Measuring Returns and Risk

1. Chapter 3 Measuring Returns and Risk

2. Measures of Investment Returns Holding Period Return (HPR) and Holding Period Return Relative (HPRR) Per-Period Return (PPR) and Return Relative (PPRR) Expected Return Annualized Return Geometric Mean (GMR) and Arithmetic Mean (AMR) Returns

3. Components of Return Periodic payments: dividends, interest, rents, or royalties Changes in market value: price appreciation or decline

4. Holding Period Return (HPR) Rate of return over some specific time period Holding Period Return Relative (HPRR) End-period value relative to beginning-of-period value for specific holding period holding period return plus one (1)

5. HPR = (Income Received + Change in Value) ? Beginning Value HPRR = (Income Received + Ending Value) ? Beginning Value HPR = HPRR � 1 or 1 + HPR = HPRR

6. Per-Period Return (PPR) Return earned for particular period (for example, annual return) Per-Period Return = (Period�s Income  + Price Change)  ? Beginning Period Value PPRR = 1 + PPR

7. Expected Return Probability Distribution E(return)=P1xR1 + P2xR2 +�+ PnxRn where P1 + P2 + � + Pn = 1

8. Practice Problem An investment has a 50% probability of a 20% rate of return, a 30% probability of a 10% rate of return, and a 20% probability of a -10% rate of return. What is its expected rate of return?

9. A Portfolio�s HPR Same formula as HPR for security, or Weighted average rate of return = W1 x R1 + W2 x R2 + . . . + Wn x Rn where Wi = % of portfolio invested in security i, Ri = the per-period return on security i, and W1 + � + Wn = 1

10. Geometric Mean Return Value of compounded per-period average rate of return of financial asset determined during specified time  HPRR = PPRR1 x PPRR2 x �x PPRRn GMR = HPRR(1 ? n) � 1

11. Practice Problem Over the last three years, an investment has provided rates of return of 20%, 10%, and -10%. What was its GMR?

12. Arithmetic Mean Return Simple average return found by dividing sum of separate per-period returns by number of periods over which they were earned

13. Why Arithmetic Mean Is a Really Bad Measure of Returns Over Time R1 = 100% R2 = �50% Arithmetic average return = +25% Geometric mean return = 0%

14. Relationship between a GMR and an AMR Only when all PPRs are identical will GMR and AMR be the same If PPRs are not identical, then GMR will always be less than the AMR Difference increases as variability among PPRs increases

15. Predicting Future Performance Based on Past Performance If predicting one period in future, use arithmetic mean If predicting T periods in future, where T = number of historical periods, use GMR If predicting H periods in the future, use: Forecast =AMR x (1 � H?T) + GMR x H?T Note: If H = T, forecast = GMR

16. Pure Risk Involves only chance of loss but no chance of gain Speculative Risk Associated with speculation in which there is some chance of gain and some chance of loss

17. Types of Risk Purchasing power (or inflation) risk Interest rate risk Market risk Business (and default) risk Liquidity risk Political (sovereign) risk Exchange rate risk Tax risk Additional commitment Risk Manager Risk

18. Purchasing Power Risk Loss of purchasing power of investment asset�s future cash flows 1 + real rate = (1 + nominal rate) ? (1 + inflation rate)real rate ~ nominal rate � inflation rate Solution: Can�t escape it.

19. Practice Problems If the nominal rate is 8% and the inflation rate is 10%, what is the real rate? If you want a real rate of return of 10% and the inflation rate is 8%, what nominal rate of return must you achieve?

20. Interest Rate Risk for debt securities, risk associated with changes in interest rates; consists of price risk and reinvestment rate risk Price Risk a change in market interest rates produces an opposite change in the value of investments Reinvestment Rate Risk risk as to what interest rate will be when income and/or principal from investments are reinvested Solution: Immunization

21. Market Risk Degree to which asset�s return is affected by events affecting entire market Also called systematic risk Solution: Low betas

22. Business (and Default) Risk Risk from events that cause changes in sales and/or expenses for a specific company Consumer preference Ineffective management Law change Foreign competition Unique for each enterprise Also called nonsystematic risk Solution: Diversification

23. Liquidity Risk Relative inability to convert an asset to cash quickly, at any time, and without any loss of principal Solution: Avoidance

24. Political (Sovereign) Risk Degree to which an investment is subject to events in foreign markets that can cause its value to drop precipitously Includes: effects of trade disputes wars, political unrest tariffs, corruption expropriation Solution: Avoidance

25. Exchange Rate Risk Degree to which investment asset affected by movements in currency exchange rates in country where investment is located Affects investments in some U.S. companies because of overseas markets, production facilities, and raw materials Solution: futures and/or forward contracts. currency options

26. Tax Risk Extent to which investment�s returns are exposed to changes in tax laws Income that is not currently taxable may be taxable later Solution: Avoid investments that may be more prone to tax risk, speculate on political developments

27. Additional Commitment Risk Degree to which investment asset may require the buyer to put additional money into that investment Inability to meet commitment might create loss of value Solution: Avoidance

28. Manager Risk Risk of a portfolio manager performing poorly Solution: Diversification

29. Measures of Risk Semivariance SEMIVAR = S Pt x (min(Ri � E(r), 0)2 Standard Deviation (Square Root of the Variance) Coefficient of Variation CV = s ? E(R) Beta

30. Why Is Risk Important? It is the driver for expected return:

31. Standard Deviation Measure of degree of dispersion of distribution Normal distribution Two times out of three actual value will be within one standard deviation on either side of mean value 19 out of 20 times will be within two standard deviations Standard deviation is square root of variance. 

32. Practice Problem An investment has an expected rate of return of 10% and a standard deviation of 12%. What is the probability of losing more than 2% in the coming period?

33. Computing Variance and Standard Deviation Formula depends on if using Historical data All observations have equal weights Historical = ex post (after the fact) Projected probability distribution Projected = ex ante (before the fact)

34. Historical Computation

35. Keystrokes & Practice Problem Enter data with S+ key. When final rate of return entered, then punch SHIFT, sxsy Practice Problem: Over the last three years, a security has had rates of returns of 20%, 10%, and -10%. What was the variance and standard deviation of returns? What was the mean rate of return?

36. Projected Computation

37. Nonnormal Distributions Skewness Positive Negative Kurtosis Leptokurtic Platykurtic Lognormal

38. Buying on Margin Margin rate: percentage of security purchase that must come from investor�s funds rather than from borrowings Initial margin rate: used when determining cash needed for new purchase or cash withdrawals Maintenance margin rate: used when determining if margin call is needed

39. Buying Power Dollar value of additional securities that can be purchased on margin with current equity in margin account

40. Net Equity Total value of account minus amount of debt outstanding (debit balance) E = MV � LOAN� where MV = market value of portfolio LOAN = current loan balance

41. Practice Problem You have a portfolio worth $100,000, with a debit balance of $34,000. If the initial margin rate is 60%, what is the buying power of the portfolio?

42. Maintenance Margin Minimum percentage of equity that ongoing margin account is required to maintain at all times MV x (1 � MMR) > LOAN where MMR = maintenance margin rate

43. Ways to Satisfy A Margin Call Deposit cash in account Add more collateral (marginable securities) to account. Sell stock from account and use proceeds to reduce the margin debt In each case, result must raise equity percentage to margin maintenance minimum to satisfy margin call.

44. Cash Necessary to Meet a Margin Call Cash added = LOAN � MV x (1 � MMR) Practice Problem: LOAN = $50,000 MV = $60,000 MMR = 30% How much cash should be added to meet a margin call?

45. Permitted Cash Withdrawals Maximum Cash Withdrawal= MV x (1 � IMR) � LOAN Practice Problem: MV = $100,000, IMR = 60%, LOAN = $15,000 What is the maximum amount of cash that could be borrowed from the account?

46. Repayment of Margin Loans Mandatory if receive a margin call No fixed schedule Interest can accrue on interest as long as minimum margin rate requirement is not violated

47. The Impact of Leverage ROA = (Ending Value � Beginning Value) ? Beginning Value ROE = {(Ending Value � Beginnning Value) � Interest Charges} ? Initial Investment

48. Broker Call-Loan Rate Interest rate charged by banks to brokers for loans that brokers use to support their margin loans to customers Usually scaled up for margin loan rate