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FINC4101 Investment Analysis

Instructor: Dr. Leng Ling

Topic: Portfolio Theory I

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Learning objectives1. Compute different measures of investment

performance: Holding-period return (HPR) Arithmetic average Geometric average Dollar-weighted return

2. Compute the expected return, variance and standard deviation of a risky investment.

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Concept Map

FI4101

Portfolio Theory

Asset Pricing

Equity

Fixed Income

Market Efficiency

Derivatives

Foreign Exchange

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Portfolio Theory I: Concept Map

Portfolio Theory I

HPR

Arithmetic,Geometric,$-weighted

Expectedreturn

Variance

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Investment return over 1 period: Holding period return (HPR)

Rate of return over a given investment (holding) period. Has two components: Price change = ending price – beginning price Cash income

Price change + Cash incomeBeginning price

Ending price - Beginning price + Cash incomeBeginning price

HPR

=

=

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Holding period return (HPR) Assumes that cash income is paid at the

end of the holding period. If cash income is received earlier,

reinvestment income is ignored. HPR can be used for different types of

investments: stock, bond, mutual fund etc. For stock, cash income = dividend For bond, cash income = coupon

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Holding period return (HPR) Stock

Price change + Cash dividendBeginning price

Price change Cash dividend +

Beginning price Beginning priceCapital gains yield + Dividend yield

HPR=

=

=

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Simple HPR example You are thinking of investing in ABC Inc’s stock. You

intend to hold the stock for 1 year. ABC’s stock is currently selling at $50 and is expected to rise to $56 by the end of the year. The company is expected to pay a per share dividend of $0.60 during the year.

Compute: HPR Capital gains yield Dividend yield. Sum up the capital gains yield and the dividend yield. Is

that the same as the HPR?

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Investment return over many periods

Three alternative ways of measuring average returns over multiple periods: Arithmetic average (arithmetic mean) Geometric average (geometric mean) Dollar-weighted return

Use the following example to illustrate each return measure.

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Example: Table 5.1 Quarterly cash flows and HPRs of a mutual fund.

1st Qtr 2nd Qtr 3rd Qtr 4th Qtr

Beg. Assets ($m) 1 1.2 2 0.8

HPR (%) 10 25 (20) 25

Assets before flows ($m) 1.1 1.5 1.6 1

Net inflow ($m) 0.1 0.5 (0.8) 0.0

End. Assets ($m) 1.2 2 0.8 1

What is the arithmetic average, geometric average and dollar-weighted return over the four quarters?

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Arithmetic average Suppose we hold an asset over N periods:

1, 2,…, and N. And we label the HPR in each period as:

r1, r2, …, rN The arithmetic average is the sum of returns in

each period divided by number of periods.

Arithmetic average = (r1 + r2 + r3 + ... rN) /N In the example,

arithmetic average = (10+25-20+25)/4 = 10%

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Geometric Average (time-weighted) Single per period return that gives the same

cumulative performance as the sequence of actual returns.

Geometric average = [(1+r1) x (1+r2) ... x (1+rN)]1/N - 1

In the example, geometric average= [(1.1) x (1.25) x (.8) x (1.25)] 1/4 - 1

= (1.375) 1/4 -1 = .0829 = 8.29%

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Dollar-weighted return (IRR) This is simply the internal rate of return (IRR) on

an investment ! IRR: the interest rate that will make the PV of

cash inflows equal to the PV of cash outflows. In other words, IRR is the discount rate such that the

NPV is 0.

0

11 00

N

tt

tN

tt

t

IRR

COF

IRR

CIF

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In the example, think in terms of capital budgeting. So, the mutual fund is a “project” from investor’s perspective.

• Initial Investment is an outflow• Ending value is an inflow• Additional investment is an outflow• Reduced investment (withdraw money) is an

inflow

Dollar-weighted return (IRR)

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Dollar-Weighted ReturnTime

0 1 2 3 4Net cash flow ($m) -1 -0.1 -0.5 0.8 1

Using the definition of the IRR,

2 3 4

2 3 4

0.1 0.5 0.8 11

1 (1 ) (1 ) (1 )0.1 0.5 0.8 1

11 (1 ) (1 ) (1 )

IRR IRR IRR IRR

IRR IRR IRR IRR

+ + = ++ + + +- -

= + + ++ + + +

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Quoting Rates of ReturnAnnual percentage rate, APR

= rate per period X n Where n = no. of compounding periods per year

Effective annual interest rate,

EAR 11

m

m

APR

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Quoting Rates of Return With continuous compounding, the relationship

between EAR and APR becomes

EAR = eAPR – 1‘e’ is the exponential function (that appears on your

financial calculator as [ex])

Equivalently,

APR = Ln(1 + EAR)Ln is the natural log function.

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HPR, APR, EAR problem Suppose you buy a bond of General Electric at a

price of $990. The bond pays coupons semi-annually, has an annual coupon rate of 6%, a face value of $1,000 and will mature in six months’ time. You intend to hold the bond till it matures.

What is the 6-month HPR? What is the APR of this investment? What is the EAR of this investment?

Another example of HPR Suppose you bought a bond of General Electric

at a price of $990 6 months ago. The bond pays coupons semi-annually, has an annual coupon rate of 6%, a face value of $1,000 and will mature in 12 months from today. Today it just paid the coupon and you intend to sell it immediately at current market price. The current YTM is 15%.

What is the current market price? What will be your HPR?

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Describing investment uncertainty: Scenario analysis

Investment is risky simply because we don’t know what will happen in the future for certain. One way of quantifying risk is through scenario analysis.

Scenario analysis: The process of devising a list of possible economic scenarios and specifying: - The likelihood (probability) of each scenario. The HPR that will be realized in each scenario.

The list of possible HPRs with associated probabilities is called the probability distribution of HPRs. This is critical in helping us to evaluate risky investments.

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Probability distribution of HPR The probability distribution provides

information for us to measure the reward and risk of an investment.

Reward of the investment: Expected return Also known as ‘mean return’, ‘mean of the

distribution of HPRs’. Risk of the investment: Variance

Let’s start with a simple scenario analysis.

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Say you want to buy Google’s stock and hold it for a year.

During this coming year, you think there are 3 possible economic scenarios: boom, normal growth, recession.

State of the Economy

Scenario Probability, p(s) HPR (%)

Boom 1 0.25 44

Normal 2 0.50 14

Recession 3 0.25 -16

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Expected Return, E(r) The weighted average of returns in all

possible scenarios, s = 1,2,…S, with weights equal to the probability of that particular scenario.

p(s): probability of scenario sr(s): HPR in scenario s

S

s

srsp

srsprprprE

1

)()(

)()(...)2()2(...)1()1()(

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Expected Return, E(r) With the formula, Google’s expected

return is:

E(r) = (0.25 x 44) + (0.5 x 14) + (0.25 x -16)

= 14%

Probability in each scenario

HPR in each scenario

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Variance, Var(r) When we talk about risk, we often think

of surprises or deviations from what we expect. Variance captures this idea.

Variance: The expected value of squared deviation from the mean.

Also known as σ2 (read as ‘sigma squared’).

S

s

rEsrsprVar1

22 )]()()[()(

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Standard deviation, SD(r)

Standard deviation: Square root of variance

Returning to the Google example,Var(r)

= 0.25(44 – 14)2 + 0.5(14 – 14)2 + 0.25(-16 -14)2

= 450

SD(r)= (450)1/2 = 21.21%

)()( rVarrSD

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How to interpret E(r), Var(r) and SD(r)

The bigger the expected return, the bigger the potential reward from the investment, vice versa.

The bigger the variance, the bigger the risk of the investment, vice versa.

The bigger the standard deviation, the bigger the risk of the investment, vice versa.

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Describing investment performance in the past

If we are interested in the rewards from investing in the past (using historical data), we can use (1) arithmetic average, (2) geometric average.

To quantify risk, use ‘historical’ or ‘sample’ variance: Arithmetic average

No. of periods

n

ii rr

n 1

22 )(1

1

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Example: S&P500 index, 1988-1992

Year HPR(%)

1988 16.9

1989 31.3

1990 -3.2

1991 30.7

1992 7.7

Compute the arithmetic average, geometric average, and variance.

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Example: S&P500 index, 1988-1992

Year (1)HPR(%)

(2) 1+HPR

(3)Deviation from

arithmetic average

(4) Squared deviation

1988 16.9 1.169 16.9 – 16.7 = 0.2 0.04

1989 31.3 1.313 31.3 – 16.7 = 14.6 213.16

1990 -3.2 0.968 -3.2 – 16.7 = -19.9 396.01

1991 30.7 1.307 30.7 – 16.7 = 14 196

1992 7.7 1.077 7.7 – 16.7 = -9 81

Total 83.4 886.21

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Example: S&P500 index, 1988-1992, cont’d

Arithmetic average = 83.4/5 =16.7%

Geometric average =

[(1.169) x (1.313) x (0.968) x (1.307) x (1.077)]1/5 – 1

= 0.15902 or 15.9%

Variance = 886.21/(5 – 1) = 221.6

Standard deviation = (221.6)1/2 =14.9%

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Risk premiums & risk aversion If you don’t want to invest in a risky asset like a

stock, what is the alternative? Risk-free assets like treasury bills.

The return you get is the risk-free rate (rf). Risk-free rate = rate of return that can be earned with

certainty. Risk premium: expected return in excess of the

risk-free rate.

Risk premium = E(r) – rf

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Risk premium depends on risk aversion and variance

Risk aversion: reluctance to accept risk. Risk premium of a portfolio, E(rp) – rf

E(rp) – rf = A x var(rp)

A = measures degree of investor’s risk aversion,

Var(rp) = variance (risk) of the portfolio

Risk premium increases if: Portfolio variance increases OR Risk-aversion, A, increases

Inferred Risk Aversion(price of risk)

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p

fp rrEA

2

)(

Average A in market

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M

fM rrEA

2

)(

Sharpe Ratio Measure the risk-return tradeoff

is the standard deviation

* It works for portfolio only, not individual security.

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p

fpp

rrES

)(

Problem sets after Chapter 5, # 11.

The expected cash flow is: (0.5 x $50,000) + (0.5 x $150,000) = $100,000

With a risk premium of 10%, the required rate of return is 15%. Therefore, if the value of the portfolio is X, then, in order to earn a 15% expected return:

X(1.15) = $100,000 X = $86,957

  If the portfolio is purchased at $86,957, and the expected payoff is $100,000, then the

expected rate of return, E(r), is:

= 0.15 = 15.0%

The portfolio price is set to equate the expected return with the required rate of return.

  If the risk premium over T-bills is now 15%, then the required return is:

5% + 15% = 20%

The value of the portfolio (X) must satisfy:

X(1.20) = $100, 000 X = $83,333

  For a given expected cash flow, portfolios that command greater risk premium must sell at

lower prices. The extra discount from expected value is a penalty for risk.37

957,86$

957,86$000,100$

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Summary Single period: holding-period return (HPR) Many periods: arithmetic average,

geometric average, and dollar-weighted return.

Expected return measures the ‘reward’ from an investment.

Variance (standard deviation) measures the ‘risk’ from an investment.

Practice 1

1.Chapter 5 problem sets : 5,6,7,8.

2. Suppose you bought a bond of BT Co. at a price of $990 6 months ago. The bond pays coupons semi-annually, has an annual coupon rate of 6%, a face value of $1,000 and will mature in 24 months from today. Today it just paid the coupon. The current YTM is 15%. What is the current market price? What is your HPR for the last 6 months? If the interest rate on the market (YTM) does not change for the next 2

years, what will be your HPR for one year if you intend to sell the bond after 6 months receiving the 2nd coupon payment? (assume all coupons do not earn any investment returns)

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Homework 11. Suppose an investor’s risk aversion A=2 and the variance of the return of the portfolio he chose is 0.0260. The risk-free rate is 2%. The value of the portfolio has 0.3 probability to go to $2000 in one year, 0.5 probability to $1500 and 0.2 probability to $1000. What is the maximum price he would pay for this portfolio?

2. Suppose you bought a bond at a price of $990 12 months ago. The bond pays coupons annually, has an annual coupon rate of 6%, a face value of $1,000 and will mature in 36 months. Today it just paid the coupon. The current YTM is 15%. Suppose the YTM will decrease to 10% after 12 months and then remains the same till expiration. What is the arithmetic average annual return for the next two years?

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