“international finance and payments” course iii: “international financial portfolios theory”...
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“International Finance and Payments”
Course III:Course III:
““International financial portfolios theoryInternational financial portfolios theory””
Lect. Cristian PĂUNLect. Cristian PĂUN
Email: Email: [email protected]
URL: http://www.finint.ase.roURL: http://www.finint.ase.ro
Academy of Economic Studies
Faculty of International Business and Economics
Course 3: International Financial Portfolios Theory
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International Financial Markets and Institutions - review
• the main components of the IFS are: financial markets, financial institutions and financial instruments;
• financial markets: money markets and capital markets (the differences);
• internal financial resources vs. external financial resources;
• direct financing vs. indirect financing;
• financial institutions: international financial institutions, government agencies, depositary institutions and investment institutions
• financial instruments: direct investment and indirect investment instruments;
• direct investment instruments: money market instruments and capital market instruments
• money market instruments: T-bills, REPO, negotiable DC, commercial papers, forward contracts;
• capital market instruments: fixed income instruments, variable income instruments, derivatives.
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Risk and return in international finance• Knight definition: risk is the decisional situation in which we can associate o probability to future events (uncertainty and certainty)
• Main critics:
• it is difficult to make a difference between the three decisional situations;
• it is very difficult to determine the risk level for an investment based on the capacity of probabilistic associations;
• a too simplistic approach;
• high subjectivity in the probabilistic association process (some investors can consider an event as a risky one and others as an uncertain one based on different analytical capacities).
• Other risk definitions:• “Risk is the possibility that the returns be lower than expected” (Mehr Hedges);
• “Risk is the possible return variability caused by an uncertain further event” (Dorfman);
• “Risk is the incertitude about future possible losses” (Redja)
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Incertitude and risk factors:• lack of perfect information;• impossibility to make a correct prediction in case of future events;• incapacity to identify all the alternatives for your decision;• the future events are usually unique;• the investors profile;• impossibility to control all factors or events;• time pressure.
•higher risk implies usually higher expected return;
• the relation between the utility of an investment return and the return is not a linear one (decreasing marginal utility - u(w)>0 and u(w)<0) – Bernoulli
• if an investor prefers an investment p instead an investment q than U(p) is higher than U(q) – Neumann & Morgenstern
• if we have to chose between two investment p an q, the utility of a linear combination is equal with the amount of the utility for each alternative:
U( p + (1- )q) = U(p) + (1-)U(q) for each (0, 1)
Risk and return relation:
Course 3: International Financial Portfolios Theory
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A. Markovitz Model
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Markovitz Model Hypothesis:
1. Expected profit is normally distributed;
2. Investors are seeking in every moment for their profit maximization;
3. Investors have a decreasing marginal utility of their wealth;
4. The variability of the probable profits is the proper measure for the level of
risk assumed by the investors;
5. The investment decision is based on risk & return profile;
6. The investors usually prefer higher profits at a given level of risk;
7. The investors usually prefer lower risk at a given level of the expected
return;
8. Investors have a limited time for their investment decision.
Course 3: International Financial Portfolios Theory
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Expected return
it
1tti
ti
1ti
P
DPPRET
qk
qk
4k3k
4k3k
2k1k
2k1kik R...
p...
RR
pp
RR
ppR
n1,ini
q1,ii
n
q1,i2iq1,i1i
q1,iiq1,ii
21
port
E(R
p
w
...E(RE(R
...pp
...ww
R
)))
q
1i iik Rp)E(R
n
1i iiport )E(Rw)E(R
- initial assumption
Course 3: International Financial Portfolios Theory
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Expected return - example
Individual expected return
p
Security A Security B
w(A) E(Rport)Ri Ri
0.02 10% 10% 0 10.00%
0.08 12% 11% 0.1 11.10%
0.11 14% 13% 0.2 13.20%
0.12 16% 17% 0.3 16.70%
0.15 18% 19% 0.4 18.60%
0.17 20% 21% 0.5 20.50%
0.13 22% 23% 0.6 22.40%
0.09 24% 24% 0.7 24.00%
0.07 26% 25% 0.8 25.80%
0.06 28% 26% 0.9 27.80%
1.00 19.2% 19.4% 1 19.24%
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Variance, covariance and correlation
)E(RR...
p...
)E(RR)E(RR
pp)E(RR
iqk
qk
i2ki1k
2k1kiik
2 iii2 RERpσ
iii RERpσ The measure of a risk in case of an individual security
Variance properties:
1. var (constant)= 0
2. var (c x z) = c2 x var (z)
3. var (x + y) = var (x) + var (y) + cov (x, y)
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Variance, covariance and correlation (cont.)
)E(RR)E(RRpCov jjx
N
1x
iixiij
Covariance properties:
1. cov(y, xi)= c1*cov(y,x1)+c2* cov(y,x2)+...cn* cov(y,xn) when
y= c1*x1+c2*x2+...cn*xn
2. cov(x,y) = cov(y,x)
3. cov(c * x, y)=c*cov(x,y)
)E(RRp)E(RRp
)E(RR)E(RRp
disp(y)disp(x)
y)cov(x,y)correl(x,
yi
yii
xi
xii
yi
yi
xi
xii
Interpretation:
• correl(x,y) = 0 – x is independent of y
• correl(x,y)=1 – x is total dependent of y
• correl(x,y) – negative means inverse relation between x and y
Course 3: International Financial Portfolios Theory
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p
Security A Security B
w(A) E(Rport) StDev(Port)Ri Ri
0.02 10.00% 10.00% 0.00% 10.00% 0.0035
0.08 12.00% 11.00% 10.00% 11.10% 0.0241
0.11 14.00% 13.00% 20.00% 13.20% 0.0320
0.12 16.00% 17.00% 30.00% 16.70% 0.0366
0.15 18.00% 19.00% 40.00% 18.60% 0.0391
0.17 20.00% 21.00% 50.00% 20.50% 0.0399
0.13 22.00% 23.00% 60.00% 22.40% 0.0391
0.09 24.00% 24.00% 70.00% 24.00% 0.0366
0.07 26.00% 25.00% 80.00% 25.80% 0.0320
0.06 28.00% 26.00% 90.00% 27.80% 0.0241
1.00 19.2% 19.4% 100.00% 19.24% 0.0037
Variance A 0.00366667
Variance B 0.00349889
Covariance 0.00317
Correlation 0.98336775
(8) covww2σw σ
(7) covww2σw σ
n
1i
n
1jijji
2i
2iportofoliu
n
1i
n
1jijji
2i
2iportofoliu
2
n
1i iiport )E(Rw)E(R
Course 3: International Financial Portfolios Theory
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Measuring the return distribution:
A B
E(Ra) E(Rb)
loss profit
loss profit
Return distribution for two investments with the same variation
3iii3 )E(rrpM
U(r) = E(r) – a0 x σ2 + a1 x M3 – a2 x M4 + a3 x M5 - ....
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Deriving Efficient Frontier:
A
B
C
A
B
C
Combination between A, B
and C
rf
Here is impossible to find a portfolio
Standard deviation
High risk / High return
Medium risk / Medium return
Low risk / Low return
Inefficient portfolios
Efficient Frontier of a market
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Optimal portfolio using Markovitz Model:
Efficient Frontier
CAL1
CAL2
CAL3
rf
Optimal portfolio
Efficient Frontier
CAL
Optimal risky portfolio
M
Investment
Debt
Risk
Expected return
Optimal portfolio:
Max{f(P)}=Max{[E(rP) - rF]/σP}
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Markovitz model (two assets example):
E(Ri) StDev Cov(A,B)Security A 7% 0.12Security B 14% 0.2
W(A) StDev E(Ri)0% 0.04 14.00%10% 0.03 13.30%20% 0.03 12.60%30% 0.02 11.90%40% 0.02 11.20%50% 0.02 10.50%60% 0.02 9.80%70% 0.01 9.10%80% 0.01 8.40%90% 0.01 7.70%
100% 0.01 7.00%
0.0087
E(Ri)
0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
12.00%
14.00%
16.00%
0.00 0.01 0.02 0.03 0.04 0.05
E(Ri)
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Optimal portfolio using utility function:
k 0.8A 5
Correl -0.7W(A) 15.57%W(B) 84.43%
Uport= E(rport) - σport2 x Aver x k,
(A)w(B)w
σσcorel2σσAk
B)corel(A,σσσAverk)E(r)E(r(A)w
optopt
BAABB2
A2
BAB2
BAopt
1
Aver – coefficient that measures the level of risk aversion for an investor
Optimal = Highest return at the lowest level of risk
More risk averse investor: Less risk averse investor:
k 0.8A 20
Correl -0.7W(A) 30.48%W(B) 69.52%
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Z1= 50000
Z2= 150000
P E(ret) U
1 50000 10.82
0.9 60000 10.93
0.8 70000 11.04
0.7 80000 11.15
0.6 90000 11.26
0.5 100000 11.37
0.4 110000 11.48
0.3 120000 11.59
0.2 130000 11.70
0.1 140000 11.81
0 150000 11.92
Risk averse investor:U(R) = Σ Pn lnRi
U(100000)=11.51
U(150000)=11.92
U(50000)=10.82
E(U(z))=11.37
z
50000 USD profit utility
50.000 USD loss utility
U(100000)=11.51
z1 z2E(z)
100.000 USD
50.000 USD
150.000 USD
p=1/2
p=1/2
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Risk lover investor:
U(150000)=2250
Profit utility
E(U(z))=1250
U(50000)=250
U(100000)=1000
E(z) z2z1
Loss utility
C(z)
Z1= 50000
Z2= 150000
P E(ret) U
1 50000 250
0.9 60000 450
0.8 70000 650
0.7 80000 850
0.6 90000 1050
0.5 100000 1250
0.4 110000 1450
0.3 120000 1650
0.2 130000 1850
0.1 140000 2050
0 150000 2250
U(100000)=1000
U(z) = K
1zp n
ii , n=2, k=10
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Markovitz Model and risk aversion:
Risk
Expected return
Risk lovers
Risk averse
P
InvestmentDebt
Efficient Frontier
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Conclusions:
• using historical data we can assign probabilities for returns in case of securities;
• expectation in case of returns are based on average value of probabilistic returns;
• we have only one optimal risky portfolio on efficient frontier;
• Markovitz Model is quite complicated to be applied on international markets or in case of complex portfolios;
• Markovitz approach improved the portfolio selection;
• the singleness of the optimal portfolio explains the development of new financial markets (pension funds, investment funds, life insurance policies)
• Markovitz Model takes into consideration the investors attitude against the risk;
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B. Capital Asset Pricing Model
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CAPM hypothesis:
• there are many investors on financial markets (incapacity to have a major influence on security pricing);
• limited time to take an investment decision (“myopic” behavior);
• we have risky instruments and risk free rate instruments;
• we have no transaction costs or taxes in case of investment transactions;
• all investors have a rational behavior (maximize their returns);
• all investors analyze investment alternatives in the same way (the return expectation are homogenous);
• market portfolio => optimal portfolio;
• risk premium = Rm – RFR
• risk measure: beta coefficient
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CAPM Equation:
Rf
Rm
Ei
Betaβi=1
Risk premium
Securities with a higher risk than market risk
Securities with a lower risk than market risk
2M
iMi
)r,R(Cov
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Problems with CAPM:
• we have a distinction between systemic and non-systemic risk;
• CAPM is a simple model and easy to use;
• we have no risk free rate instruments on a financial market;
• it is difficult to construct the optimal risky portfolio (market portfolio);
• market is not the only risk factor with impact on the expected return (company dimension, transaction costs);
• market index approximation;
• CAPM is a static model focused on expected return at a moment t;
• there was a lot of tests on the relevance of the CAPM (Roll, Fama & MacBeth, Banz, Jensen);
• is difficult to develop a global CAPM;
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Alternatives of the CAPM:
),(FRRRE Mimfiii A. CAPM and transaction costs:
B. CAPM and companies dimension (Banz, 1981):
C. CAPM and non-tradable assets (Mayers, 1972):
D. Single factor model:
iMiii RR
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C. Arbitrage Pricing Theory
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APT Model Hypothesis:
• factorial models can explain expected returns;
• arbitrage opportunities = zero investment portfolios;
• arbitrage opportunities occur when the law of one price is violated
• financial markets are perfect with a law volatility;
• rational equilibrium market prices move to rule out arbitrage opportunities;
• violation of the no arbitrage condition is the strongest form of market irrationality
• the way of exploiting arbitrage opportunities does not depend on risk aversion
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APT Equation:
i
K
1k
kikii
iKiK22i11iii
Fr
F...FFr
KiK22i11iii ...)r(E
Risk factors:
1. Chen, Ross and Roll APT Model
2. Fama & French APT Model
3. Morgan Stanley APT Model
4. Salomon Smith Barney APT Model
Systematic risk Non-systematic risk
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Chen, Ross and Roll APT Model (original APT):1. Industrial production (reflects changes in cash flow expectations)2. Yield spread btw high risk and low risk corporate bonds (reflects
changes in risk preferences)3. Difference between short-and long-term interest rate (reflects shifts in
time preferences)4. Unanticipated inflation5. Expected inflation (less important)
Fama & French APT Model :1. Market
2. Company size
3. Book-to-market factor
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Morgan Stanley APT Model:
Salomon Smith Barney APT Model
GDP growth1. Long-term interest rates2. Foreign exchange (Yen, Euro, Pound basket)3. Market Factor4. Commodities or oil price index
1. Market trend2. Economic growth3. Credit quality4. Interest rates5. Inflation shocks6. Small cap premium
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Problems with APT:
• Existence of arbitrage opportunities (single price violation);
• It is difficult to identify a proper set of risk factors (this factors should be uncorrelated and expected returns for all securities should be sensitive to them);
• Singleness of the risk factors selection;
• The applicability of the model to real world;
• The stability of the relationship between expected return and risk factors during a longer period of time;
• The independence between risk factors for a determined period of time;
• Modification in terms of expected return sensitivity to risk factors;
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Final Conclusions:• the most important financial resources are obtained trough international capital markets by issuing bonds or stocks;
• when a company chose to issue securities on international markets it is very important to understand investors behavior;
• investment decision is based on risk and expected return analysis;
• we have different theories in case of risk & return valuation:
• Markovitz: mean for expected return and variance for risk
• CAPM: linear relation between expected return and risk, risk is evaluated based on a specific indicator;
• APT: linear relation between expected return and a lot of risk factors;
• we have an utility function associated to expected returns;
• we have different risk attitudes (aversion, preference, indifference);
• we have arbitrage opportunities when we have different prices and return to the same categories of securities.