FINANCIAL ECONOMICS I

RETURN AND RISK CONCEPTS

A GLOSSARY OF BASIC CONCEPTS

2

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

RETURN

Return is gain or loss over time, usually expressed as

an annual percentage of initial investment

Holding period rate of return, 𝐻𝑃𝑅 =𝑃1−𝑃0+𝐶

𝑃0

Effective annual rate (EAR) of return is useful for

comparing investments with different horizons

❖ (1 + 𝐸𝐴𝑅)𝑇= 1 + 𝐻𝑃𝑅, where 𝑇 is a fraction of 1 year

Annual percentage rate (APR) of return is the rate of

return for each period times number of periods in a year

❖ 𝐴𝑃𝑅 = 𝑚 × 𝐻𝑃𝑅 (Note: 𝑚 = 1/𝑇)3

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

CONTINUOUS COMPOUNDING

Recall: 𝐴𝑃𝑅 = 𝑚 × 𝐻𝑃𝑅 =1

𝑇𝐻𝑃𝑅

This implies that 𝐻𝑃𝑅 = 𝑇 × 𝐴𝑃𝑅

So that (1 + 𝐸𝐴𝑅)𝑇= 1 + (𝑇 × 𝐴𝑃𝑅)

Alternatively 1 + 𝐸𝐴𝑅 = 1 + (𝑇 × 𝐴𝑃𝑅) 1/𝑇

When T → 0, we approach continuous compounding:

❖ 1 + 𝐸𝐴𝑅 = 𝑒𝑅𝐶𝐶, where 𝑒 ≈ 2.718282

❖ It follows that 1 + 𝐸𝐴𝑅 𝑇 = 𝑒𝑇×𝑅𝐶𝐶

𝑒𝑇×𝑅𝐶𝐶 is the total rate of return for a period, given the continuous compounding rate 𝑅𝐶𝐶 4

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

RISK

Expected return, 𝐸 𝑅 = σ𝑖=1𝑛 𝑃𝑖𝑅𝑖

Standard deviation of returns, 𝑆𝐷 𝑅 = 𝑉𝑎𝑟(𝑅) 1/2

Variance of returns, 𝑉𝑎𝑟 𝑅 = σ𝑖=1𝑛 𝑃𝑖 𝑅𝑖 − 𝐸 𝑅 2

Expected holding period return can be broken down thus:

𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑟𝑒𝑡𝑢𝑟𝑛 =𝑅𝑖𝑠𝑘 𝑓𝑟𝑒𝑒𝑟𝑎𝑡𝑒

+𝑅𝑖𝑠𝑘

𝑃𝑟𝑒𝑚𝑖𝑢𝑚

Therefore, risk premium (or excess return) is the

holding period return minus risk-free rate of return 5

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

MARKOWITZ PORTFOLIO OPTIMIZATION

6

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

RISK AND RETURN OF A PORTFOLIO

The expected return on a portfolio is simply the

weighted average of the expected returns on assets

comprising the portfolio, with the weights being

the relative proportions of each asset in the

portfolio:

𝐸 𝑅𝑃 =

𝑖=1

𝑛

𝑤𝑖𝐸 𝑅𝑖 = 𝑤1𝐸 𝑅1 +𝑤2𝐸 𝑅2 + …+𝑤𝑛𝐸 𝑅𝑛

Unlike the expected return on a portfolio, the

standard deviation of returns on a portfolio is not

merely a weighted average of risks on the assets

comprising the portfolio 7

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

RISK AND RETURN OF A PORTFOLIO

Rather, the extent of association among returns on the assets comprising the portfolio matters ad must be incorporated in the computation of portfolio’s total risk

Therefore, portfolio risk (standard deviation) is computed as

𝑆𝐷 𝑅𝑃 =

𝑗=1

𝑛

𝑘=1

𝑛

𝑤𝑗𝑤𝑘𝐶𝑜𝑣 𝑅𝑗 , 𝑅𝑘

The covariance is obtained as

𝐶𝑜𝑣 𝑅𝑗 , 𝑅𝑘 =

𝑖=1

𝑛

𝑃𝑖 𝑅𝑖𝑗 − 𝐸 𝑅𝑗 𝑅𝑖𝑘 − 𝐸 𝑅𝑘

Or

𝐶𝑜𝑣 𝑅𝑗 , 𝑅𝑘 = 𝑆𝐷 𝑅𝑗 𝑆𝐷 𝑅𝑘 𝐶𝑜𝑟𝑟 𝑅𝑗 , 𝑅𝑘

8

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

RISK AND RETURN OF A PORTFOLIO

Substituting the covariance formula into the portfolio standard deviation formula gives

𝑆𝐷 𝑅𝑃 =

𝑗=1

𝑛

𝑘=1

𝑛

𝑤𝑗𝑤𝑘𝑆𝐷 𝑅𝑗 𝑆𝐷 𝑅𝑘 𝐶𝑜𝑟𝑟 𝑅𝑗 , 𝑅𝑘

Suppose that 𝑛 = 2. the portfolio risk is computed as

𝑤1𝑤1𝑆𝐷 𝑅1 𝑆𝐷 𝑅1 𝐶𝑜𝑟𝑟 𝑅1, 𝑅1 𝑤1𝑤2𝑆𝐷 𝑅1 𝑆𝐷 𝑅2 𝐶𝑜𝑟𝑟 𝑅1, 𝑅2𝑤2𝑤1𝑆𝐷 𝑅1 𝑆𝐷 𝑅2 𝐶𝑜𝑟𝑟 𝑅2, 𝑅1 𝑤2𝑤2𝑆𝐷 𝑅2 𝑆𝐷 𝑅2 𝐶𝑜𝑟𝑟 𝑅2, 𝑅2

𝑤12𝑉𝑎𝑟 𝑅1 𝑤1𝑤2𝑆𝐷 𝑅1 𝑆𝐷 𝑅2 𝐶𝑜𝑟𝑟 𝑅1, 𝑅2

𝑤2𝑤1𝑆𝐷 𝑅2 𝑆𝐷 𝑅1 𝐶𝑜𝑟𝑟 𝑅2, 𝑅1 𝑤22𝑉𝑎𝑟 𝑅2

𝑆𝐷 𝑅𝑃 = 𝑤12𝑉𝑎𝑟 𝑅1 +𝑤2

2𝑉𝑎𝑟 𝑅2 + 2𝑤1𝑤2𝑆𝐷 𝑅1 𝑆𝐷 𝑅2 𝐶𝑜𝑟𝑟 𝑅1, 𝑅29

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

RISK AVERSION AND UTILITY VALUES

Investors are risk averse if they demand positive risk

premium for a given quantity of risk

Investment opportunities (assets) are attractive if

they have low risk and high expected returns

If risk increases with return, investors must trade off

risk for return

The trade-off is achieved by assigning utility to the

assets based on their risk and expected return

10

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

RISK AVERSION AND UTILITY VALUES

The following utility function has gained wide usage in evaluating investment opportunities:

𝑈 = 𝐸 𝑅 − ൗ1 2𝐴𝜎2

where 𝑈 is the utility value, 𝐴 is investor’s risk aversion index; Τ1 2 is a scaling convention

More risk averse investors have large values of 𝐴 which penalizes risky investments more severely

The utility score can be interpreted as the certainty equivalent rate of return

(the rate of return that a risk-free asset must earn to be as competitive as the risky portfolio in question)

11

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

MEAN-VARIANCE CRITERION

Mean-variance dominance criterion states that asset A

dominates asset B if

𝐸 𝑅𝐴 ≥ 𝐸 𝑅𝐵

and

𝑉𝐴𝑅 𝑅𝐴 ≤ 𝑉𝑎𝑟 𝑅𝐵

At least one equality must be strict to rule out

indifference

12

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

INDIFFERENCE CURVES

Indifference curves are expected return-standard

deviation trade-off function (yield constant utility)

Consider portfolios P and Q in the figure below

13

I

IV

II

III

𝐸 𝑅𝑃

𝜎𝑃

Indifference curve

P

Q

Utility increases

Uti

lity

in

crease

s

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

PORTFOLIOS OF ONE RISKY ASSET AND

ONE RISK-FREE ASSET

Assume an investor holds the proportion 𝑦 of her wealth

in a risky portfolio, P, and (1 − 𝑦) in a risk-free asset*, F

Thus, the complete portfolio’s riskiness will vary with 𝑦

Let’s define terms as follows:

𝑟𝑃 – the rate of return on the risky portfolio, 𝑃

𝐸 𝑟𝑃 – the expected return on risky portfolio, 𝑃

𝜎𝑃 – the standard deviation of risky portfolio, 𝑃

𝑟𝑓 – return on risk-free asset, 𝐹14

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

PORTFOLIOS OF ONE RISKY ASSET AND

ONE RISK-FREE ASSET CONT’D

Thus, the expected return, E 𝑟𝐶 , on the complete

portfolio, 𝐶, is

E 𝑟𝐶 = 𝑦𝐸 𝑟𝑃 + (1 − 𝑦)𝑟𝑓

= 𝑟𝑓 + 𝑦 𝐸 𝑟𝑃 − 𝑟𝑓 … … … …(1)

Because asset 𝐹 is risk-free, the standard deviation of

the returns on portfolio 𝐶 is simply:

𝜎𝐶 = 𝑦𝜎𝑃 … … … … … … … … (2)

15

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

PORTFOLIOS OF ONE RISKY ASSET AND

ONE RISK-FREE ASSET CONT’D

Solving for 𝑦 in eq. (2), substituting the result

( Τ𝑦 = 𝜎𝐶 𝜎𝑃) in equation (1) and rearranging:

E 𝑟𝐶 = 𝑟𝑓 + 𝜎𝐶𝐸 𝑟𝑃 − 𝑟𝑓

𝜎𝑃… … … … … …(3)

Equation (3) is called the capital allocation line (CAL*)

The expected excess return on portfolio 𝐶, per unit of

additional risk is the slope of the CAL, also called the

reward-to-volatility ratio, (or the Sharpe ratio):

𝑆 =𝐸 𝑟𝑃 − 𝑟𝑓

𝜎𝑃

16

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

PORTFOLIOS OF ONE RISKY ASSET AND

ONE RISK-FREE ASSET CONT’D

Lending and Borrowing

Consider the graph of equation (3):

17

P𝐸 𝑟𝑃

𝜎𝑃

CAL

𝑆 =𝐸 𝑟𝑃 − 𝑟𝑓

𝜎𝑃

𝑟𝑓

C

T

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

PORTFOLIOS OF ONE RISKY ASSET AND

ONE RISK-FREE ASSET CONT’D

Suppose P is the investor’s ‘desired’ risky portfolio

Thus, she can invest in 𝑃 and also invest (lend) in the

risk-free asset, 𝐹, to form their optimal portfolio, (e.g. 𝐶)

Investors can also borrow at the risk-free rate, if

possible, to form a portfolio to the right of 𝑃 (e.g. 𝑇)

However, since the investor has borrowed, the

proportion of her wealth in portfolio 𝑃, 𝑦 > 1

❖ Accordingly, (1 − 𝑦)must be negative

Clearly, portfolio 𝑇 must be riskier than portfolio 𝐶18

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

PORTFOLIOS OF ONE RISKY ASSET AND

ONE RISK-FREE ASSET CONT’D

In practice, private (nongovernmental) investors

usually borrow at above the risk-free rate (say, 𝑟𝑓𝐵)

Private investors borrow through margin accounts

Thus, their capital allocation line will be kinked at P:

19

𝑅𝑓𝐵

𝑅𝑓

P𝐸 𝑅𝑃

𝜎𝑃

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

RISK TOLERANCE AND ASSET ALLOCATION

The CAL gives provides investors with the set of all feasible capital allocation choices (opportunity set)

We can determine the portfolio (combination of risky and

risk-free asset) at which investors maximize utility thus:

𝑀𝑎𝑥 𝑈 = 𝐸 𝑅𝐶 − ൗ1 2𝐴𝜎𝐶2 = 𝑟𝑓 + 𝑦 𝐸 𝑟𝑃 − 𝑟𝑓 − Τ1 2𝐴𝑦2𝜎𝑃

2

Differentiating 𝑈 w.r.t. 𝑦, setting the derivative equal to 0 and solving for 𝑦 gives

𝑦∗ = Τ𝐸 𝑟𝑃 − 𝑟𝑓 𝐴𝜎𝑃2

Thus, the optimal portfolio is inversely proportional to risk and risk tolerance coefficient but directly proportional to the risk premium

20

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

RISK TOLERANCE AND ASSET ALLOCATION

Theoretically, individuals choose portfolios from the

opportunity set based on their degrees of risk aversion

(less risk-averse investors hold less of the risk-free

asset etc.)

The degree of risk aversion is reflected in the slope of

each investor’s indifference curve

The slope of the indifference curve is steeper the higher

is the investor’s risk aversion coefficient, A

See illustration next slide and in Excel 21

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

RISK TOLERANCE AND ASSET ALLOCATION

– INDIFFERENCE CURVES

220.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

A = 4

A = 2

U = 0.15

U = 0.05

More desirable portfolios

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

CAPITAL ALLOCATION USING INDIFFERENCE

CURVES

Assume 𝜎𝑃 = 15%; 𝐸 𝑅𝑃 = 35%; and 𝑅𝑓 = 3%

230.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

CAL

Optimal portfolio, C

Risky

portfolio, P

𝑅𝑓

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

PASSIVE (RISKY) PORTFOLIOS

Portfolio P above is formed by the investor after researching each asset’s promised returns and risk

An alternative is to invest in a passive portfolio (which do not require research and regular rebalancing)

Passive portfolios are available in some of Africa’s stock markets [e.g. South Africa’s index funds and/or ETFs (Satrix, Stanlib etc.)]

Advantages of passive portfolios

1. Lower cost

2. Free-rider benefit

A capital allocation line representing a strategy with a passive risky portfolio is called the capital market line (CML) 24

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

DIVERSIFICATION

The risk of an asset can be split into two components:

1. Systematic risk: the general component attributable to broad macroeconomic factors (e.g. GDP)

2. Non-systematic or diversifiable risk: related to factors specific to the asset issuer (e.g. managerial changes)

Non-systematic risk can be reduced (possibly eliminated) by investing in several non-related assets

❖ This strategy is called diversification 25

Systematic risk

Non-systematic risk

No. of securities

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

PORTFOLIOS OF TWO RISKY ASSETS

Assume an investor holds a portfolio of two assets, 1 and

2 with expected returns 𝐸 𝑅1 and 𝐸 𝑅2 respectively

The proportion of the investor’s wealth in asset 1 is 𝑤1and the proportion in asset 2 is 𝑤2 (where 𝑤2 = 1 − 𝑤1)

The expected return on the two-asset portfolio is

𝐸 𝑟𝑃 = 𝑤1𝐸 𝑟1 +𝑤2𝐸 𝑟2

The variance of the two-asset portfolio is

𝑉𝑎𝑟 𝑅𝑃 = 𝑤12𝑉𝑎𝑟 𝑅1 +𝑤2

2𝑉𝑎𝑟 𝑅2 + 2𝑤1𝑤2𝐶𝑜𝑣 𝑅1, 𝑅226

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

PORTFOLIOS OF TWO RISKY ASSETS CONT’D

The covariance of returns can be expressed as

𝐶𝑜𝑣 𝑅1, 𝑅2 = 𝑆𝐷 𝑅1 𝑆𝐷 𝑅2 𝐶𝑜𝑟𝑟(𝑅1, 𝑅2)

Thus, the two-asset portfolio variance is

𝜎𝑃2 = 𝑤1

2𝜎12 +𝑤2

2𝜎22 + 2𝑤1𝑤2𝜎1𝜎2𝜌12

Note:

1. Portfolio risk is usually less than the weighted average of risks of individual assets comprising it

2. The upper bound portfolio risk is realized when the correlation coefficient is +1

3. The lower bound is realized when the correlation coefficient is: -1 27

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

MINIMUM VARIANCE PORTFOLIO

The combination of assets 1 and 2 that gives a portfolio

with the minimum variance can be obtained by

differentiating the variance equation with respect to 𝑤1,

setting the result equal to zero, then solving for 𝑤1∗:

𝑑𝑉𝑎𝑟 𝑅𝑃𝑑𝑤1

= 2𝑤1𝜎𝑋2 − 2𝜎𝑌

2 + 2𝑤1𝜎𝑌2 + 2𝑟𝑋,𝑌𝜎𝑋𝜎𝑌 − 4𝑤1𝑟𝑋,𝑌𝜎𝑋𝜎𝑌 = 0

2𝑤1 𝜎𝑋2 + 𝜎𝑌

2 − 2𝑟𝑋,𝑌𝜎𝑋𝜎𝑌 + 2𝑟𝑋,𝑌𝜎𝑋𝜎𝑌 − 2𝜎𝑌2 = 0

Dividing through by 2 and solving for 𝑤1∗, we obtain

𝑤1∗ =

𝜎𝑌2− 𝑟𝑋,𝑌𝜎𝑋𝜎𝑌

𝜎𝑋2+ 𝜎𝑌

2− 2𝑟𝑋,𝑌𝜎𝑋𝜎𝑌

28

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

PORTFOLIOS OF TWO RISKY ASSETS

CONT’D

Consider two assets 𝑋 and 𝑌 :

Suppose you can form ten portfolios with weights 100%,

90%, 80%, 70%, 60%, 50%, 40%, 30%, 20%, 10% and 0%

respectively in asset 𝑋

The expected returns and standard deviations can be

obtained under various correlation assumptions 29

Asset Expected return Standard deviation

X 10% 3.5%

Y 8.5% 2%

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

PORTFOLIO OPPORTUNITY SET

30

8.4

8.6

8.8

9

9.2

9.4

9.6

9.8

10

10.2

0 0.5 1 1.5 2 2.5 3 3.5 4

Ex

pe

cte

d r

etu

rn

(%

)

Standard deviation of returns (%)

Lower bound Upper bound Port Opp Set

More desirable portfolios

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

PORTFOLIO OPPORTUNITY SET

Notice that the benefits of diversification are greater

the lower the correlation coefficient

Negative correlations give deeper curvatures and give

more desirable portfolios

Notice, too, that perfect positive correlation is not

useful for diversification

31

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

PORTFOLIO CHOICE

32

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

EFFICIENT FRONTIER WITH 𝑵 RISKY ASSETS

Consider the portfolio opportunity set (minimum

variance frontier) of 𝑁 risky assets in Figure 1

B

𝐸(𝑅𝐷)

𝜎 𝜎𝐴,𝐷,𝐸

𝜎𝐴,𝐷,𝐸𝜎𝐴,𝐷,𝐸

D

C 𝐸(𝑅𝐶,𝐸)

E

𝐸(𝑅𝐴) A

∎

∎

∎

∎

∎

Figure 1: The opportunity set of risky portfolios

∎Y

∎X

33

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

On Figure 1

The area enclosed by the curvature plus the boundary represents portfolios of risky assets

The opportunity set is clearly an infinite set

Point B is the minimum variance portfolio

Portfolios along the frontier 𝐵𝑌 will be preferred by investors to portfolios below the frontier

Accordingly, the frontier 𝐵𝑌 is known as the efficient frontier or efficient set

The efficient frontier is the market determined investors’ marginal rate of transformation (MRT) between risk and return

❖ It is the set of portfolios that offers the highest possible expected return for any given level of risk or entails the lowest risk for any given level of expected return 34

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

Because all portfolios on the frontier 𝐵𝑌 are efficient, rational investors can choose any one of them

❖ Each individual’s exact portfolio choice is informed by the individual’s subjective marginal rate of substitution (MRS) between risk and return

❖ The MRS is determined by investors’ degree of risk aversion (slope of the investor’s indifference curves)

Indifference curves are maps of the individual’s risk-return trade-off that yield the same total utility

The individual’s total utility is maximized when his/her MRS exactly equals MRT

Thus, the point of tangency between each individual’s highest indifference curve and the efficient frontier is the location of the individual’s optimal portfolio: the point of tangency therefore sets the individual’s subjective price of risk

35

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

B

𝐸(𝑅𝐿)

𝜎 𝜎𝐿

𝜎𝐴,𝐷,𝐸𝜎𝐴,𝐷,𝐸

𝑶𝑯 𝐸(𝑅𝐻) ∎

∎

∎

Figure 2: Optimal portfolio choice

∎Y

𝑶𝑳

𝜎𝐻

𝜎𝐴,𝐷,𝐸𝜎𝐴,𝐷,𝐸

OPTIMAL PORTFOLIO CHOICE

36

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

On Figure 2

The figure shows the optimal choices of two

investors with dissimilar degrees of risk aversion

Investor 𝐻, who has a relatively high degree of risk

aversion (steep indifference curve), chooses his

optimal portfolio on the lower segment of the

efficient frontier, portfolio 𝑂𝐻,

Investor 𝐿 with relatively low degree of risk

aversion chooses her optimal portfolio on the upper

part of the frontier, point 𝑂𝐿, where she bears more

risk but enjoys the prospect of higher returns

37

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

RISK-FREE ASSETS AND PORTFOLIO CHOICE

Consider an investor who buys a risky portfolio in the proportion 𝑤1 of her wealth and invests the remaining proportion (1 − 𝑤1) in a risk-free asset

An asset is risk-free if it offers the same return possibility in all states of nature

The expected return on the resulting portfolio is

𝐸 𝑅𝑃 = 𝑤1𝐸 𝑅1 + 1 − 𝑤1 𝑅𝑓 (1)

where 𝐸 𝑅1 is the expected return on the risky-asset portfolio and 𝑅𝑓 is the return on the risk-free asset

The variance of the resulting portfolio is

𝑉𝑎𝑟 𝑅𝑃 = 𝑤12𝜎1

2 + 1 − 𝑤12𝜎𝑓

2 + 2𝑤1 1 − 𝑤1 𝑟1,𝑓𝜎1𝜎𝑓 = 𝑤12𝜎1

2

(2) 38

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

From equation (2), the standard deviation of the portfolio is

𝑆𝐷 𝑅𝑃 = 𝑤1𝜎1 (3)

Equations (2) and (3) show that the risk-free asset does not contribute to the total risk of the resulting portfolio

The portfolio that results from the combination of a risky-asset portfolio and a risk-free asset has return characteristics that are linear in 𝐸(𝑅𝑃) and 𝜎𝑃 space

Proof of linearity is straightforward:

𝑑𝐸 𝑅𝑃

𝑑𝑤1= 𝐸 𝑅1 − 𝑅𝑓

𝑑𝑆𝐷 𝑅𝑃

𝑑𝑤1= 𝜎1

❖ Therefore, the slope of the function that describes the return and risk of the portfolio is:

𝑑𝐸 𝑅𝑃

𝑑 𝑆𝐷 𝑅𝑃=

Τ𝑑𝐸 𝑅𝑃 𝑑𝑤1

Τ𝑑 𝑆𝐷 𝑅𝑃 𝑑𝑤1=

𝐸 𝑅1 −𝑅𝑓

𝜎1(4) 39

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

COMBINING THE RISK-FREE ASSET WITH

A RISKY PORTFOLIO

Suppose that the market is frictionless (no

transaction costs and other inefficiencies) so that

investors can lend unlimited amounts of money at

the risk-free rate, 𝑅𝑓

Since the return-risk combination that results is a

straight-line function (already proved), we can

draw a straight line from the risk-free rate to any

risky-asset portfolio on the efficient frontier

Points along the straight lines represent various

possible portfolio combinations – see Figure 340

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

COMBINING THE RISK-FREE ASSET WITH

A RISKY PORTFOLIO

B

𝜎

∎

∎

∎

Figure 3: Combining the risky-asset portfolio with the risk-free asset

∎Y

𝑅𝑓

M

C

∎X

A ∎

41

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

1 2

COMBINING THE RISK-FREE ASSET WITH

A RISKY PORTFOLIO

Notice that portfolios along the line 𝑅𝑓𝑀 dominate all

the portfolios along the lines below it

Notice, too, that portfolios on the curvature 𝑀𝑋, originally on the efficient frontier, are also dominated by the portfolios on the line 𝑅𝑓𝑀 and are therefore no

longer efficient

Accordingly, when the risk-free asset with return, 𝑅𝑓, is

introduced, all rational investors will choose portfolios from the line segment 𝑅𝑓𝑀 and the curvature 𝑀𝑌

(That is, the new efficient frontier is 𝑅𝑓𝑀𝑌)

Each investor’s optimal portfolio will, again, be determined by their degrees of risk aversion

42

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

COMBINING THE RISK-FREE ASSET WITH

A RISKY PORTFOLIO

𝜎

∎

Figure 4: Portfolio choice with lending at the risk-free rate

∎Y

𝑅𝑓

M

∎X

∎

∎

𝑶𝑳

𝑶𝑯

43

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

QUESTIONS FOR SELF-STUDY

44

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

SELF-STUDY QUESTION

You are presented with the following information

relating to the expected performance of the stock

of company J after one year

Required

1. Determine the expected return, the standard

deviation of returns and coefficient of variation

for the above stock 45

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

Possible Return (GhC) 100 120 130 150 180 220

Probability 0.05 0.14 0.20 0.36 0.20 0.05

SELF-STUDY QUESTION

Suppose you have decided to invest in the above

stock, based on the values computed in (i).

However, you have read in the financial press

about the risk reduction benefits of diversification

and decided to combine stock J with other

securities.

Your Financial Analyst has provided you with the

following information relating to two stocks of

companies K and L, and an outstanding bond

issued by company M. Further the correlation

coefficients between the returns on pair-wise

combinations of the four securities are provided: 46

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

SELF-STUDY QUESTION

Your maximum acceptable standard deviation on the resulting portfolio is GhC 130 and you have a total of GhC 1,500,000 to invest in the above securities. Your financial analyst has suggested that GhC 300,000 and GhC 500,000 should be invested in stock L and bond M, respectively

1. How much money must you invest in each of the remaining securities to attain your maximum risk target?

2. Compute the coefficient of variation of the resulting portfolio

47

(C) A

ER

C F

ina

ncia

l Eco

nom

ics

Security

Expected

Return (GhC)

Standard

Deviation (GhC)

Correlation With

J K L

K 250 50 0.50

L 80 10 0.10 -0.20

M 100 20 0.15 0.10 0.30