hoc manner:

Abu Dhabi
Argentina
China
Czech Republic
India
Kuwait
Mexico
Morocco
New Zealand
Portugal
Saudi Arabia
South Africa * prospective member on 8 July 2022
Trinidad and Tobago
Turkey

Development and History of Paris Club

Early Stage

In 1956, the world economy was emerging from the aftermath of the Second World War. The Bretton Woods institutions were in the early stages of their existence, international capital flows were scarce, and exchange rates were fixed. Few African countries were independent and the world was divided along Cold War lines. Yet there was a strong spirit of international cooperation in the Western world and, when Argentina voiced the need to meet its sovereign creditors to prevent a default, France offered to host an exceptional three-day meeting in Paris that took place from 14 to 16 May 1956.

Dealing with the Debt Crisis (1981-1996)

1981 marked a turning point in Paris Club activity. The number of agreements concluded per year rose to more than ten and even to 24 in 1989. This was the famous “debt crisis” of the 1980s, triggered by Mexico defaulting on its sovereign debt in 1982 and followed by a long period during which many countries negotiated multiple debt agreements with the Paris Club, mainly in sub-Saharan Africa and Latin America, but also in Asia (the Philippines), the Middle East (Egypt and Jordan) and Eastern Europe (Poland, Yugoslavia and Bulgaria). Following the collapse of the Soviet Union in 1992, Russia joined the list of countries that have concluded an agreement with the Paris Club. So by the 1990s, Paris Club activity had become truly international.

Debt Burden Enlarges for some Countries

In 1996, the international financial community realized that the external debt situation of a number of mostly African low-income countries had become extremely difficult. This was the starting point of the Heavily Indebted Poor Countries (HIPC) Initiative.

The HIPC Initiative demonstrated the need for creditors to take a more tailored approach when deciding on debt treatment for debtor countries. Hence in October 2003, Paris Club creditors adopted a new approach to non-HIPCs: the “Evian Approach”.

Evian Approach

General frame of the Evian approach

  1. Analysis the sustability

    When a country approaches the Paris Club, the sustainability of its debt would be examined, before the financing assurances are requested, in coordination with the IMF according to its standard debt sustainability analysis to see whether there might be a sustainability concern in addition to financing needs. Specific attention would be paid to the evolution of debt ratios over time as well as to the debtor country’s economic potential; its efforts to adjust fiscal policy; the existence, durability and magnitude of an external shock; the assumptions and variables underlying the IMF baseline scenario; the debtor’s previous recourse to Paris Club and the likelihood of future recourse. If a sustainability issue is identified, Paris Club creditors will develop their own view on the debt sustainability analysis in close coordination with the IMF.

  2. if face liquidity problem

    For countries who face a liquidity problem but are considered to have sustainable debt going forward, the Paris Club would design debt treatments on the basis of the existing terms. However, Paris Club creditors agreed that the rationale for the eligibility to these terms would be carefully examined, and that all the range built-into the terms including through shorter grace period and maturities, would be used to adapt the debt treatment to the financial situation of the debtor country. Countries with the most serious debt problems will be dealt with more effectively under the new options for debt treatments. For other countries, the most generous implementation of existing terms would only be used when justified.

  3. if not sustainable or need special treatment

    For countries whose debt has been agreed by the IMF and the Paris Club creditor countries to be unsustainable, who are committed to policies that will secure an exit from the Paris Club in the framework of their IMF arrangements, and who will seek comparable treatment from their other external creditors, including the private sector, Paris Club creditors agreed that they would participate in a comprehensive debt treatment. However, according to usual Paris Club practices, eligibility to a comprehensive debt treatment is to be decided on a case-by-case basis.

    In such cases, debt treatment would be delivered according to a specific process designed to maintain a strong link with economic performance and public debt management. The process could have three stages. In the first stage, the country would have a first IMF arrangement and the Paris Club would grant a flow treatment. This stage, whose length could range from one to three years according to the past performance of the debtor country, would enable the debtor country to establish a satisfactory track record in implementing an IMF program and in paying Paris Club creditors. In the second stage, the country would have a second arrangement with the IMF and could receive the first phase of an exit treatment granted by the Paris Club. In the third stage, the Paris Club could complete the exit treatment based on the full implementation of the successor IMF program and a satisfactory payment record with the Paris Club. The country would thus only fully benefit from the exit treatment if it maintains its track record over time.

Data

There data in the website yoy.

Insights

Refer to Horn et al., (2021) figure 9 in page 13, Paris Club seems played important role during 2010s.

tbc

Reference

https://clubdeparis.org/en

CFA Learning Notes and Materials

11th April 2024

I have passed the CFA III level exam, and been granted the chart.

For any errates and insights, please free to contact to me.

Here below are my learning footprints for CFA level III. All files are converted to .html as you will find in the following . If you need the raw markdown codes, please move to my Github Repo.

P.S. there are typos and miswritten parts in the notes. Welcome to find me and help me update those mistakes. Or, probably I will update them if I fail the level III exam (in that I would review those notes). 🙂

Best Wishes
FZ

  1. CME
    P.S. BehaviouralFinance
  2. AssetAllocation
  3. Derivatives&Exchange
  4. Fixed-Income
  5. Equity
    P.S. Equity-Active
  6. Fixed-Income
  7. Alternatives
  8. PrivateWealthManagement
  9. InstitutionalInvestors
  10. TradingEvaluationManagerSelection
    P.S. TradingAdditional
  11. Ethics
    P.S. Ethics_from_Level_II_Code_n_Standards

Two Approaches for Forecasting Exchange Rate

The first approach is that analysts focus on flows of export and imports to establish what the net trade flows are and how large they are relative to the economy and other, potentially larger financing and investment flows. The approach also considers differences between domestic and foreign inflation rates that relate to the concept of purchasing power parity. Under PPP, the expected percentage change in the exchange rate should equal the difference between inflation rates. The approach also considers the sustainability of current account imbalances, reflecting the difference between national saving and investment.

The second approach is that the analysis focuses on capital flows and the degree of capital mobility. It assumes that capital seeks the highest risk-adjusted return. The expected changes in the exchange rate will reflect the differences in the respective countries’ assets’ characteristics such as relative short-term interest rates, term, credit, equity and liquidity premiums. The approach also considers hot money flows and the fact that exchange rates provide an across the board mechanism for adjusting the relative sizes of each country’s portfolio of assets.

Source by CFA reading materials

Least Squares Method – Intro to Kalman Filter

Consider a Linear Equation,

$$ y_i = \sum_{j=1}^n C_{i,j} x_j +v_i,\quad i=1,2,…$$

, where C_{i,j} are scalars and v_i\in \mathbb{R} is the measurement noise. The noise is unknown, while we assume it follows certain patterns (the assumptions are due to some statistical properties of the noise). We assume v_i, v_j are independent for i\neq j. Properties are mean of zero, and variance equals sigma squared.

$$\mathbb{E}(v_i)=0$$

$$\mathbb{E}(v_i^2) = \sigma_i^2$$

We can rewrite y_i = \sum_{j=1}^n C_{i,j} x_j +v_i as,

$$ \begin{pmatrix} y_1 \ y_2 \ \vdots\ y_s\end{pmatrix} = \begin{pmatrix} C_{11} & C_{12} & \cdots & C_{1n} \ C_{21} & C_{22}& \cdots & C_{2n} \ \vdots & \vdots & \cdots & \vdots \ C_{s1} & C_{s2} & \cdots & C_{sn}\end{pmatrix} \begin{pmatrix} x_1 \ x_2 \ \vdots\ x_n\end{pmatrix} + \begin{pmatrix} v_1 \ v_2 \ \vdots\ v_s\end{pmatrix} $$

, in a matrix form,

$$ \vec{y} = C \vec{x} + \vec{v} $$

, but I would write in a short form,

$$ y= C x +v$$

We solve for the least squared estimator from the optimisation problem, (there is a squared L2 norm)

$$ \min_x || y-Cx ||_2^2 $$

Recursive Least Squared Method

The classic least squared estimator might not work well when data evolving. So, there emerges a Recursive Least Squared Method to deal with the discrete-time instance. Let’s say, for a discrete-time instance k, y_k \in \mathbb{R}’ is within a set of measurements group follows,

$$y_k = C_k x + v_k$$

, where C_k \in \mathbb{R}^{l\times n}, and v_k \in \mathbb{R}^l is the measurement noise vector. We assume that the covariance of the measurement noise is given by,

$$ \mathbb{E}[v_k v_k^T] = R_k$$

, and

$$\mathbb{E}[v_k]=0$$

The recursive least squared method has the following form in this section,

$$\hat{x}k = \hat{x}{k-1} + K_k (y_k – C_k \hat{x}_{k-1})$$

, where \hat{x}k and \hat{x}{k-1} are the estimates of the vector x at the discrete-time instants k and k-1, and K_k \in \mathbb{R}^{n\times l} is the gain matrix that we need to determine. K_k is coined the ‘Gain Matrix’

The above equation updates the estimate of x at the time instant k on the basis of the estimate \hat{x}_{k-1} at the previous time instant k-1 and on the basis of the measurement y_k obtained at the time instant k, as well as on the basis of the gain matrix K_k computed at the time instant k.

Notation

$\hat{x}$ is the estimate.

$$ \hat{x}k = \begin{pmatrix} \hat{x}{1,k} \ \hat{x}{2,k} \ \vdots \\hat{x}{n,k} \end{pmatrix} $$

, which is corresponding with the true vector x.

$$x = \begin{pmatrix} x_1 \ x_2 \ \vdots \ x_n \end{pmatrix}$$

The estimation error, \epsilon_{i,k} = x_i – \hat{x}_{i,k} \quad i=1,2,…,n.

$$\epsilon_k = \begin{pmatrix} \epsilon_{1,k} \ \epsilon_{2,k} \ \vdots \\epsilon_{n,k} \end{pmatrix} = x – \hat{x}_k = \begin{pmatrix} x_1-\hat{x}_{1,k} \ x_2 – \hat{x}_{2,k} \ \vdots \x_n-\hat{x}_{n,k} \end{pmatrix} $$

The gain K_k is computed by minimising the sum of variances of the estimation errors,

$$ W_k = \mathbb{E}(\epsilon_{1,k}^2) + \mathbb{E}(\epsilon_{2,k}^2) + \cdots + \mathbb{E}(\epsilon_{n,k}^2) $$

Next, let’s show the cost function could be represented as follows, (tr(.) is the trace of a matrix)

$$ W_k = tr(P_k) $$

, and P_k is the estimation error covariance matrix defined by

$$ P_k = \mathbb{E}(\epsilon_k \epsilon_k^T )$$

Or, says,

$$ K_k = arg\min_{K_k} W_k = tr\bigg( \mathbb{E}(\epsilon_k \epsilon_k^T ) \bigg)$$

Why is that?

$$\epsilon_k \epsilon_k^T = \begin{pmatrix} \epsilon_{1,k} \ \epsilon_{2,k} \\vdots \ \epsilon_{n,k} \end{pmatrix} \begin{pmatrix} \epsilon_{1,k} & \epsilon_{2,k} & \cdots & \epsilon_{n,k} \end{pmatrix}$$

$$ = \begin{pmatrix} \epsilon_{1,k}^2 & \cdots & \epsilon_{1,k}\epsilon_{n,k} \ \vdots & \epsilon_{i,k}^2 & \vdots \ \epsilon_{1,k}\epsilon_{n,k} & \cdots & \epsilon_{n,k}^2\end{pmatrix} $$

So,

$$ P_k = \mathbb{E}[\epsilon_k \epsilon_k^T] $$

$$tr(P_k) = \mathbb{E}(\epsilon_{1,k}^2) + \mathbb{E}(\epsilon_{2,k}^2) + \cdots + \mathbb{E}(\epsilon_{n,k}^2)$$

Optimisation

$$ K_k = arg\min_{K_k} W_k = tr\bigg( \mathbb{E}(\epsilon_k \epsilon_k^T ) \bigg) = tr(P_k)$$

Let’s derive the optimisation problem.

$$\epsilon_k = x-\hat{x}_k$$

$$ =x-\hat{x}{k-1} – K_k(y_k – C_k \hat{x}{k-1}) $$

$$ = x- \hat{x}{k-1} – K_k (C_k x + v_k – C_k \hat{x}{k-1}) $$

$$ = (I – K_k C_k)(x-\hat{x}_{k-1}) – K_k v_k $$

$$ =(I-K_k C_k )\epsilon_{k-1} – K_k v_k $$

Recall y_k = C_k x + v_k and \hat{x}k = \hat{x}{k-1} + K_k (y_k – C_k \hat{x}_{k-1})

So, \epsilon_k \epsilon_k^T would be,

$$\epsilon_k \epsilon_k^T = \bigg((I-K_k C_k )\epsilon_{k-1} – K_k v_k\bigg)\bigg((I-K_k C_k )\epsilon_{k-1} – K_k v_k\bigg)^T$$

$P_k = \mathbb{E}(\epsilon_k \epsilon_k^T)$, and $P_{k-1} = \mathbb{E}(\epsilon_{k-1} \epsilon_{k-1}^T)$.

$\mathbb{E}(\epsilon_{k-1} v_k^T) = \mathbb{E}(\epsilon_{k-1}) \mathbb{E}(v_k^T) =0$ by the white noise property of $\epsilon$ and $v$. However, $\mathbb{E}(v_k v_k^T) = R_k$. Substituting all those into $P_k$, we would get,

$$P_k = (I – K_k C_k)P_{k-1}(I – K_k C_k)^T + K_k R_k K_k^T$$

$$ P_k = P_{k-1} – P_{k-1} C_k^T K_k^T – K_k C_k P_{k-1} + K_k C_k P_{k-1}C_k^T K_k^T + K_k R_k K_k^T $$

$$W = tr(P_k)= tr(P_{k-1}) – tr(P_{k-1} C_k^T K_k^T) – tr(K_k C_k P_{k-1}) + tr(K_k C_k P_{k-1}C_k^T K_k^T) + tr(K_k R_k K_k^T) $$

We take F.O.C. to solve for K_k = arg\min_{K_k} W_k = tr\bigg( \mathbb{E}(\epsilon_k \epsilon_k^T ) \bigg) = tr(P_k), by letting \frac{\partial W_k}{\partial K_k} = 0. See the Matrix Cookbook and find how to do derivatives w.r.t. K_k.

$$\frac{\partial W_k}{\partial K_k} = -2P_{k-1} C_k^T + 2K_k C_k P_{k-1} C_k^T + 2K_k R_k = 0$$

We solve for K_k,

$$ K_k = P_{k-1} C_k^T (R_k + C_k P_{k-1} C_k^T)^{-1}$$

, we let L_k = R_k + C_k P_{k-1} C_k^T, and L_k has the following property L_k = L_k^T and L_k^{-1} = (L_k^{-1})^T

$$ K_k = P_{k-1} C_k^T L_k^{-1} $$

Plug K_k = P_{k-1} C_k^T K_k^{-1} back into P_k.

$$ P_k = P_{k-1} – K_kC_k P_{k-1} = (I-K_k C_k)P_{k-1} $$

Summary

In the end, the Recursive Least Squared Method could be summarised as the following three equations.

  • 1. Update the Gain Matrix.

$$ K_k = P_{k-1} C_k^T (R_k + C_k P_{k-1} C_k^T)^{-1}$$

  • 2. Update the Estimate.

$$\hat{x}_k = \hat{x}_{k-1} + K_k (y_k – C_k \hat{x}_{k-1})$$

  • 3. Propagation of the estimation error covariance matrix by using this equation.

(I-K_k C_k)P_{k-1}

Reference

Introduction to Kalman Filter: Derivation of the Recursive Least Squares Method

Sigmoid & Logistic

Sigmoid function is largely used for the binary classification, in either machine learning algorithm or econometrics.

Why the Sigmoid Function shapes in this form?

Firstly, let’s introduce the odds.

Odds provide a measure of the likelihood of a particular outcome. They are calculated as the ratio of the number of outcomes that produce that outcome to the number that do not.

Odds also have a simple relation with probability: the odds of an outcome are the ratio of the probability that the outcome occurs to the probability that the outcome does not occur. In mathematical terms, p is the probability of the outcome, and 1-p is the probability of not occurring.

$$ odds = \frac{p}{1-p} $$

Odd and Probability

Let’s find some insights behind the probability and the odd. Probability links with the outcomes in that for each outcomes, the probability give its specific corresponding probability. Pr(Y), where Y is the outcome, and Pr(\cdot) is the probability density function that project outcomes to it’s prob.

What about the odds? Odds is more like a ratio that is calculated by the probability as the formula says.

Implication: Compared to the probability, odds provide more about how the binary classification is balanced or not, but the probability distribution.

Example

Rolling a six-side die. The probability of rolling 6 is 1/6, but the odd is $1/5.

Formula

$$ odd = \frac{Pr(Y)}{1-Pr(Y)} $$

, where Y is the outcomes.

Logit

As the probability Pr(Y) is always between [0,1], the odds must be non-negative, odd \in [0,\infty]. We may want to apply a monotonic transformation to re-gauge that range of odds. We will apply on the logarithm.

$$ Sigmoid/Logistic := log(odds) =log\bigg( \frac{Pr(Y)}{1-Pr(Y)} \bigg) $$

We then get the Sigmoid function.

As the transformation we apply on is monotonic, the Sigmoid function remains the similar properties as the odd. The Sigmoid function keeps the similar implication, representing the balance of the binary outcomes.

Then, we bridge Y = f(X), the outcome Y is a function of events X. Here, we assume a linear form as Y = X\beta. The sigmoid function would then become a function of X.

$$g(X) = log\bigg( \frac{Pr(X\beta)}{1-Pr(X\beta)} \bigg) $$

$$ e^g = \frac{p}{1-p} $$

$$ p = \frac{e^g}{e^g+1}=\frac{1}{1+e^{-g}}$$

$$ p = \frac{1}{1+e^{-X\beta}}$$

We finally get out logistic sigmoid function as above.

Dirac Delta Function

The Dirac Delta Function could be applied to simplify the differential equation. There are three main properties of Dirac Delta Function.

$$\delta (x-x’) =\lim_{\tau\to0}\delta (x-x’)$$

such that,

$$ \delta (x-x’) = \begin{cases} \infty & x= x’ \ 0 & x\neq x’ \end{cases} $$

$$\int_{-\infty}^{\infty} \delta (x-x’)\ dx =1$$

Three Properties:

  • Property 1:

$$\delta(x-x’)=0 \quad \quad ,x\neq x’ $$

  • Property 2:

$$ \int_{x’-\epsilon}^{x’+\epsilon} \delta (x-x’)dx =1\quad \quad ,\epsilon >0 $$

  • Property 3:

$$\int_{x’-\epsilon}^{x’+\epsilon} f(x)\ \delta (x-x’)dx = f(x’)$$

At x=x’ the Dirac Delta function is sometimes thought of has having an “infinite” value. So, the Dirac Delta function is a function that is zero everywhere except one point and at that point it can be thought of as either undefined or as having an “infinite” value.

Girsanov’s Theorem

Statement

We can change the probability measure, and then make a random variable follows a certain probability measure.

  • Radon-Nikodym Derivative:

$$Z(\omega) = \frac{\tilde{P}(\omega)}{P(\omega)}$$

  • $\tilde{P}(\omega)$ is the risk-neutral probability measure.
  • ${P}(\omega)$ is the actual probability measure.
  • Properties:
    • $Z(\omega)>0$
    • $\mathbb{E}(Z)=1$
    • As \tilde{P}(\omega) = Z(\omega) P(\omega), so if Z(\omega), then \tilde{P}(\omega)>P(\omega). vice versa.

We can calculate that,

$$ \underbrace{\tilde{\mathbb{E}}(X)}_{\text{Expectation under Risk-neutral Probability Measure}} = \underbrace{\mathbb{E}(ZX)}_{\text{Expectation under Actual Probability Measure}} $$

Proof & Example

Under (\Omega,\mathcal{F},P), A\in \mathcal{F}, let X be a random variable X\sim N(0,1). \mathbb{E}(X)=0, and \mathbb{Var}(X)=1.

$Y=X+\theta$, $\mathbb{E}(Y)=\theta$, and $\mathbb{Var}(Y)=1$.

$X$ here is s.d. normal under the actual probability measure.

However, Y here is not standard normal under the current probability P(.), because \mathbb{E}(Y)\neq0.

What do we do?

We change the probability measure from P(.)\to\tilde{P}(.) to let Y be standard normal under the new probability measure!

We set the Radon-Nikodym Derivative,

$$Z(\omega) = exp\{ -\theta\ X(\omega) – \frac{1}{2}\theta^2 \}$$

Now, we can create the probability measure \tilde{P}(A), A={ \omega;Y(\omega)\leq b) }

$$\tilde{P}(A) = \int_A Z(\omega)\ dP(\omega)$$

such that Y=X+\theta would be standard normal distributed under the new probability measure \tilde{P}(A).

$$\tilde{P}(A) = \tilde{P}(Y(\omega \leq b)$$

$$ = \int_{{ Y(\omega)\leq b } } exp{ -\theta\ X(\omega) – \frac{1}{2}\theta^2 } \ dP(\omega)$$

, then change the integral range from the set A to \Omega by multiplying that indicator.

$$ = \int_{\Omega }\mathbb{1}_{ Y(\omega)\leq b }\ exp{ -\theta\ X(\omega) – \frac{1}{2}\theta^2 } \ dP(\omega)$$

, change from dP to dX,

$$ = \int_{-\infty }^{\infty }\mathbb{1}_{ b-\theta}\ exp{ -\theta\ X(\omega) – \frac{1}{2}\theta^2 } \ \frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}X^2(\omega)} \ dX(\omega)$$

$$ =\frac{1}{\sqrt{2\pi}} \int_{-\infty }^{b-\theta}\ exp{ -\theta\ X(\omega) – \frac{1}{2}\theta^2- \frac{1}{2}X^2(\omega)} \ dX(\omega)$$

$$ =\frac{1}{\sqrt{2\pi}} \int_{-\infty }^{b-\theta}\ exp\Bigg\{ -\frac{1}{2}\bigg(\theta+ X(\omega)\bigg)^2\Bigg\} \ dX(\omega)$$

, as Y=X+\theta, dY = dX, we now change dX to dY,

$$ =\frac{1}{\sqrt{2\pi}} \int_{-\infty }^{b}\ exp\big\{ -\frac{1}{2}Y(\omega)^2\big\} \ dY(\omega)$$

, the above is now a standard normal distribution for Y(\omega).

Dutch Disease

In Dutch Disease, certain sectors have enormous exports demand, which would drive the demand of currency for that country. Its currency appreciates. However, the rest sectors that may not have such huge amount of exports demand would also have to undergo an appreciation of currency. Export demands for goods and services in the rest sectors would decrease even severe.

The Impact of Balance of Payments Flows

As noted earlier, the parity conditions may be appropriate for assessing fair value for currencies over long horizons, but they are of little use as a real-time gauge of value. There have been many attempts to find a better framework for determining a currency’s short-run or long-run equilibrium value. Let’s now examine the influence of trade and capital flows.

A country’s balance of payments consists of its (1) current account as well as its (2) capital and (3) financial account. The official balance of payments accounts make a distinction between the “capital account” and the “financial account” based on the nature of the assets involved. For simplicity, we will use the term “capital account” here to reflect all investment/financing flows. Loosely speaking, the current account reflects flows in the real economy, which refers to that part of the economy engaged in the actual production of goods and services (as opposed to the financial sector). The capital account reflects financial flows. Decisions about trade flows (the current account) and investment/financing flows (the capital account) are typically made by different entities with different perspectives and motivations. Their decisions are brought into alignment by changes in market prices and/or quantities. One of the key prices—perhaps the key price—in this process is the exchange rate.

Countries that import more than they export will have a negative current account balance and are said to have current account deficits. Those with more exports than imports will have a current account surplus. A country’s current account balance must be matched by an equal and opposite balance in the capital account. Thus, countries with current account deficits must attract funds from abroad in order to pay for the imports (i.e., they must have a capital account surplus).

When discussing the effect of the balance of payments components on a country’s exchange rate, one must distinguish between short-term and intermediate-term influences on the one hand and longer-term influences on the other. Over the long term, countries that run persistent current account deficits (net borrowers) often see their currencies depreciate because they finance their acquisition of imports through the continued use of debt. Similarly, countries that run persistent current account surpluses (net lenders) often see their currencies appreciate over time.

However, investment/financing decisions are usually the dominant factor in determining exchange rate movements, at least in the short to intermediate term. There are four main reasons for this:

  • Prices of real goods and services tend to adjust much more slowly than exchange rates and other asset prices.
  • Production of real goods and services takes time, and demand decisions are subject to substantial inertia. In contrast, liquid financial markets allow virtually instantaneous redirection of financial flows.
  • Current spending/production decisions reflect only purchases/sales of current production, while investment/financing decisions reflect not only the financing of current expenditures but also the reallocation of existing portfolios.
  • Expected exchange rate movements can induce very large short-term capital flows. This tends to make the actualexchange rate very sensitive to the currency views held by owners/managers of liquid assets.

Current Account Imbalances and the Determination of Exchange Rates

Current account trends influence the path of exchange rates over time through several mechanisms:

  • The flow supply/demand channel
  • The portfolio balance channel
  • The debt sustainability channel

Let’s briefly discuss each of these mechanisms next.

The Flow Supply/Demand Channel

The flow supply/demand channel is based on a fairly simple model that focuses on the fact that purchases and sales of internationally traded goods and services require the exchange of domestic and foreign currencies in order to arrange payment for those goods and services. For example, if a country sold more goods and services than it purchased (i.e., the country was running a current account surplus), then the demand for its currency should rise, and vice versa. Such shifts in currency demand should exert upward pressure on the value of the surplus nation’s currency and downward pressure on the value of the deficit nation’s currency.

Hence, countries with persistent current account surpluses should see their currencies appreciate over time, and countries with persistent current account deficits should see their currencies depreciate over time. A logical question, then, would be whether such trends can go on indefinitely. At some point, domestic currency strength should contribute to deterioration in the trade competitiveness of the surplus nation, while domestic currency weakness should contribute to an improvement in the trade competitiveness of the deficit nation. Thus, the exchange rate responses to these surpluses and deficits should eventually help eliminate—in the medium to long run—the source of the initial imbalances.

The amount by which exchange rates must adjust to restore current accounts to balanced positions depends on a number of factors:

  • The initial gap between imports and exports
  • The response of import and export prices to changes in the exchange rate
  • The response of import and export demand to changes in import and export prices

If a country imports significantly more than it exports, export growth would need to far outstrip import growth in percentage terms in order to narrow the current account deficit. A large initial deficit may require a substantial depreciation of the currency to bring about a meaningful correction of the trade imbalance.

A depreciation of a deficit country’s currency should result in an increase in import prices in domestic currency terms and a decrease in export prices in foreign currency terms. However, empirical studies often find limited pass-through effects of exchange rate changes on traded goods prices. For example, many studies have found that for every 1% decline in a currency’s value, import prices rise by only 0.5%—and in some cases by even less—because foreign producers tend to lower their profit margins in an effort to preserve market share. In light of the limited pass-through of exchange rate changes into traded goods prices, the exchange rate adjustment required to narrow a trade imbalance may be far larger than would otherwise be the case.

Many studies have found that the response of import and export demand to changes in traded goods prices is often quite sluggish, and as a result, relatively long lags, lasting several years, can occur between (1) the onset of exchange rate changes, (2) the ultimate adjustment in traded goods prices, and (3) the eventual impact of those price changes on import demand, export demand, and the underlying current account imbalance.

The Portfolio Balance Channel

The second mechanism through which current account trends influence exchange rates is the so-called portfolio balance channel. Current account imbalances shift financial wealth from deficit nations to surplus nations. Countries with trade deficits will finance their trade with increased borrowing. This behaviour may lead to shifts in global asset preferences, which in turn could influence the path of exchange rates. For example, nations running large current account surpluses versus the United States might find that their holdings of US dollar–denominated assets exceed the amount they desire to hold in a portfolio context. Actions they might take to reduce their dollar holdings to desired levels could then have a profound negative impact on the dollar’s value.

“Shifts in Global Asset Preferences” means would alter the components of assets allocation in the portfolio.

The Debt Sustainability Channel

The third mechanism through which current account imbalances can affect exchange rates is the so-called debt sustainability channel. According to this mechanism, there should be some upper limit on the ability of countries to run persistently large current account deficits. If a country runs a large and persistent current account deficit over time, eventually it will experience an untenable rise in debt owed to foreign investors. If such investors believe that the deficit country’s external debt is rising to unsustainable levels, they are likely to reason that a major depreciation of the deficit country’s currency will be required at some point to ensure that the current account deficit narrows significantly and that the external debt stabilises at a level deemed sustainable.

The existence of persistent current account imbalances will tend to alter the market’s notion of what exchange rate level represents the true, long-run equilibrium value. For deficit nations, ever-rising net external debt levels as a percentage of GDP should give rise to steady (but not necessarily smooth) downward revisions in market expectations of the currency’s long-run equilibrium value. For surplus countries, ever-rising net external asset levels as a percentage of GDP should give rise to steady upward revisions of the currency’s long-run equilibrium value. Hence, one would expect currency values to move broadly in line with trends in debt and/or asset accumulation.

Reference

CFA Readings

Value at Risk & Expected Shortfalls

Value at Risk – VaR

VaR is a probability statement about the potential change in the value of a portfolio.

Notation

$$Porb(x\leq VaR(X))= 1-c$$

$$ Prob\bigg(z \leq \frac{VaR(X)-\mu}{\sigma}\bigg)=1-c $$

  • $c$ – confidence interval, i.e. $c=99\%$. Then $1-c = 1\% $
  • $\mu$ and $\sigma$ are for $X$.
    • For Example, if X is yearly return, then \mu_{252days}=252\cdot\mu_{1day}, and \sigma_{252days}=\sqrt{252}\cdot\sigma_{1day}
  • $x$ here is the return. So, $c$ is the confidence interval, i.e. 99%.
    • VaR focus on the tail risks. If x stands for return, then tail risk is on the left tail, z_{1-c}.
  • If x is the loss, the tail risk is on the right tail. z_c

$$VaR(X) = \mu + \sigma\cdot \Phi^{-1}(1-c)$$

$$VaR(X) = \mu + \sigma\cdot z_{1-c}$$

  • I.E.

​ If c=99\%, then 1-c=1\%, so z_{1-c}=z_{0.01} \approx -2.33

​ VaR(X) = \mu – 2.33\cdot \sigma

P.S.

​ The unit of VaR is the amount of loss, so it should be monetary amount. For example, if the total amount of portfolio is USD 1 million, then VaR = \$1m \cdot (\mu – 2.33\cdot \sigma).

Loss Distribution

Remember X is a distribution of loss. If we know the distribution of Portfolio Return R, R\sim N(\mu, \sigma^2), then what is the dist for X?

$$X \sim N(-\mu, \sigma^2)$$

Right! Loss is just the negative return. Also, the volatility would not be affected by plus / minus.

Expected Shortfall (ES)

Expected Shortfall states the Expected Loss during time T conditional on the loss being greater than the c^{th} percentile of the loss distribution.

Notation

$$ ES_c (X) = \mathbb{E}\bigg[ X|X\leq VAR_c(X) \bigg] $$

  • Be attention here, X is a r.v., and x stands for return here! while the only variable in the ES_c(X) is c, the confidence level, instead of X.
  • $c$ is the confidence level, i.e. $c$ = 99%.
  • If x stands for return, then the VaR is the left-tail, z_{1-c}.

$$ ES_c (X) = \mathbb{E}\bigg[ X|X\geq VAR_c(X) \bigg] $$

  • If x stands for loss (, which is the negative of return ), then the VaR is the right-tail, z_{c}.

Derivation

Notation Form

Consider, x is the return, then ES_c (X) = \mathbb{E}\bigg[ X|X\leq VAR_c(X) \bigg], and VaR_c(x)= \mu + z_{1-c}\sigma, where c is the confidence level c=99\% for example.

$$ES_c(X) = \frac{\int_{-\infty}^{VaR} xf(x)dx }{\int_{-\infty}^{VaR} f(x)dx } = \frac{\int_{-\infty}^{VaR} x \phi(x)dx }{\int_{-\infty}^{VaR} \phi(x)dx } =\frac{\int_{-\infty}^{VaR} x \phi(x)dx }{ \Phi(VaR) – \Phi(-\infty)} $$

$$= \frac{1}{ \Phi(VaR) – \Phi(-\infty) }\int_{-\infty}^{VaR}x \frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} dx $$

Replace z = \frac{x-\mu}{\sigma}, then x = \mu + z \sigma, and dx = \sigma dz

$$ = \frac{1}{\Phi(VaR)} \int_{-\infty}^{VaR}(\mu + z\sigma) \frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{z^2}{2}}\sigma dz $$

$$ = \frac{1}{\Phi(VaR)}\mu \int_{-\infty}^{VaR}\frac{1}{\sqrt{2\pi }} e^{-\frac{z^2}{2}} dz + \sigma^2\int_{-\infty}^{VaR} z \frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{z^2}{2}} dz $$

$$ = \frac{1}{\Phi(VaR)}\mu \Phi(VaR) – \frac{\sigma^2}{\Phi(VaR)}\int_{-\infty}^{VaR} \frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{z^2}{2}} d(-\frac{z^2}{2}) $$

$$ = \mu – \frac{\sigma^2}{\Phi(VaR)} \frac{1}{\sqrt{2\pi \sigma^2}} \int_{-\infty}^{VaR} e^{-\frac{z^2}{2}} d(-\frac{z^2}{2}) $$

$$ = \mu – \frac{\sigma}{\Phi(VaR)} \frac{1}{\sqrt{2\pi }} e^{-\frac{z^2}{2}} |_{-\infty}^{VaR} $$

$$ = \mu – \frac{\sigma}{\Phi(VaR)} \frac{1}{\sqrt{2\pi }} e^{-\frac{VaR^2}{2}}= \mu – \frac{\sigma}{\Phi(VaR)} \phi(VaR)$$

Recall, VaR_c(x)= \mu + z_{1-c}\sigma, so \phi(VaR_c(x))= \phi(\mu + z_{1-c}\sigma) \leftrightarrow \phi(z_{1-c}) = \phi\bigg( \Phi^{-1}(1-c) \bigg), and \Phi(VaR_c(x))= \Phi(\mu + z_{1-c}\sigma) \leftrightarrow \phi(z_{1-c}) = \Phi\bigg( \Phi^{-1}(1-c) \bigg) = 1-c.

Thus,

$$ ES_c(X) =\mu – \frac{\sigma}{\Phi(VaR)} \phi(VaR)=\mu -\sigma \frac{\phi\big( \Phi^{-1}(1-c) \big)}{1-c}$$

VaR Form

we ‘sum up’ (integrate) the VaR from c to 1, conditional on 1-c.

$$ES_c(X) = \frac{1}{1-c} \int_c^1 VaR_u(X)du$$

$$ ES_c(X) = \frac{1}{1-c} \int_c^1 \bigg( \mu + \sigma\cdot \Phi^{-1}(1-u) \bigg) du $$

$$ =\mu + \frac{\sigma}{1-c} \int^1_c \Phi^{-1}(1-u) du $$

We let u = \Phi(Z), where Z \sim N(0,1). Then,

  • $du =d(\Phi(z)) =\phi(z) dz$.
  • $u\in (c,1)$, so $z = \Phi^{-1}(u)\in (z_c \ , \infty)$

Thus,

$$ ES_c(X) =\mu + \frac{\sigma}{1-c} \int^{\infty}_{z_c} \Phi^{-1}\big(1-\Phi(z)\big)\phi(z) dz $$

As 1-\Phi(z) = \Phi(-z)

$$ ES_c(X) =\mu + \frac{\sigma}{1-c} \int^{\infty}_{z_c} \Phi^{-1}(\Phi(-z))\phi(z) dz = \mu – \frac{\sigma}{1-c} \int^{\infty}_{z_c} z\phi(z) dz $$

$ \int_{z_c}^{\infty} z \phi(z)dz = \int_{z_c}^{\infty} z \frac{1}{\sqrt{2\pi}}e^{-\frac{z^2}{2}}dz = -\frac{1}{\sqrt{2\pi}} \int_{z_c}^{\infty} -e^{\frac{z^2}{2}}d(e^{-\frac{z^2}{2}})$

$=\frac{1}{\sqrt{2\pi}}e^{-\frac{z_c^2}{2}}=\phi(z_c)=\phi\big(\Phi^{-1}(c)\big)$, bring it back to $ES_c(X)$

$$ES_c(X) = \mu – \sigma\frac{ \phi\big(\Phi^{-1}(c)\big)}{1-c}$$

Morden Portfolio Theory

  • $x$ – vector weights
  • $R$ – vector of all assets’ returns
  • $\mu = \mathbb{E}(R)$ – mean return of all assets
  • $\Sigma = \mathbb{E}\bigg[ (R-\mu)(R-\mu)^T \bigg]$ – var-cov matrix of all assets

So,

  • $\mu_x = x^T \mu$ – becomes a scalar now
  • $\sigma^2 = x^T \Sigma x$ – collapse to be a scalar

Optimisation

  • Maximise Expected Return s.t. volatility constraint.

$$ \max_{x} \mu_x \quad s.t. \quad \sigma_x \leq \sigma^* $$

  • Minimise Volatility s.t. return constraint.

$$ \min_{x} \sigma_x \quad s.t. \quad \mu_x \geq \mu^* $$

Portfolio Risk Measures

By definition, the loss of a portfolio is the negative of return, L(x) = -R(x).

The Loss distribution becomes the same normal distribution with x-axis reversed.

  • Volatility of Loss: \sigma(L(x)) = \sigma_x, the minus does not matter in the s.d.
  • Standard Deviation-based risk measure: =\mathbb{E}(L(x)) + cz_{c}\sigma(L(x)), x-axis is revered, so z_{1-c} for return becomes z_c for loss.
  • VaR: VaR_{\alpha}(x)=inf\bigg{ \mathscr{l}:Prob\big[ L(x)\leq \mathscr{l} \geq\big] \alpha \bigg}
  • Expected Shortfall: ES_{\alpha}(x) = \frac{1}{1-\alpha} \int_{\alpha}^1 VaR_u(x) du. In other form, ES_{\alpha}(x)=\mathbb{E}\bigg( L(x)| L(x)\geq VaR_{\alpha}(x) \bigg)

As R \sim N(\mu, \Sigma),

  • for our portfolio with weights x, mean = \mu, and \sigma_x = \sqrt{x^T \Sigma x}.
  • for the loss, mean = -\mu, and \sigma_x = \sqrt{x^T \Sigma x}.