Finance 300Financial Markets

Lecture 2

Fall, 2001©

Professor J. Petry

http://www.cba.uiuc.edu/broker/fin300/fin300pp.htm

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Chapter II-Portfolio Theory

1. Measuring Portfolio Risk & Return2. Diversification3. Capital Asset Pricing Model (CAPM)4. Arbitrage Pricing Theory (APT)

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Measuring Portfolio Returns

• Holding Period Rate of Return (HPR)• Cash Flow Adjusted Rate of Return (CFA)• Statistical (arithmetic) Rate of Return• Time-Weighted (geometric) Rate of Return• Internal Rate of Return (IRR)

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Holding Period Rate of Return

V

VHPR0

1

HPR = Holding Period Return

V1 = Ending value

V0 = Beginning value

Used to calculate the total return from an investment over any set time period (day, month, year, 3 weeks). Includes all sources of income (capital gain, dividends).

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ExamplesHPR with V0 of 1,000,000 and V1 of 1,300,000

R = 1,300,000/1,000,000 = 1.30 = 1 + 30%

HPR with V0 of 200,000 and V1 of 134,000

R = 134,000/200,000 = 0.67 = 1 - 33%

HPR with V0 of 2,000,000 and V1 of:

2,124,770 HPR = 1,843,748 HPR = 2,000,000 HPR =

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Cash Flow Adjusted Rate of Return

V cashflowdate

cashflowdate

VR

303030

0

1

V0 = Beginning value V1 = Ending value

date = number of days prior to cash flow timing

Monthly CFA used to correct returns for entrance and exit of investable resources under management.

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ExamplesHPR of 1 month investment with V0 of 2,000,000 and V1 of

2,575,000. R = 1 + 28.75%

And if 500,000 was invested on the last day of the month?

000,000,2

000,075,2

000,000,2 000,500303030

000,5003030

000,575,2

R

R = 1 + 3.75%

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Examples-CFA Rate of ReturnThree investment managers (Ralph, Joe & Madonna) start

the month with 250,000. Each is given 150,000 more to invest (Ralph at the outset, Joe on the 20th and Madonna on the 30th). They ended with 412,500; 410,000, and 410,050 respectively. Who should you invest with?

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Cash Flow Adjusted Rate of Return

V cashflow

cashflowVR5.

5.

0

1

When there are many cash flows spread throughout the month or they are small relative to the value of the portfolio, this formula is simplified to assume the flows occur mid-month.

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Cash Flow Adjusted Rate of Return

Many investments yield dividend payments, which are often reinvested by the portfolio manager. We would NOT consider this a cash flow which should be adjusted in the fashion described by CFA return.

Use the data for Ralph, Joe and Madonna in this simplified formula. How different are your answers from before? Which makes each portfolio manager look better?

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Statistical (arithmetic) Rate of Return

Assumes a constant amount reinvested every period. When looking at the return on an investment without reference to when an investor bought, sold or reinvested money, this is the appropriate method to use. (i.e.starting fresh every period)

Single best forecast for future one-period returnsDoes not consider compoundingUses arithmetic total and mean to perform calculations.

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Examples-Statistical Rate of Return Fred earned the following returns:

Jan 3.4%Feb 5.2%March -3.5%

Fred’s total return = 3.4 + 5.2 - 3.5 = 5.1% for the quarterFred’s average return = 5.1 / 3 = 1.7% per monthFred’s annualized return is 1.7 x 12 = 20.4%

If Billy Bob earned 13.4% in Q1, and -5.0% in Q2, what was his average return, semi-annual return and annualized returns?

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Time-Weighted Rate of ReturnThe time-weighted rate of return assumes that whatever amount you ended with last period, you reinvest at the beginning of this period. This is the method to use when tracking exactly the amount bought, sold, reinvested by a particular client. Referred to as “geometric”. Compounds, but doesn’t consider amount invested. Provides same result as HPR.

Rudolph had returns of 3.3%, -2.5%, 5.0% in 1st quarter.

Total Return =(1+.033)(1-.025)(1+.05)=1+5.753375%

Average Return =

Annualized Return=

%882.1105753375.13 R

%0769.25105753375.112

3 R

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Examples of Time-Weighted R of R

Use the data for BillyBob and Fred and the Time-weighted (geometric) rate of return. How do your answers compare to the statistical return? Why?

Do “Things-To-Do” II-4.

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Returns Using Arithmetic and Geometric Averaging

Arithmeticra = (r1 + r2 + r3 + ... rn) / n

ra = (.10 + .25 - .20 + .25) / 4

= .10 or 10%Geometricrg = {[(1+r1) (1+r2) .... (1+rn)]} 1/n - 1

rg = {[(1.1) (1.25) (.8) (1.25)]} 1/4 - 1

= (1.5150) 1/4 -1 = .0829 = 8.29%

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Dollar Weighted ReturnsInternal Rate of Return (IRR) - the discount rate that results in

present value of the future cash flows being equal to the investment amount.

Approaches rate of return like a capital budgeting problem in corporate finance. – Considers changes in investment– Initial Investment is an outflow– Ending value is considered as an inflow– Additional investment is a negative flow– Reduced investment is a positive flow

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Examples-Dollar Weighted ReturnsYou are given a trust of 100,000 to manage. You must

pay-out $5,000 at the end of each of the next three years. The trust is terminated at a target value of $110,000 at end of year three. Verify that you must earn a constant return of 8.1% to meet the demands of the trust.

3321 081.1

074,110

081.1

5000$

081.1

5000$

081.1

5000$000,100$

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Examples-Dollar Weighted ReturnsWhat rate of return would be required if the payment were 4,000

and the terminal value of the trust was required to be 110,000?

What rate of return would be required if the payments were 3,000 and the terminal value of the trust was required to be 115,000?