International Financial Markets 1. How Capital Markets Work

Lecture Notes: E-Mail: Colloquium:

www.rainer-maurer.de [email protected] Friday 17.15 - 18.45 (room W4.1.03)

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Prof. Dr. Rainer Maurer

1. How Capital Markets Work 1. How Capital Markets Work 1.1. Supply and Demand on Capital Markets 1.1.1. Why People Save 1.1.2. Why People Invest 1.1.3. Investor and Saver Surplus 1.2. Capital Markets and Risk 1.2.1. Why People Don’t Like Risk 1.2.2. How People Handle Risk 1.3. Basic Evaluation Techniques for Capital Markets 1.3.1. The Discounted Cash-Flow Method 1.3.2. The Internal Rate of Return Method 1.3.3. Risk and Return: The Sharpe Ratio 2. Questions for Review Literature:1) ◆ Chapter 4, 25, Mankiw, N.G. (2001): Principles of Economics, Harcourt Coll. Publ., Orlando. ◆ Chapter 7, Mankiw, N.G. (2002): Macroeconomics, Worth Publishers, New York. 1) The

recommended literature typically includes more content than necessary for an understanding of this chapter. Relevant for the examination is the content of this chapter as presented in the lectures. Prof. Dr. Rainer Maurer

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1. How Capital Markets Work 1.1.1. Why People Save 1. How Capital Markets Work 1.1. Supply and Demand on Capital Markets 1.1.1. Why People Save 1.1.2. Why People Invest 1.2. Capital Markets and Risk 1.2.1. Why People Don’t Like Risk 1.2.2. How People Handle Risk 1.3. Basic Evaluation Techniques for Capital Markets 1.3.1. The Discounted Cash-Flow Method 1.3.2. The Internal Rate of Return Method 1.3.3. Risk and Return: The Sharpe Ratio -3-

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1. How Capital Markets Work 1.1.1. Why People Save

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1. How Capital Markets Work 1.1.1. Why People Save ➤ Why do people save? ■ Making savings means ◆ “consumption today” is postponed in favor of ◆ “consumption in the future” ■ Why are people willing to give up “consumption today” in favor of “consumption in the future”? ■ Because they receive interest payments for their savings. ■ The standard assumption is therefore that the willingness to save depends positively on the interest rate:

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The Slope of the Savings Curve % Why do people save more, when they receive higher interest payments?

Savings = S(i)

Higher interest payments allow for “higher consumption in the future”. This compensates for the “lower consumption today”.

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1. How Capital Markets Work 1.1.1. Why People Save ➤ How does this affect consumption of households? ➤ The relationship between “savings today” and “consumption today” is inverse. ➤ The budget constraint of a household shows this. If we neglect the necessity to pay taxes, the simplest form a budget constraint is given by the equation: Income = Savings + Consumption Y = S + C C = Y – S C(i ) = [ Y – S(i ) ] -

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The Slope of the Savings Curve %

As a consequence, people consume less, if the interest rate is high.

Consumption = C(i)

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1. How Capital Markets Work 1.1.1. Why People Save ➤ How does an increase of permanent income affect the savings function? Y = S + C ➤ It must increase savings and/or consumption. ➤ Most likely is that it increases both savings and consumption at the same time, because a permanent increase of income means that higher income will also be available in future periods. So people have no reason to postpone current consumption into the future.

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The Slope of the Savings Curve %

How does an increase of permanent income “y” change the willingness to save?

Savings = S(i, y1) Savings = S(i, y2)

If the permanent income of households y2 > y1 grows, households will typically save more!

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The Slope of the Savings Curve % Savings = S(i, y1) Savings = S(i, y2)

Therefore household permanent income “y” is a shift parameter of the savings function!

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The Slope of the Savings Curve %

Savings = S(i, y2) Savings = S(i, y1)

If the permanent income of households y2 < y1 decreases, households will typically save less!

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1. How Capital Markets Work 1.1.1. Why People Save ➤ How will an increase of future unemployment risk (=“worsening of the economic outlook”) affect a household’s willingness to save today? ➤ How will this affect consumption today? ➤ How will this affect the business cycle? ➤ How will this affect the risk of becoming unemployed?

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1. How Capital Markets Work 1.1.1. Why People Save 1. How Capital Markets Work 1.1. Supply and Demand on Capital Markets 1.1.1. Why People Save 1.1.2. Why People Invest 1.2. Capital Markets and Risk 1.2.1. Why People Don’t Like Risk 1.2.2. How People Handle Risk 1.3. Basic Evaluation Techniques for Capital Markets 1.3.1. The Discounted Cash-Flow Method 1.3.2. The Internal Rate of Return Method 1.3.3. Risk and Return: The Sharpe Ratio

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1. How Capital Markets Work 1.1.2. Why People Invest ➤ Why do people invest? ■ Investment means ◆ to spend money for “economic activities today”, which are assumed to yield a “return in the future” ■ Investment projects can be ranked according to their expected return. ■ This yields the following curve:

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The Slope of the Investment Curve %

Investment volume of the first project

Expected return

Available investment projects depending on their expected return and investment volume

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The Slope of the Investment Curve % Interest rate: 8%

If the market interest rate is 8%, only the first investment project is profitable! All other investment projects are not undertaken!

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The Slope of the Investment Curve %

If the market interest rate is 2%, only the first five investment projects are profitable!

Interest rate: 2%

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%

If we addThe the Slope savings to the curveCurve of available ofcurve the Investment investment projects we recognize, how many investment projects savers are willing to finance: Savings = S(i)

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%

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If we addThe the Slope savings to the curveCurve of available ofcurve the Investment investment projects we recognize, how many investment projects saver are willing to finance: Savings = S(i)

Investor surplus

Equilibrium interest rate

=> An exchange of savings at the resulting equilibrium interest rate is “mutual beneficial”! Saver surplus

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In the The following, Slope ofwe thewillInvestment for simplicity Curve approximate the curve of investment projects with a straight line: % Savings = S(i)

Investment = I(i)

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The Slope ofwe thewillInvestment Curve In the following, for simplicity approximate the curve of investment projects with a straight line: % Savings = S(i)

Investor surplus

Market interest rate

Investment = I(i)

Saver surplus

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The Slope of the Investment Curve %

Contrary to the savings curve, the investment curve depends on the negatively interest rate!

The investment curve is also influenced by shift parameters, e.g. the expected return of investment projects, r1!

Investment = I(i)

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The Slope of the Investment Curve %

Investment = I(i,r2)

If firms expect on average a higher return on investment r1r2 (e.g. because of a lower demand for their goods), they will typically want to invest less.

Investment = I(i,r1) Investment = I(i,r2)

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The Capital Market % S(i,y)

Combination of the savings supply curve and investment demand curve

i1* Equilibrium Interest Rate

I(i)

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S1*

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The Capital Market % y1 < y2

S1(i,y1) S2(i,y2)

i1* i2*

I(i)

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S1* S2*

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The Capital Market % r1 < r2

S(i)

i2* i1* I2(i, r2) I1(i , r1)

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S1* S2*

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1. How Capital Markets Work 1.2.1. Why People Don’t Like Risk 1. How Capital Markets Work 1.1. Supply and Demand on Capital Markets 1.1.1. Why People Save 1.1.2. Why People Invest 1.2. Capital Markets and Risk 1.2.1. Why People Don’t Like Risk 1.2.2. How People Handle Risk 1.3. Basic Evaluation Techniques for Capital Markets 1.3.1. The Discounted Cash-Flow Method 1.3.2. The Internal Rate of Return Method 1.3.3. Risk and Return: The Sharpe Ratio -29-

Prof. Dr. Rainer Maurer

1. How Capital Markets Work 1.2.1. Why People Don’t Like Risk ➤ Do you like risk? ➤ Experiment I: What do you take (a) or (b)?

◆ (a) You receive 3 €. ◆ (b) You receive 3 €. You will get additional 2 € with a

probability of 50% and you will have to pay 3 € with a probability of 50%.

Option (a): Option (b): EV: (0.5*(3+2) + 0.5*(3-3) = 2.5 Prof. Dr. Rainer Maurer

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1. How Capital Markets Work 1.2.1. Why People Don’t Like Risk ➤ Do you like risk? ➤ Experiment II: What do you take (a) or (b)?

◆ (a) You receive 3 €. ◆ (b) You receive 3 €. You will get additional 2 € with a

probability of 50% and you will have to pay 2 € with a probability of 50%.

Option (a): Option (b): EV: (0.5*(3+2) + 0.5*(3-2) = 3 Prof. Dr. Rainer Maurer

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1. How Capital Markets Work 1.2.1. Why People Don’t Like Risk ➤ Do you like risk? ➤ Experiment III: What do you take (a) or (b)?

◆ (a) You receive 3 €. ◆ (b) You receive 3 €. You will get additional 7 € with a

probability of 50% and you will have to pay 1 € with a probability of 50%.

Option (a): Option (b): EV: (0.5*(3+7) + 0.5*(3-1) = 6 Prof. Dr. Rainer Maurer

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1. How Capital Markets Work 1.2.1. Why People Don’t Like Risk ➤ What does the experiment show?

■ Most people prefer a certain payment over a risky payment. ■ A risky payment is accepted only if it includes a premium, which is “high enough”. In economics this premium is called “risk premium”. ■ The magnitude of this “risk premium” individually differs from person to person. ■ However, the existence of a risk premium shows that people generally do not like risk: They are willing to accept risk only, if they are compensated for the risk by a higher payment! ■ In economics we call this “being risk averse”.

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Source: Deutsche Bundesbank

Empirical example for a risk premium:

= Corporate Bonds Prof. Dr. Rainer Maurer

= Government Bonds - 34 -

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Interest Spreads between German Government Bonds and Government Bonds of Other Countries (Maturity 5 years): Interest Spread = Interest Rate of Bond of Country x – Interest Rate of German Bond

Empirical example for a risk premium:

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Empirical example for a risk premium:

Arbitrage relation between credit default swaps and government bonds:

irisky = irisk free + srate + irisk free* srate Prof. Dr. Rainer Maurer

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1. How Capital Markets Work 1.2.1. Why People Don’t Like Risk ➤ Why do people demand a risk premium?

■ Our "self-experiment" and empirical data from financial markets clearly show that people are risk averse and demand risk premiums for risky investments. ■ Now the question is, why do people behave this way? ■ Is it "irrational fear" to be "risk averse" or can we explain it? ■ The next slides show the standard microeconomic explanation for risk averse behavior. ◆ Standard microeconomics derives the explanation from a quite plausible property of the utility function of people: Decreasing marginal utility of consumption. ◆ The next slide gives an explanation of this property:

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1. How Capital Markets Work 1.2.1. Why People Don’t Like Risk Utility from the Consumption of Cookies per Day Utility = U(Cookies)

…and so on Utility of the 2nd Cookie

Utility from the Consumption of 16 Cookies = 12 “Utils”

Utility of the 1st Cookie

Quantity of Cookies (kg) Prof. Dr. Rainer Maurer

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12*50% + 4*50% = 8

1. How Capital Markets Work 1.2.1. Why People Don’t Like Risk Experiment II: (a) You receive 3 € with a probability of 100%. (b) You receive 3 €. You will get additional 2 € Utility Units with a probability of 50% and you will have to pay 2 € with a probability of 50%. Expected utility from the uncertain payment is lower than the expected utility from the certain payment => A person with this utility function will prefer the certain payment!

Value of Consumption Goods (€)

(a) 9,5 Expected Utility Units for the Certain Payment (b) 8 Expected Utility Units for the Uncertain Payment

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1. How Capital Markets Work 1.2.1. Why People Don’t Like Risk

Utility Gain: 2

Experiment II: (a) You receive 3 € with a probability of 100%. (b) You receive 3 €. You will get additional 2 € Utility Units with a probability of 50% and you will have to pay 2 € with a probability of 50%. The reason for the lower expected utility is the stronger change of utility in case of a loss compared to the case of a gain, because of decreasing marginal utility!

Utility Loss: 5,5

Income Income Loss of 2 € Gain of 2 € (a) 9,5 Expected Utility Units for the Certain Payment (b) 8 Expected Utility Units for the Uncertain Payment Prof. Dr. Rainer Maurer

Value of Consumption Goods (€)

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1. How Capital Markets Work 1.2.1. Why People Don’t Like Risk Experiment II: (a) You receive 3 € with a probability of 100%. (b) You receive 3 €. You will get additional 2 € Utility Units with a probability of 50% and you will have to pay 2 € with a probability of 50%. In case of a utility function with constant marginal utility, the utility gain in case of an income gain would be equal to the utility loss in case of an income loss and hence expected utility in case of a certain payment would be equal to expected utility of an uncertain payment!

Utility Gain: 4 Utility Loss: 4

Income Income Loss of 2 € Gain of 2 €

Value of Consumption Goods (€)

(a) 6 Expected Utility Units for the Certain Payment (b) 6 Expected Utility Units for the Uncertain Payment Prof. Dr. Rainer Maurer

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1. How Capital Markets Work 1.2.1. Why People Don’t Like Risk ➤ Consequently,

■ decreasing marginal utility causes the utility loss in case of an income loss of 2 € to be larger than a utility gain in case of an income gain of 2 €.

➤ Therefore,

■ in case of decreasing marginal utility, expected utility of an uncertain payment is always lower than expected utility of a certain payment.

➤ Therefore,

■ in case of an uncertain payment, the income gain must always be larger than the income loss, in order to make people accept the uncertain payment. = A risk premium must be paid!

➤ Obviously,

■ this is the case in our “Experiment III” as the following shows:

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15*50% + 7*50% = 11

1. How Capital Markets Work 1.2.1. Why People Don’t Like Risk Experiment III: (a) You receive 3 € with a probability of 100%. (b) You receive 3 €. You will get additional 7 € with a probability of 50% and you will have Utility Units to pay 1 € with a probability of 50%. Utility Gain: 5,5 Utility Loss: 2,5

Expected utility from this uncertain payment is higher than the expected utility from the certain payment => A person with this utility function will prefer the uncertain payment!

Income Income Gain of 7 € Loss of 1 € (a) 9,5 Expected Utility Units for the Certain Payment (b) 11 Expected Utility Units for the Uncertain Payment

Value of Consumption Goods (€)

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1. How Capital Markets Work 1.2.1. Why People Don’t Like Risk

Utility Units

Risk Neutral Utility Function

Normal Person (Risk Averse) Utility Function

Gambler (Risk Lover) Utility Function

Value of Consumption Goods (€)

Since we know from experiments that most people are risk averse, we can draw the conclusion that most people have a utility function with decreasing marginal utility! -44-

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1. How Capital Markets Work 1.2.1. Why People Don’t Like Risk Risk aversion is not specific for human behavior. Experiments with plants show that plants also show risk averse behavior:

Dener,Kacelnik,Shemesh (2016), Pea Plants Show Risk Sensitivity, Current Biology, Volume 26, Issue 13, p. 1763–1767, 11 July 2016

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1. How Capital Markets Work 1.2.2. How People Handle Risk 1. How Capital Markets Work 1.1. Supply and Demand on Capital Markets 1.1.1. Why People Save 1.1.2. Why People Invest 1.2. Capital Markets and Risk 1.2.1. Why People Don’t Like Risk 1.2.2. How People Handle Risk 1.3. Basic Evaluation Techniques for Capital Markets 1.3.1. The Discounted Cash-Flow Method 1.3.2. The Internal Rate of Return Method 1.3.3. Risk and Return: The Sharpe Ratio Prof. Dr. Rainer Maurer

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1. How Capital Markets Work 1.2.2. How People Handle Risk ➤ We have already seen, how normal people handle risk: ■ They demand a risk premium!

➤ Financial markets offer a possibility to eliminate risk: ■ Hedging!

➤ The following tables illustrate the principle of hedging based on several numeric examples:

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1. How Capital Markets Work 1.2.2. How People Handle Risk

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1. How Capital Markets Work 1.2.2. How People Handle Risk

➤ This example shows: ■ If the return of one stock goes up exactly when the return of the other stock goes down, a portfolio of both stocks completely eliminates the risk! ■ Consequently, investing your money in a portfolio of both stocks implies no risk, while investing your money in one of both stocks only implies a lot of risk! ■ Note: In case of a perfect hedge, the correlation coefficient equals exactly -1! -49-

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1. How Capital Markets Work 1.2.2. How People Handle Risk

➤ This example shows: ■ If the return of one stock goes up exactly when the return of the other stock goes up, a portfolio of both stocks does not affect risk at all! ■ Consequently, investing your money in a portfolio of both stocks implies the same risk, as investing your money in one of both stocks only! ■ Note: In case of a no hedge, the correlation coefficient equals exactly 1! Prof. Dr. Rainer Maurer

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1. How Capital Markets Work 1.2.2. How People Handle Risk

The Miracle of Hedging!

➤ This example shows:

■ If the return of one stock goes up when the return of the other stock goes up, but not by exactly the same degree, a portfolio of both stocks can reduce risk somewhat but not completely eliminate it! ■ Consequently, investing your money in a portfolio of both stocks implies a lower risk, as investing your money in one of both stocks only! ■ Note: In case of a normal hedge, the correlation coefficient lies between 0 and 1! -51-

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1. How Capital Markets Work 1.2.2. How People Handle Risk ➤ In the real world, perfect hedges are as rare as no hedges! ➤ Fortunately, normal (imperfect) hedges are the rule, so that investing in portfolios generally makes more sense than investing in single stock! ➤ Why are stocks so often imperfect hedges? ■ On one hand, there are a lot of common economic factors that effect all stocks in the same way, causing a positive correlation of returns: ◆ The business cycle, prices of raw materials, wages, tax reforms… ■ To the other hand, every firm has its own markets segments and these segments often react in a different way to these common economic factors: ◆ For example, a discounter like Aldi profits from high consumer confidence as well as a luxury store like KaDeWe, but suffers less from a recession, because of its cheaper range of goods... Prof. Dr. Rainer Maurer

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1. How Capital Markets Work 1.2.2. How People Handle Risk ➤ As the examples have shown, we can comfortably measure the hedge quality of two kind of stocks by the correlation coefficient. ➤ How do we compute the correlation coefficient?

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1. How Capital Markets Work 1.2.2. How People Handle Risk ➤ How do we compute the variance?

➤ How do we compute the covariance?

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1. How Capital Markets Work 1.2.2. How People Handle Risk ➤ Interpretation of the Correlation Coefficient:

➤ A correlation coefficient of -1 indicates that the value of two stocks moves through time with exactly opposite fluctuations: ■ If the stock of Raincoat Corp. displays a positive deviation from its mean value, the stock of Sunglasses International displays a negative deviation form its mean value. ■ If the stock of Raincoat Corp. displays a negative deviation from its mean value, the stock of Sunglasses International displays a positive deviation form its mean value. -55-

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1. How Capital Markets Work 1.2.2. How People Handle Risk ➤ Interpretation of the Correlation Coefficient:

➤ A correlation coefficient of 1 indicates that the value of two stocks moves through time with exactly the same fluctuations: ■ If the stock of Raincoat Corp. displays a positive deviation from its mean value, the stock of Umbrella Unlimited displays a positive deviation form its mean value too. ■ If the stock of Raincoat Corp. displays a negative deviation from its mean value, the stock of Umbrella Unlimited displays a negative deviation form its mean value too. Prof. Dr. Rainer Maurer

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1. How Capital Markets Work 1.2.2. How People Handle Risk ➤ Interpretation of the Correlation Coefficient:

➤ A correlation coefficient between 0 and 1 indicates that the value of two stocks moves through time with rather similar but not exactly the same fluctuations: ■ If the stock of Sunglasses Int. displays a positive deviation from its mean value, the stock of United Steel displays most of the time a positive deviation form its mean value too – but not always. ■ If the stock of Sunglasses Int. displays a negative deviation from its mean value, the stock of United Steel International displays most of the time a negative deviation form its mean value too – but not always. -57-

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1. How Capital Markets Work 1.2.2. How People Handle Risk ➤ Now it’s up to you: ■ As portfolio manager you have to decide in the following cases, whether to invest in single stock or in a portfolio. ■ What do you recommend?

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1. How Capital Markets Work 1.2.2. How People Handle Risk

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1. How Capital Markets Work 1.2.2. How People Handle Risk ➤ The Market-Beta:

■ As already seen, it is almost impossible to find perfectly negatively correlated stocks – so that all portfolios end up with a risk that cannot be eliminated – the so called “market risk”. ■ The market risk is measured by the variance of the average return of the market portfolio, i,e. the risk that cannot be eliminated by investing in the market portfolio. ■ The tendency of the return of a stock to move with the average return of the market portfolio is called its market-beta. ■ The market-beta is a measure of the relative volatility of a stock return compared to the return of the total stock market (= a portfolio consisting of all stocks minus the one stock whose beta is measured) as a whole.

◆ A beta of 1 means that a stock return moves exactly as the market does. ◆ A beta of 2 means that if a stock market return moves up by 10 %, the stock return moves up by 20 %.

=> Adding a high (low) beta-stock to a portfolio means increasing (reducing) the portfolio risk.

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1. How Capital Markets Work 1.2.2. How People Handle Risk ➤ How do we compute the market-beta?

➤ As we will see in chapter 2, the market-beta plays a central role for the evaluation model of stocks – the so called CAPModel (Capital Asset Pricing Model). ➤ The following graph visualizes its interpretation.

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1. How Capital Markets Work 1.2.2. How People Handle Risk β=1 => Stock A is as volatile as the market.

Return of Stock A

=> Adding Stock A to the market portfolio does not change portfolio risk

β=1

“Average β-Stock“

Return of the Market Portfolio

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1. How Capital Markets Work 1.2.2. How People Handle Risk β=0,5 => Stock A is less volatile than the market.

Return of Stock A

“Low β-Stock“ β=0,5

=> Adding Stock A to the market portfolio does reduce portfolio risk Return of the Market Portfolio

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1. How Capital Markets Work 1.2.2. How People Handle Risk β=1,5 => Stock A is more volatile than the market. => Adding Stock A to the market portfolio does increase portfolio risk

Return of Stock A

β=1,5

“High β-Stock“

Return of the Market Portfolio

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1. How Capital Markets Work 1.2.2. How People Handle Risk

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1. How Capital Markets Work 1.2.2. How People Handle Risk

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1. How Capital Markets Work 1.2.2. How People Handle Risk ➤ To sum up: ■ If two stocks display a perfect negative correlation of returns, it is possible to completely eliminate their return risks by holding a portfolio of both stocks instead of holding only one stock. ■ To measure the correlation of returns, we use the “correlation coefficient”. ■ The miracle of hedging: It is possible to reduce risks by holding a portfolio of stocks, even if the correlation of all stocks is positive but not perfect (i.e. if the correlation coefficient lies between 0 and 1). ■ The correlation of the return of a stock with the market portfolio (=the portfolio consisting of all stocks dealt with in the market) is called “market-beta” of a stock. -69-

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1. How Capital Markets Work 1.3.1. The Discounted Cash-Flow Method 1. How Capital Markets Work 1.1. Supply and Demand on Capital Markets 1.1.1. Why People Save 1.1.2. Why People Invest 1.2. Capital Markets and Risk 1.2.1. Why People Don’t Like Risk 1.2.2. How People Handle Risk 1.3. Basic Evaluation Techniques for Capital Markets 1.3.1. The Discounted Cash-Flow Method 1.3.2. The Internal Rate of Return Method 1.3.3. Risk and Return: The Sharpe Ratio Prof. Dr. Rainer Maurer

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1. How Capital Markets Work

1.3.1. The Discounted Cash-Flow Method ➤ As the next chapter will show, very different kind of assets are traded on capital markets. ➤ Two technical procedures are important for the evaluation of these different assets: ■

Discounted Cash-Flow Method

■

Internal Rate of Return Method

➤ Before we apply these procedures to the various types of assets in the next chapter, we will analyze them in some detail in the following: ➤ We start with the discounted cash-flow method: -71-

Prof. Dr. Rainer Maurer

1. How Capital Markets Work

1.3.1. The Discounted Cash-Flow Method ➤ What do you prefer: 1 € today or 1 € in one year? ➤ The basic idea of the discounted cash-flow method is: ■

Determining the present value of a flow of future payments – either from an investment project or a financial market asset.

■

Technically, payments of different points in time are made comparable by evaluating each payment with a time specific discount factor and adding up these comparable payments to the “present value” of the payment flow.

➤ The following examples show, how this works:

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1. How Capital Markets Work 1.3.1. The Discounted Cash-Flow Method ➤ Discounting a payment flow:

➤

The present value of these four payments is determined by discounting each single payment (positive or negative) with the discount factor for the corresponding year.

➤

Example: The discount factor for a payment three years from now for a discount rate of 5% equals: (1,05)^(-3) = 0,86 Hence the present value of a payment three years form now is worth 86% of its future value.

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1. How Capital Markets Work 1.3.1. The Discounted Cash-Flow Method ➤

Discounting a payment flow:

➤

This payment flow with equal annual payments of 100 per year clearly shows that payments further in the future a more discounted than payments closer to the present.

➤

A comparison with the following table shows that a lower discount rate increases the present value:

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1. How Capital Markets Work 1.3.1. The Discounted Cash-Flow Method ➤ Problem of the discounting approach: ■

An appropriate discount rate has to be chosen!

➤ The appropriate discount rate should reflect the risk related to a payment flow. ■

Uncertain payment flows, whose payments are based on estimated forecasts only, should be discounted with a higher discount rate than secure payment flows.

■

Hence, the discount rate should include an appropriate risk premium.

➤ How can this be done?

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1. How Capital Markets Work

1.3.1. The Discounted Cash-Flow Method ➤ How to find the appropriate discount rate? ■

In many cases it is possible to find a market interest rate for a payment flow of the same risk class: ◆ If the payment flow is nearly certain, the market interest rate of a fixed rate security with the same risk structure, for example a bond of a government with high creditworthiness, should be chosen as discount rate.

■

However, for many uncertain payment flows, it is difficult to find a market interest rate of the same risk class.

◆ If the payment flow is uncertain and based on a forecast (for

example the dividend payments from a stock company) a market interest rates for exactly the same risk class are hard to find.

◆ In this case, concepts like the Capital Asset Pricing Model (CAPM) can to be employed to calculate an appropriate discount rate. Chapter 2 will show, how this works.

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1. How Capital Markets Work 1.3.1. The Discounted Cash-Flow Method Details If the payment flow consists of equal payments (also called “annuities”) we can use the annuity present value formula:

If the flow of equal payments holds on for all eternity (≈200 years) this formula simplifies to:

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1. How Capital Markets Work 1.3.2. The Internal Rate of Return Method 1. How Capital Markets Work 1.1. Supply and Demand on Capital Markets 1.1.1. Why People Save 1.1.2. Why People Invest 1.2. Capital Markets and Risk 1.2.1. Why People Don’t Like Risk 1.2.2. How People Handle Risk 1.3. Basic Evaluation Techniques for Capital Markets 1.3.1. The Discounted Cash-Flow Method 1.3.2. The Internal Rate of Return Method 1.3.3. Risk and Return: The Sharpe Ratio

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1. How Capital Markets Work 1.3.2. The Internal Rate of Return Method ➤ The basic idea of the internal rate of return method is: ■

Very often one knows the market price of an asset or investment project and its payment flow.

■

In such cases it is possible to determine the internal rate of return of the asset or investment project.

■

Technically, this amounts to finding a discount rate that equals the present value of the payment flow with the market price of the asset.

➤ The following examples shows, how this works:

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1. How Capital Markets Work 1.3.2. The Internal Rate of Return Method t+1

➤ If the payment flow consists of one period only, it is easy to calculate the IRR (=internal rate of return) by hand:

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1. How Capital Markets Work 1.3.2. The Internal Rate of Return Method

➤

If the payment flow consists of more than one period, for example 5 periods, calculating the IRR implies solving a polynomial of degree 5. This involves the following problem: ■

Formulas for analytical solutions exist only for polynomials lower degree 4.

➤ Therefore, numerical solutions methods must be applied. ➤ One such method is available for Excel (the IKV() Function). -81-

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1. How Capital Markets Work 1.3.2. The Internal Rate of Return Method Details Calculating the IRR with the numeric iteration procedure of Excel: An often used tool to calculate the yield to maturity is the Excel-formula “IKV(rage of cells containing the flow of payments; cell containing the estimated value)”. This formula calculates the “internal rate of return” (IRR) of a flow of payments (in the order of appearance; at least one negative payment (=investment=purchase of bond=market value of bond) is necessary) based on a numerical iteration procedure. It starts the iteration process with a value for “i” you can specify as “estimated value” (default value is 10%) and changes this value until it reaches a solution value for the exact formula with an accuracy of 0,00001 percent. If it does not find such a solution after 20 calculation steps it delivers the error message “#value!”. For an example see my Excel-file “Computing yield to maturity”, 2nd sheet.

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1. How Capital Markets Work 1.3.2. The Internal Rate of Return Method ➤ Consequently, calculating the IRR typically implies the usage of a computer. ➤ If the IRRs of several payment flows are calculated, it is in principle possible to select the payment flow with the highest IRR as the most profitable one. ➤ However, one has to take care of the risk implied by each payment flow! ➤Since an uncertain payment flow is riskier than a certain payment flow, the uncertain flow must offer a risk premium (for people with normal utility function…) ➤One often used measure, which takes care of both return and risk, is the so called Sharpe Ratio, which was proposed by William F. Sharpe (1966). Prof. Dr. Rainer Maurer

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1. How Capital Markets Work 1.3.3. Risk and Return: The Sharpe Ratio 1. How Capital Markets Work 1.1. Supply and Demand on Capital Markets 1.1.1. Why People Save 1.1.2. Why People Invest 1.2. Capital Markets and Risk 1.2.1. Why People Don’t Like Risk 1.2.2. How People Handle Risk 1.3. Basic Evaluation Techniques for Capital Markets 1.3.1. The Discounted Cash-Flow Method 1.3.2. The Internal Rate of Return Method 1.3.3. Risk and Return: The Sharpe Ratio

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1. How Capital Markets Work 1.3.3. Risk and Return: The Sharpe Ratio ➤ The basic idea of the Sharpe Ratio is to determine the risk premium paid per unit of risk. ➤ Therefore, the formula of the Sharpe Ratio relates the risk premium of the internal rate of return of a specific investment j (= the difference between this expected return, E(ij), and the return of a risk free market interest rate, ro, e.g. a government bond of high creditworthiness), E(ij)- ro, to the standard deviation of the return of the specific investment (= (variance(ij))^0,5 ):

➤ The higher the Sharpe Ratio, the higher is the risk premium per unit risk and hence the more attractive is the investment. -85-

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1. How Capital Markets Work 1.3.3. Risk and Return: The Sharpe Ratio ➤ Which asset A or B offers the best relation between risk and return? Calculate the Sharp Ratio for a risk free return of 2 %.

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1. How Capital Markets Work 1.3.3. Risk and Return: The Sharpe Ratio ➤ Problem with the Sharpe Ratio: ■

The standard deviation of an investment, i.e. the measure for its risk, is typically not known, but has to be estimated based on forecasted future returns. In our above example we simply assumed to know them!

■

For financial market assets, e.g. the return of stocks, such kind of forecasts are typically highly inaccurate.

■

Therefore, very often, the standard deviation of a stock is estimated based on its past returns.

■

Even though such a calculation is easily done and seems to be highly “accurate and plausible”, one has to be aware that the application of such “historic” standard deviations for future investment decisions, implies the “hidden assumption” that the “future” will similar to the “past”.

■

For many kind of stocks, this assumption has proven wrong!

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1. How Capital Markets Work 1.2.2. How People Handle Risk ➤ To sum up: ■ The discounted cash-flow and the internal rate of return method are methods to calculate the profitability of different kind of investments. ■ The Sharpe-Ratio is a measure for the relation of return and risk of an investment. ■ All these measures rely very often to a high degree on estimations and forecasts. ■ In the next chapter we will apply these methods to the standard assets of the major capital market segments.

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Chapter 1: Questions

You should be able to answer the following questions at the end of this chapter. If you have difficulties in answering a question, discuss this question with me during or at the end of the next lecture or attend my colloquium.

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Chapter 1: Questions for Review 1.

Why can “saving” increase personal utility? Give a graphical and verbal explanation. 2. How does saving behavior affects consumption behavior? 3. If saving increases utility, why do savers demand interest? 4. Why depends the willingness to save positively on the interest rate? 5. What is a “production function”? 6. Explain the motive for investment. 7. Why depends the willingness to invest negatively on the interest rate? 9. What insures the equality of saving and investment in a market equilibrium? Explain your answer based on the a diagram of the capital market. 10. How does an increase in savings supply affect the interest rate?

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Chapter 1: Questions for Review 11. How does an increase in investment demand affect the interest rate? 12. Explain the meaning of “risk averse” and “risk neutral”. 13. Why are most people risk averse? Give a graphical and verbal explanation. 14. What is a “risk premium”? 15. Why do people demand a risk premium? 16. What is “hedging”? 17. What property must two stocks have to be a “perfect hedge”? 18. Is it possible to hedge risk with two stocks, if there return is positively correlated? 19. Why are perfect hedges rare? 20. What is a “normal hedge”? 21. How do you explain that so many stocks are “normal hedges”?

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Chapter 1: Questions for Review 22. Calculate the variances and the correlation coefficient of the following stocks. Can they be used to hedge each other?

23. Calculate the variances and the correlation coefficient of the following stocks. Can they be used to hedge each other?

24. What is the “market-beta”? 25. What is meant by “low beta stock” and “high beta stock”? Prof. Dr. Rainer Maurer

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Chapter 1: Questions for Review 26. Give a verbal explanation of the discounted cash flow and the internal rate of return method.

27. 28. 29. 30.

Which criteria should an appropriate discount rate fulfill? What is the definition of the Sharpe Ratio? Give a verbal interpretation of the Sharpe Ratio. What are the two major criteria for an investment decision?

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Chapter 1: Questions for Review 31. What asset is the most attractive? Base your decision on the Sharpe Ratio.

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Portfolio Theory

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Lecture Notes: E-Mail: Colloquium:

www.rainer-maurer.de [email protected] Friday 17.15 - 18.45 (room W4.1.03)

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Prof. Dr. Rainer Maurer

1. How Capital Markets Work 1. How Capital Markets Work 1.1. Supply and Demand on Capital Markets 1.1.1. Why People Save 1.1.2. Why People Invest 1.1.3. Investor and Saver Surplus 1.2. Capital Markets and Risk 1.2.1. Why People Don’t Like Risk 1.2.2. How People Handle Risk 1.3. Basic Evaluation Techniques for Capital Markets 1.3.1. The Discounted Cash-Flow Method 1.3.2. The Internal Rate of Return Method 1.3.3. Risk and Return: The Sharpe Ratio 2. Questions for Review Literature:1) ◆ Chapter 4, 25, Mankiw, N.G. (2001): Principles of Economics, Harcourt Coll. Publ., Orlando. ◆ Chapter 7, Mankiw, N.G. (2002): Macroeconomics, Worth Publishers, New York. 1) The

recommended literature typically includes more content than necessary for an understanding of this chapter. Relevant for the examination is the content of this chapter as presented in the lectures. Prof. Dr. Rainer Maurer

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1. How Capital Markets Work 1.1.1. Why People Save 1. How Capital Markets Work 1.1. Supply and Demand on Capital Markets 1.1.1. Why People Save 1.1.2. Why People Invest 1.2. Capital Markets and Risk 1.2.1. Why People Don’t Like Risk 1.2.2. How People Handle Risk 1.3. Basic Evaluation Techniques for Capital Markets 1.3.1. The Discounted Cash-Flow Method 1.3.2. The Internal Rate of Return Method 1.3.3. Risk and Return: The Sharpe Ratio -3-

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1. How Capital Markets Work 1.1.1. Why People Save

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1. How Capital Markets Work 1.1.1. Why People Save ➤ Why do people save? ■ Making savings means ◆ “consumption today” is postponed in favor of ◆ “consumption in the future” ■ Why are people willing to give up “consumption today” in favor of “consumption in the future”? ■ Because they receive interest payments for their savings. ■ The standard assumption is therefore that the willingness to save depends positively on the interest rate:

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Prof. Dr. Rainer Maurer

The Slope of the Savings Curve % Why do people save more, when they receive higher interest payments?

Savings = S(i)

Higher interest payments allow for “higher consumption in the future”. This compensates for the “lower consumption today”.

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1. How Capital Markets Work 1.1.1. Why People Save ➤ How does this affect consumption of households? ➤ The relationship between “savings today” and “consumption today” is inverse. ➤ The budget constraint of a household shows this. If we neglect the necessity to pay taxes, the simplest form a budget constraint is given by the equation: Income = Savings + Consumption Y = S + C C = Y – S C(i ) = [ Y – S(i ) ] -

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The Slope of the Savings Curve %

As a consequence, people consume less, if the interest rate is high.

Consumption = C(i)

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1. How Capital Markets Work 1.1.1. Why People Save ➤ How does an increase of permanent income affect the savings function? Y = S + C ➤ It must increase savings and/or consumption. ➤ Most likely is that it increases both savings and consumption at the same time, because a permanent increase of income means that higher income will also be available in future periods. So people have no reason to postpone current consumption into the future.

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Prof. Dr. Rainer Maurer

The Slope of the Savings Curve %

How does an increase of permanent income “y” change the willingness to save?

Savings = S(i, y1) Savings = S(i, y2)

If the permanent income of households y2 > y1 grows, households will typically save more!

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The Slope of the Savings Curve % Savings = S(i, y1) Savings = S(i, y2)

Therefore household permanent income “y” is a shift parameter of the savings function!

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The Slope of the Savings Curve %

Savings = S(i, y2) Savings = S(i, y1)

If the permanent income of households y2 < y1 decreases, households will typically save less!

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1. How Capital Markets Work 1.1.1. Why People Save ➤ How will an increase of future unemployment risk (=“worsening of the economic outlook”) affect a household’s willingness to save today? ➤ How will this affect consumption today? ➤ How will this affect the business cycle? ➤ How will this affect the risk of becoming unemployed?

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Prof. Dr. Rainer Maurer

1. How Capital Markets Work 1.1.1. Why People Save 1. How Capital Markets Work 1.1. Supply and Demand on Capital Markets 1.1.1. Why People Save 1.1.2. Why People Invest 1.2. Capital Markets and Risk 1.2.1. Why People Don’t Like Risk 1.2.2. How People Handle Risk 1.3. Basic Evaluation Techniques for Capital Markets 1.3.1. The Discounted Cash-Flow Method 1.3.2. The Internal Rate of Return Method 1.3.3. Risk and Return: The Sharpe Ratio

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1. How Capital Markets Work 1.1.2. Why People Invest ➤ Why do people invest? ■ Investment means ◆ to spend money for “economic activities today”, which are assumed to yield a “return in the future” ■ Investment projects can be ranked according to their expected return. ■ This yields the following curve:

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The Slope of the Investment Curve %

Investment volume of the first project

Expected return

Available investment projects depending on their expected return and investment volume

€ Prof. Dr. Rainer Maurer

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The Slope of the Investment Curve % Interest rate: 8%

If the market interest rate is 8%, only the first investment project is profitable! All other investment projects are not undertaken!

€ Prof. Dr. Rainer Maurer

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The Slope of the Investment Curve %

If the market interest rate is 2%, only the first five investment projects are profitable!

Interest rate: 2%

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%

If we addThe the Slope savings to the curveCurve of available ofcurve the Investment investment projects we recognize, how many investment projects savers are willing to finance: Savings = S(i)

€ Prof. Dr. Rainer Maurer

%

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If we addThe the Slope savings to the curveCurve of available ofcurve the Investment investment projects we recognize, how many investment projects saver are willing to finance: Savings = S(i)

Investor surplus

Equilibrium interest rate

=> An exchange of savings at the resulting equilibrium interest rate is “mutual beneficial”! Saver surplus

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In the The following, Slope ofwe thewillInvestment for simplicity Curve approximate the curve of investment projects with a straight line: % Savings = S(i)

Investment = I(i)

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The Slope ofwe thewillInvestment Curve In the following, for simplicity approximate the curve of investment projects with a straight line: % Savings = S(i)

Investor surplus

Market interest rate

Investment = I(i)

Saver surplus

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The Slope of the Investment Curve %

Contrary to the savings curve, the investment curve depends on the negatively interest rate!

The investment curve is also influenced by shift parameters, e.g. the expected return of investment projects, r1!

Investment = I(i)

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The Slope of the Investment Curve %

Investment = I(i,r2)

If firms expect on average a higher return on investment r1r2 (e.g. because of a lower demand for their goods), they will typically want to invest less.

Investment = I(i,r1) Investment = I(i,r2)

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The Capital Market % S(i,y)

Combination of the savings supply curve and investment demand curve

i1* Equilibrium Interest Rate

I(i)

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S1*

€

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The Capital Market % y1 < y2

S1(i,y1) S2(i,y2)

i1* i2*

I(i)

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€

S1* S2*

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The Capital Market % r1 < r2

S(i)

i2* i1* I2(i, r2) I1(i , r1)

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S1* S2*

€

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1. How Capital Markets Work 1.2.1. Why People Don’t Like Risk 1. How Capital Markets Work 1.1. Supply and Demand on Capital Markets 1.1.1. Why People Save 1.1.2. Why People Invest 1.2. Capital Markets and Risk 1.2.1. Why People Don’t Like Risk 1.2.2. How People Handle Risk 1.3. Basic Evaluation Techniques for Capital Markets 1.3.1. The Discounted Cash-Flow Method 1.3.2. The Internal Rate of Return Method 1.3.3. Risk and Return: The Sharpe Ratio -29-

Prof. Dr. Rainer Maurer

1. How Capital Markets Work 1.2.1. Why People Don’t Like Risk ➤ Do you like risk? ➤ Experiment I: What do you take (a) or (b)?

◆ (a) You receive 3 €. ◆ (b) You receive 3 €. You will get additional 2 € with a

probability of 50% and you will have to pay 3 € with a probability of 50%.

Option (a): Option (b): EV: (0.5*(3+2) + 0.5*(3-3) = 2.5 Prof. Dr. Rainer Maurer

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1. How Capital Markets Work 1.2.1. Why People Don’t Like Risk ➤ Do you like risk? ➤ Experiment II: What do you take (a) or (b)?

◆ (a) You receive 3 €. ◆ (b) You receive 3 €. You will get additional 2 € with a

probability of 50% and you will have to pay 2 € with a probability of 50%.

Option (a): Option (b): EV: (0.5*(3+2) + 0.5*(3-2) = 3 Prof. Dr. Rainer Maurer

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1. How Capital Markets Work 1.2.1. Why People Don’t Like Risk ➤ Do you like risk? ➤ Experiment III: What do you take (a) or (b)?

◆ (a) You receive 3 €. ◆ (b) You receive 3 €. You will get additional 7 € with a

probability of 50% and you will have to pay 1 € with a probability of 50%.

Option (a): Option (b): EV: (0.5*(3+7) + 0.5*(3-1) = 6 Prof. Dr. Rainer Maurer

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1. How Capital Markets Work 1.2.1. Why People Don’t Like Risk ➤ What does the experiment show?

■ Most people prefer a certain payment over a risky payment. ■ A risky payment is accepted only if it includes a premium, which is “high enough”. In economics this premium is called “risk premium”. ■ The magnitude of this “risk premium” individually differs from person to person. ■ However, the existence of a risk premium shows that people generally do not like risk: They are willing to accept risk only, if they are compensated for the risk by a higher payment! ■ In economics we call this “being risk averse”.

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Source: Deutsche Bundesbank

Empirical example for a risk premium:

= Corporate Bonds Prof. Dr. Rainer Maurer

= Government Bonds - 34 -

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Interest Spreads between German Government Bonds and Government Bonds of Other Countries (Maturity 5 years): Interest Spread = Interest Rate of Bond of Country x – Interest Rate of German Bond

Empirical example for a risk premium:

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Prof. Dr. Rainer Maurer

Empirical example for a risk premium:

Arbitrage relation between credit default swaps and government bonds:

irisky = irisk free + srate + irisk free* srate Prof. Dr. Rainer Maurer

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1. How Capital Markets Work 1.2.1. Why People Don’t Like Risk ➤ Why do people demand a risk premium?

■ Our "self-experiment" and empirical data from financial markets clearly show that people are risk averse and demand risk premiums for risky investments. ■ Now the question is, why do people behave this way? ■ Is it "irrational fear" to be "risk averse" or can we explain it? ■ The next slides show the standard microeconomic explanation for risk averse behavior. ◆ Standard microeconomics derives the explanation from a quite plausible property of the utility function of people: Decreasing marginal utility of consumption. ◆ The next slide gives an explanation of this property:

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Prof. Dr. Rainer Maurer

1. How Capital Markets Work 1.2.1. Why People Don’t Like Risk Utility from the Consumption of Cookies per Day Utility = U(Cookies)

…and so on Utility of the 2nd Cookie

Utility from the Consumption of 16 Cookies = 12 “Utils”

Utility of the 1st Cookie

Quantity of Cookies (kg) Prof. Dr. Rainer Maurer

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12*50% + 4*50% = 8

1. How Capital Markets Work 1.2.1. Why People Don’t Like Risk Experiment II: (a) You receive 3 € with a probability of 100%. (b) You receive 3 €. You will get additional 2 € Utility Units with a probability of 50% and you will have to pay 2 € with a probability of 50%. Expected utility from the uncertain payment is lower than the expected utility from the certain payment => A person with this utility function will prefer the certain payment!

Value of Consumption Goods (€)

(a) 9,5 Expected Utility Units for the Certain Payment (b) 8 Expected Utility Units for the Uncertain Payment

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Prof. Dr. Rainer Maurer

1. How Capital Markets Work 1.2.1. Why People Don’t Like Risk

Utility Gain: 2

Experiment II: (a) You receive 3 € with a probability of 100%. (b) You receive 3 €. You will get additional 2 € Utility Units with a probability of 50% and you will have to pay 2 € with a probability of 50%. The reason for the lower expected utility is the stronger change of utility in case of a loss compared to the case of a gain, because of decreasing marginal utility!

Utility Loss: 5,5

Income Income Loss of 2 € Gain of 2 € (a) 9,5 Expected Utility Units for the Certain Payment (b) 8 Expected Utility Units for the Uncertain Payment Prof. Dr. Rainer Maurer

Value of Consumption Goods (€)

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1. How Capital Markets Work 1.2.1. Why People Don’t Like Risk Experiment II: (a) You receive 3 € with a probability of 100%. (b) You receive 3 €. You will get additional 2 € Utility Units with a probability of 50% and you will have to pay 2 € with a probability of 50%. In case of a utility function with constant marginal utility, the utility gain in case of an income gain would be equal to the utility loss in case of an income loss and hence expected utility in case of a certain payment would be equal to expected utility of an uncertain payment!

Utility Gain: 4 Utility Loss: 4

Income Income Loss of 2 € Gain of 2 €

Value of Consumption Goods (€)

(a) 6 Expected Utility Units for the Certain Payment (b) 6 Expected Utility Units for the Uncertain Payment Prof. Dr. Rainer Maurer

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1. How Capital Markets Work 1.2.1. Why People Don’t Like Risk ➤ Consequently,

■ decreasing marginal utility causes the utility loss in case of an income loss of 2 € to be larger than a utility gain in case of an income gain of 2 €.

➤ Therefore,

■ in case of decreasing marginal utility, expected utility of an uncertain payment is always lower than expected utility of a certain payment.

➤ Therefore,

■ in case of an uncertain payment, the income gain must always be larger than the income loss, in order to make people accept the uncertain payment. = A risk premium must be paid!

➤ Obviously,

■ this is the case in our “Experiment III” as the following shows:

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15*50% + 7*50% = 11

1. How Capital Markets Work 1.2.1. Why People Don’t Like Risk Experiment III: (a) You receive 3 € with a probability of 100%. (b) You receive 3 €. You will get additional 7 € with a probability of 50% and you will have Utility Units to pay 1 € with a probability of 50%. Utility Gain: 5,5 Utility Loss: 2,5

Expected utility from this uncertain payment is higher than the expected utility from the certain payment => A person with this utility function will prefer the uncertain payment!

Income Income Gain of 7 € Loss of 1 € (a) 9,5 Expected Utility Units for the Certain Payment (b) 11 Expected Utility Units for the Uncertain Payment

Value of Consumption Goods (€)

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Prof. Dr. Rainer Maurer

1. How Capital Markets Work 1.2.1. Why People Don’t Like Risk

Utility Units

Risk Neutral Utility Function

Normal Person (Risk Averse) Utility Function

Gambler (Risk Lover) Utility Function

Value of Consumption Goods (€)

Since we know from experiments that most people are risk averse, we can draw the conclusion that most people have a utility function with decreasing marginal utility! -44-

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1. How Capital Markets Work 1.2.1. Why People Don’t Like Risk Risk aversion is not specific for human behavior. Experiments with plants show that plants also show risk averse behavior:

Dener,Kacelnik,Shemesh (2016), Pea Plants Show Risk Sensitivity, Current Biology, Volume 26, Issue 13, p. 1763–1767, 11 July 2016

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1. How Capital Markets Work 1.2.2. How People Handle Risk 1. How Capital Markets Work 1.1. Supply and Demand on Capital Markets 1.1.1. Why People Save 1.1.2. Why People Invest 1.2. Capital Markets and Risk 1.2.1. Why People Don’t Like Risk 1.2.2. How People Handle Risk 1.3. Basic Evaluation Techniques for Capital Markets 1.3.1. The Discounted Cash-Flow Method 1.3.2. The Internal Rate of Return Method 1.3.3. Risk and Return: The Sharpe Ratio Prof. Dr. Rainer Maurer

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1. How Capital Markets Work 1.2.2. How People Handle Risk ➤ We have already seen, how normal people handle risk: ■ They demand a risk premium!

➤ Financial markets offer a possibility to eliminate risk: ■ Hedging!

➤ The following tables illustrate the principle of hedging based on several numeric examples:

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1. How Capital Markets Work 1.2.2. How People Handle Risk

Prof. Dr. Rainer Maurer

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1. How Capital Markets Work 1.2.2. How People Handle Risk

➤ This example shows: ■ If the return of one stock goes up exactly when the return of the other stock goes down, a portfolio of both stocks completely eliminates the risk! ■ Consequently, investing your money in a portfolio of both stocks implies no risk, while investing your money in one of both stocks only implies a lot of risk! ■ Note: In case of a perfect hedge, the correlation coefficient equals exactly -1! -49-

Prof. Dr. Rainer Maurer

1. How Capital Markets Work 1.2.2. How People Handle Risk

➤ This example shows: ■ If the return of one stock goes up exactly when the return of the other stock goes up, a portfolio of both stocks does not affect risk at all! ■ Consequently, investing your money in a portfolio of both stocks implies the same risk, as investing your money in one of both stocks only! ■ Note: In case of a no hedge, the correlation coefficient equals exactly 1! Prof. Dr. Rainer Maurer

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1. How Capital Markets Work 1.2.2. How People Handle Risk

The Miracle of Hedging!

➤ This example shows:

■ If the return of one stock goes up when the return of the other stock goes up, but not by exactly the same degree, a portfolio of both stocks can reduce risk somewhat but not completely eliminate it! ■ Consequently, investing your money in a portfolio of both stocks implies a lower risk, as investing your money in one of both stocks only! ■ Note: In case of a normal hedge, the correlation coefficient lies between 0 and 1! -51-

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1. How Capital Markets Work 1.2.2. How People Handle Risk ➤ In the real world, perfect hedges are as rare as no hedges! ➤ Fortunately, normal (imperfect) hedges are the rule, so that investing in portfolios generally makes more sense than investing in single stock! ➤ Why are stocks so often imperfect hedges? ■ On one hand, there are a lot of common economic factors that effect all stocks in the same way, causing a positive correlation of returns: ◆ The business cycle, prices of raw materials, wages, tax reforms… ■ To the other hand, every firm has its own markets segments and these segments often react in a different way to these common economic factors: ◆ For example, a discounter like Aldi profits from high consumer confidence as well as a luxury store like KaDeWe, but suffers less from a recession, because of its cheaper range of goods... Prof. Dr. Rainer Maurer

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1. How Capital Markets Work 1.2.2. How People Handle Risk ➤ As the examples have shown, we can comfortably measure the hedge quality of two kind of stocks by the correlation coefficient. ➤ How do we compute the correlation coefficient?

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1. How Capital Markets Work 1.2.2. How People Handle Risk ➤ How do we compute the variance?

➤ How do we compute the covariance?

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1. How Capital Markets Work 1.2.2. How People Handle Risk ➤ Interpretation of the Correlation Coefficient:

➤ A correlation coefficient of -1 indicates that the value of two stocks moves through time with exactly opposite fluctuations: ■ If the stock of Raincoat Corp. displays a positive deviation from its mean value, the stock of Sunglasses International displays a negative deviation form its mean value. ■ If the stock of Raincoat Corp. displays a negative deviation from its mean value, the stock of Sunglasses International displays a positive deviation form its mean value. -55-

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1. How Capital Markets Work 1.2.2. How People Handle Risk ➤ Interpretation of the Correlation Coefficient:

➤ A correlation coefficient of 1 indicates that the value of two stocks moves through time with exactly the same fluctuations: ■ If the stock of Raincoat Corp. displays a positive deviation from its mean value, the stock of Umbrella Unlimited displays a positive deviation form its mean value too. ■ If the stock of Raincoat Corp. displays a negative deviation from its mean value, the stock of Umbrella Unlimited displays a negative deviation form its mean value too. Prof. Dr. Rainer Maurer

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1. How Capital Markets Work 1.2.2. How People Handle Risk ➤ Interpretation of the Correlation Coefficient:

➤ A correlation coefficient between 0 and 1 indicates that the value of two stocks moves through time with rather similar but not exactly the same fluctuations: ■ If the stock of Sunglasses Int. displays a positive deviation from its mean value, the stock of United Steel displays most of the time a positive deviation form its mean value too – but not always. ■ If the stock of Sunglasses Int. displays a negative deviation from its mean value, the stock of United Steel International displays most of the time a negative deviation form its mean value too – but not always. -57-

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1. How Capital Markets Work 1.2.2. How People Handle Risk ➤ Now it’s up to you: ■ As portfolio manager you have to decide in the following cases, whether to invest in single stock or in a portfolio. ■ What do you recommend?

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1. How Capital Markets Work 1.2.2. How People Handle Risk

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1. How Capital Markets Work 1.2.2. How People Handle Risk ➤ The Market-Beta:

■ As already seen, it is almost impossible to find perfectly negatively correlated stocks – so that all portfolios end up with a risk that cannot be eliminated – the so called “market risk”. ■ The market risk is measured by the variance of the average return of the market portfolio, i,e. the risk that cannot be eliminated by investing in the market portfolio. ■ The tendency of the return of a stock to move with the average return of the market portfolio is called its market-beta. ■ The market-beta is a measure of the relative volatility of a stock return compared to the return of the total stock market (= a portfolio consisting of all stocks minus the one stock whose beta is measured) as a whole.

◆ A beta of 1 means that a stock return moves exactly as the market does. ◆ A beta of 2 means that if a stock market return moves up by 10 %, the stock return moves up by 20 %.

=> Adding a high (low) beta-stock to a portfolio means increasing (reducing) the portfolio risk.

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1. How Capital Markets Work 1.2.2. How People Handle Risk ➤ How do we compute the market-beta?

➤ As we will see in chapter 2, the market-beta plays a central role for the evaluation model of stocks – the so called CAPModel (Capital Asset Pricing Model). ➤ The following graph visualizes its interpretation.

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1. How Capital Markets Work 1.2.2. How People Handle Risk β=1 => Stock A is as volatile as the market.

Return of Stock A

=> Adding Stock A to the market portfolio does not change portfolio risk

β=1

“Average β-Stock“

Return of the Market Portfolio

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1. How Capital Markets Work 1.2.2. How People Handle Risk β=0,5 => Stock A is less volatile than the market.

Return of Stock A

“Low β-Stock“ β=0,5

=> Adding Stock A to the market portfolio does reduce portfolio risk Return of the Market Portfolio

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1. How Capital Markets Work 1.2.2. How People Handle Risk β=1,5 => Stock A is more volatile than the market. => Adding Stock A to the market portfolio does increase portfolio risk

Return of Stock A

β=1,5

“High β-Stock“

Return of the Market Portfolio

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1. How Capital Markets Work 1.2.2. How People Handle Risk

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1. How Capital Markets Work 1.2.2. How People Handle Risk

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1. How Capital Markets Work 1.2.2. How People Handle Risk

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1. How Capital Markets Work 1.2.2. How People Handle Risk ➤ To sum up: ■ If two stocks display a perfect negative correlation of returns, it is possible to completely eliminate their return risks by holding a portfolio of both stocks instead of holding only one stock. ■ To measure the correlation of returns, we use the “correlation coefficient”. ■ The miracle of hedging: It is possible to reduce risks by holding a portfolio of stocks, even if the correlation of all stocks is positive but not perfect (i.e. if the correlation coefficient lies between 0 and 1). ■ The correlation of the return of a stock with the market portfolio (=the portfolio consisting of all stocks dealt with in the market) is called “market-beta” of a stock. -69-

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1. How Capital Markets Work 1.3.1. The Discounted Cash-Flow Method 1. How Capital Markets Work 1.1. Supply and Demand on Capital Markets 1.1.1. Why People Save 1.1.2. Why People Invest 1.2. Capital Markets and Risk 1.2.1. Why People Don’t Like Risk 1.2.2. How People Handle Risk 1.3. Basic Evaluation Techniques for Capital Markets 1.3.1. The Discounted Cash-Flow Method 1.3.2. The Internal Rate of Return Method 1.3.3. Risk and Return: The Sharpe Ratio Prof. Dr. Rainer Maurer

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1. How Capital Markets Work

1.3.1. The Discounted Cash-Flow Method ➤ As the next chapter will show, very different kind of assets are traded on capital markets. ➤ Two technical procedures are important for the evaluation of these different assets: ■

Discounted Cash-Flow Method

■

Internal Rate of Return Method

➤ Before we apply these procedures to the various types of assets in the next chapter, we will analyze them in some detail in the following: ➤ We start with the discounted cash-flow method: -71-

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1. How Capital Markets Work

1.3.1. The Discounted Cash-Flow Method ➤ What do you prefer: 1 € today or 1 € in one year? ➤ The basic idea of the discounted cash-flow method is: ■

Determining the present value of a flow of future payments – either from an investment project or a financial market asset.

■

Technically, payments of different points in time are made comparable by evaluating each payment with a time specific discount factor and adding up these comparable payments to the “present value” of the payment flow.

➤ The following examples show, how this works:

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1. How Capital Markets Work 1.3.1. The Discounted Cash-Flow Method ➤ Discounting a payment flow:

➤

The present value of these four payments is determined by discounting each single payment (positive or negative) with the discount factor for the corresponding year.

➤

Example: The discount factor for a payment three years from now for a discount rate of 5% equals: (1,05)^(-3) = 0,86 Hence the present value of a payment three years form now is worth 86% of its future value.

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1. How Capital Markets Work 1.3.1. The Discounted Cash-Flow Method ➤

Discounting a payment flow:

➤

This payment flow with equal annual payments of 100 per year clearly shows that payments further in the future a more discounted than payments closer to the present.

➤

A comparison with the following table shows that a lower discount rate increases the present value:

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1. How Capital Markets Work 1.3.1. The Discounted Cash-Flow Method ➤ Problem of the discounting approach: ■

An appropriate discount rate has to be chosen!

➤ The appropriate discount rate should reflect the risk related to a payment flow. ■

Uncertain payment flows, whose payments are based on estimated forecasts only, should be discounted with a higher discount rate than secure payment flows.

■

Hence, the discount rate should include an appropriate risk premium.

➤ How can this be done?

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1. How Capital Markets Work

1.3.1. The Discounted Cash-Flow Method ➤ How to find the appropriate discount rate? ■

In many cases it is possible to find a market interest rate for a payment flow of the same risk class: ◆ If the payment flow is nearly certain, the market interest rate of a fixed rate security with the same risk structure, for example a bond of a government with high creditworthiness, should be chosen as discount rate.

■

However, for many uncertain payment flows, it is difficult to find a market interest rate of the same risk class.

◆ If the payment flow is uncertain and based on a forecast (for

example the dividend payments from a stock company) a market interest rates for exactly the same risk class are hard to find.

◆ In this case, concepts like the Capital Asset Pricing Model (CAPM) can to be employed to calculate an appropriate discount rate. Chapter 2 will show, how this works.

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1. How Capital Markets Work 1.3.1. The Discounted Cash-Flow Method Details If the payment flow consists of equal payments (also called “annuities”) we can use the annuity present value formula:

If the flow of equal payments holds on for all eternity (≈200 years) this formula simplifies to:

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1. How Capital Markets Work 1.3.2. The Internal Rate of Return Method 1. How Capital Markets Work 1.1. Supply and Demand on Capital Markets 1.1.1. Why People Save 1.1.2. Why People Invest 1.2. Capital Markets and Risk 1.2.1. Why People Don’t Like Risk 1.2.2. How People Handle Risk 1.3. Basic Evaluation Techniques for Capital Markets 1.3.1. The Discounted Cash-Flow Method 1.3.2. The Internal Rate of Return Method 1.3.3. Risk and Return: The Sharpe Ratio

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1. How Capital Markets Work 1.3.2. The Internal Rate of Return Method ➤ The basic idea of the internal rate of return method is: ■

Very often one knows the market price of an asset or investment project and its payment flow.

■

In such cases it is possible to determine the internal rate of return of the asset or investment project.

■

Technically, this amounts to finding a discount rate that equals the present value of the payment flow with the market price of the asset.

➤ The following examples shows, how this works:

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1. How Capital Markets Work 1.3.2. The Internal Rate of Return Method t+1

➤ If the payment flow consists of one period only, it is easy to calculate the IRR (=internal rate of return) by hand:

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1. How Capital Markets Work 1.3.2. The Internal Rate of Return Method

➤

If the payment flow consists of more than one period, for example 5 periods, calculating the IRR implies solving a polynomial of degree 5. This involves the following problem: ■

Formulas for analytical solutions exist only for polynomials lower degree 4.

➤ Therefore, numerical solutions methods must be applied. ➤ One such method is available for Excel (the IKV() Function). -81-

Prof. Dr. Rainer Maurer

1. How Capital Markets Work 1.3.2. The Internal Rate of Return Method Details Calculating the IRR with the numeric iteration procedure of Excel: An often used tool to calculate the yield to maturity is the Excel-formula “IKV(rage of cells containing the flow of payments; cell containing the estimated value)”. This formula calculates the “internal rate of return” (IRR) of a flow of payments (in the order of appearance; at least one negative payment (=investment=purchase of bond=market value of bond) is necessary) based on a numerical iteration procedure. It starts the iteration process with a value for “i” you can specify as “estimated value” (default value is 10%) and changes this value until it reaches a solution value for the exact formula with an accuracy of 0,00001 percent. If it does not find such a solution after 20 calculation steps it delivers the error message “#value!”. For an example see my Excel-file “Computing yield to maturity”, 2nd sheet.

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1. How Capital Markets Work 1.3.2. The Internal Rate of Return Method ➤ Consequently, calculating the IRR typically implies the usage of a computer. ➤ If the IRRs of several payment flows are calculated, it is in principle possible to select the payment flow with the highest IRR as the most profitable one. ➤ However, one has to take care of the risk implied by each payment flow! ➤Since an uncertain payment flow is riskier than a certain payment flow, the uncertain flow must offer a risk premium (for people with normal utility function…) ➤One often used measure, which takes care of both return and risk, is the so called Sharpe Ratio, which was proposed by William F. Sharpe (1966). Prof. Dr. Rainer Maurer

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1. How Capital Markets Work 1.3.3. Risk and Return: The Sharpe Ratio 1. How Capital Markets Work 1.1. Supply and Demand on Capital Markets 1.1.1. Why People Save 1.1.2. Why People Invest 1.2. Capital Markets and Risk 1.2.1. Why People Don’t Like Risk 1.2.2. How People Handle Risk 1.3. Basic Evaluation Techniques for Capital Markets 1.3.1. The Discounted Cash-Flow Method 1.3.2. The Internal Rate of Return Method 1.3.3. Risk and Return: The Sharpe Ratio

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1. How Capital Markets Work 1.3.3. Risk and Return: The Sharpe Ratio ➤ The basic idea of the Sharpe Ratio is to determine the risk premium paid per unit of risk. ➤ Therefore, the formula of the Sharpe Ratio relates the risk premium of the internal rate of return of a specific investment j (= the difference between this expected return, E(ij), and the return of a risk free market interest rate, ro, e.g. a government bond of high creditworthiness), E(ij)- ro, to the standard deviation of the return of the specific investment (= (variance(ij))^0,5 ):

➤ The higher the Sharpe Ratio, the higher is the risk premium per unit risk and hence the more attractive is the investment. -85-

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1. How Capital Markets Work 1.3.3. Risk and Return: The Sharpe Ratio ➤ Which asset A or B offers the best relation between risk and return? Calculate the Sharp Ratio for a risk free return of 2 %.

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1. How Capital Markets Work 1.3.3. Risk and Return: The Sharpe Ratio ➤ Problem with the Sharpe Ratio: ■

The standard deviation of an investment, i.e. the measure for its risk, is typically not known, but has to be estimated based on forecasted future returns. In our above example we simply assumed to know them!

■

For financial market assets, e.g. the return of stocks, such kind of forecasts are typically highly inaccurate.

■

Therefore, very often, the standard deviation of a stock is estimated based on its past returns.

■

Even though such a calculation is easily done and seems to be highly “accurate and plausible”, one has to be aware that the application of such “historic” standard deviations for future investment decisions, implies the “hidden assumption” that the “future” will similar to the “past”.

■

For many kind of stocks, this assumption has proven wrong!

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1. How Capital Markets Work 1.2.2. How People Handle Risk ➤ To sum up: ■ The discounted cash-flow and the internal rate of return method are methods to calculate the profitability of different kind of investments. ■ The Sharpe-Ratio is a measure for the relation of return and risk of an investment. ■ All these measures rely very often to a high degree on estimations and forecasts. ■ In the next chapter we will apply these methods to the standard assets of the major capital market segments.

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Chapter 1: Questions

You should be able to answer the following questions at the end of this chapter. If you have difficulties in answering a question, discuss this question with me during or at the end of the next lecture or attend my colloquium.

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Chapter 1: Questions for Review 1.

Why can “saving” increase personal utility? Give a graphical and verbal explanation. 2. How does saving behavior affects consumption behavior? 3. If saving increases utility, why do savers demand interest? 4. Why depends the willingness to save positively on the interest rate? 5. What is a “production function”? 6. Explain the motive for investment. 7. Why depends the willingness to invest negatively on the interest rate? 9. What insures the equality of saving and investment in a market equilibrium? Explain your answer based on the a diagram of the capital market. 10. How does an increase in savings supply affect the interest rate?

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Chapter 1: Questions for Review 11. How does an increase in investment demand affect the interest rate? 12. Explain the meaning of “risk averse” and “risk neutral”. 13. Why are most people risk averse? Give a graphical and verbal explanation. 14. What is a “risk premium”? 15. Why do people demand a risk premium? 16. What is “hedging”? 17. What property must two stocks have to be a “perfect hedge”? 18. Is it possible to hedge risk with two stocks, if there return is positively correlated? 19. Why are perfect hedges rare? 20. What is a “normal hedge”? 21. How do you explain that so many stocks are “normal hedges”?

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Chapter 1: Questions for Review 22. Calculate the variances and the correlation coefficient of the following stocks. Can they be used to hedge each other?

23. Calculate the variances and the correlation coefficient of the following stocks. Can they be used to hedge each other?

24. What is the “market-beta”? 25. What is meant by “low beta stock” and “high beta stock”? Prof. Dr. Rainer Maurer

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Chapter 1: Questions for Review 26. Give a verbal explanation of the discounted cash flow and the internal rate of return method.

27. 28. 29. 30.

Which criteria should an appropriate discount rate fulfill? What is the definition of the Sharpe Ratio? Give a verbal interpretation of the Sharpe Ratio. What are the two major criteria for an investment decision?

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Chapter 1: Questions for Review 31. What asset is the most attractive? Base your decision on the Sharpe Ratio.

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Portfolio Theory

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