Solution Manual For Investments, 12th Edition

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CHAPTER 1: THE INVESTMENT ENVIRONMENT1-1CHAPTER 1:THEINVESTMENTENVIRONMENTPROBLEM SETS1.While it is ultimatelytrue that real assets determine the material well-being of aneconomy, financial innovation in the form of bundling and unbundling securitiescreates opportunities for investors to form more efficient portfolios.Bothinstitutional andindividualinvestorscan benefit when financial engineering createsnew products that allow them to manage their portfolios of financial assets moreefficiently. Bundling and unbundling create financial products with new propertiesand sensitivities to various sources of riskthatallowsinvestors toreduce volatilitybyhedgingparticular sources of risk more efficiently.2.Securitization requires access to a large number of potential investors.To attractthese investors, the capital market needs:1.a safe system of business laws and low probability of confiscatorytaxation/regulation;2.a well-developed investment banking industry;3.a well-developed system of brokerage and financial transactions;and4.well-developed media, particularly financial reporting.These characteristics are found in (indeed make for) a well-developed financialmarket.3.Securitization leads to disintermediation; that is, securitization provides a meansfor market participants to bypass intermediaries.For example, mortgage-backedsecurities channel funds to the housing market without requiring that banks orthrift institutions make loans from their own portfolios.Securitization works welland can benefit many, but only if the market for these securities is highly liquid.As securitization progresses,however, andfinancial intermediariesloseopportunities, theymust increase otherrevenue-generatingactivities such asproviding short-term liquidity to consumers and small business and financialservices.4.The existence of efficient capital markets and the liquid trading of financial assetsmakeit easy for large firms to raise the capital needed to finance their investmentsin real assets.IfFord, for example, could not issue stocks or bonds to the generalpublic, it would have a far more difficult time raising capital.Contraction of thesupply of financial assets would make financing more difficult, thereby increasingthe cost of capital.A higher cost of capitalresults inless investment and lowerreal growth.

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CHAPTER 1: THE INVESTMENT ENVIRONMENT1-25.Even if the firm does not need to issue stock in any particular year, the stock marketis still important to the financial manager.The stock price provides importantinformation about how the market values the firm's investment projects.For example,if the stock price rises considerably, managers might conclude that the marketbelieves the firm's future prospects are bright.This might be a useful signal to thefirm to proceed with an investment such as an expansion of the firm's business.In addition, sharesthatcan be traded in the secondary marketare moreattractive toinitialinvestors sincethey know thatthey will be able to sell their shares.This inturn makes investors more willing to buy shares in a primary offering and thusimproves the terms on which firms can raise money in the equity market.Remember that stock exchanges like those in New York, London, and Paris are theheart of capitalism, in which firms can raise capital quickly in primary marketsbecause investors know there are liquid secondary markets.6.a.No.The increase in price did not add tothe productive capacity of theeconomy.b.Yes, the value of the equity held in theseassetshasincreased.c.Future homeownersas a wholeareworse off, sincemortgage liabilities havealso increased.In addition,this housing price bubble will eventually burst andsociety as a whole (and most likelytaxpayers) willsufferthe damage.7.a.The bank loan isa financial liability for Lanni, and a financial asset for the bank.The cash Lanni receives is a financial asset.The new financial asset created isLanni's promissory noteto repay the loan.b.Lanni transfers financial assets (cash) to the software developers. In return,Lannireceives the completed software package, which isa real asset.Nofinancial assets are created or destroyed; cash is simply transferred from one partyto another.c.Lanniexchangesthe real asset (the software) for a financial asset,which is1,250shares of Microsoft stock. If Microsoft issues new shares in order to pay Lanni,then this would represent the creation of new financial assets.d.By selling its shares in Microsoft,Lanni exchanges one financial asset (1,250shares of stock) for another ($125,000in cash). Lanniusesthefinancial assetof$50,000incash torepaythe bank andretire its promissory note. The bank mustreturn its financial asset to Lanni.The loan is "destroyed" in the transaction, since itis retired when paid off and no longer exists.

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CHAPTER 1: THE INVESTMENT ENVIRONMENT1-38.a.AssetsLiabilities &Shareholders’EquityCash$ 70,000Bank loan$ 50,000Computers30,000Shareholders’ equity50,000Total$100,000Total$100,000Ratio of realassetsto total assets = $30,000/$100,000 = 0.30b.AssetsLiabilities &Shareholders’EquitySoftware product*$ 70,000Bank loan$ 50,000Computers30,000Shareholders’ equity50,000Total$100,000Total$100,000*Valued at costRatio of realassetsto total assets = $100,000/$100,000 = 1.0c.AssetsLiabilities &Shareholders’EquityMicrosoft shares$125,000Bank loan$ 50,000Computers30,000Shareholders’ equity105,000Total$155,000Total$155,000Ratio of realassetsto total assets = $30,000/$155,000 = 0.19Conclusion: when the firm starts up and raises working capital, it ischaracterized bya low ratio of realassetsto total assets.When it is in full production, it hasa highratio of real assetsto total assets.When the project "shuts down" and the firm sells itoff for cash, financial assets once again replace real assets.9.a.For commercial banks, the ratio is:$134.3/$17,532.8= 0.0077b.For nonfinancial firms, the ratio is:$23,678/$45,464= 0.5208c.The difference should be expectedprimarily because the bulk of thebusiness of financial institutions is to makeloans and the bulk ofthebusiness ofnon-financial corporations is to invest in equipment,manufacturing plants, and property. The loans are financial assets forfinancial institutions, but the investments of non-financial corporations arereal assets.10.a.Primary-market transactionin which gold certificates are being offered topublic investors for the first time by an underwriting syndicate led by JW KorthCapital.

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CHAPTER 1: THE INVESTMENT ENVIRONMENT1-4b.The certificates are derivative assetsbecause they represent an investment inphysical gold, but each investor receives a certificate and no gold. Note thatinvestors can convert the certificate into gold during the four-year period.11.a.A fixed salary means that compensation is (at least in the short run)independent of the firm's success.This salary structure does not tie the manager’simmediate compensation to the success of the firm, so a manager might not feeltoo compelled to work hard to maximize firm value.However, the managermight view this as the safest compensation structure and therefore value it morehighly.b.A salary that is paid in the form of stock in the firm means that the manager earnsthe most when the shareholders’ wealth is maximized.Five years of vesting helpsalign the interests of the employee with the long-term performance of the firm.Thisstructure is therefore most likely to align the interests of managers and shareholders.If stock compensation is overdone, however, the manager might view it as overlyrisky since the manager’s career is already linked to the firm, and this undiversifiedexposure would be exacerbated with a large stock position in the firm.c.Aprofit-linked salarycreatesgreat incentives for managers to contribute to thefirm’s success.However, a managerwhose salary is tied to short-term profits will berisk seeking, especially if these short-term profits determine salaryor if thecompensation structure doesnotbear thefullcost of theproject’srisks.Shareholders,in contrast, bear the losses as well as the gains on the project and might be lesswilling to assume that risk.12.Even if an individual shareholder could monitor and improve managers’ performanceand thereby increase the value of the firm, the payoff would be small, since theownership share in a large corporation would be very small.For example, if you own$10,000 ofFordstock and can increase the value of the firm by 5%, a very ambitiousgoal, you benefit by only:0.05$10,000 = $500. The cost, both personal andfinancial to an individual investor, is likely to be prohibitive and would typicallyeasily exceed any accrued benefits, in this case $500.In contrast, acreditor, such as abank,that has a multimillion-dollar loan outstandingto the firm has a big stake in making sure that the firm can repay the loan.It is clearlyworthwhile for the bank to spend considerable resources to monitor the firm.13.Mutual funds accept funds from small investors and invest, on behalf of theseinvestors, in thedomesticand international securities markets.Pension funds accept funds and then investin a wide range of financial securities, onbehalf of current and future retirees, thereby channeling funds from one sector of theeconomy to another.

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CHAPTER 1: THE INVESTMENT ENVIRONMENT1-5Venture capital firms pool the funds of private investors and invest in start-up firms.Banks accept deposits from customers and loan those funds to businesses or use thefunds to buy securities of large corporations.14.Treasury bills serve a purpose for investors who prefer a low-risk investment.The lower average rate of return compared to stocks is the price investors payfor predictability of investment performance and portfolio value.15.Witha top-down investing style, you focus on asset allocation or the broadcomposition of the entire portfolio, which is the major determinant of overallperformance.Moreover, top-down management is the natural way to establish aportfolio with a level of risk consistent with your risk tolerance.The disadvantage ofanexclusiveemphasis on top-down issues is that you may forfeit the potential highreturns that could result from identifying and concentrating in undervalued securitiesor sectors of the market.With abottom-up investing style, you try to benefit from identifying undervaluedsecurities.The disadvantage is thatinvestors mighttend to overlook the overallcomposition of your portfolio, which may result in a non-diversified portfolio or aportfolio with a risk level inconsistent withthe appropriatelevel of risk tolerance.Inaddition, this technique tends to require more active management, thus generatingmore transaction costs.Finally,the bottom-upanalysis may be incorrect, in which casethere will be afruitlessly expended effort and money attempting to beat a simple buy-and-hold strategy.16.You should be skeptical.If the author actually knows how to achieve such returns, onemust question why the author would then be so ready to sell the secret to others.Financial markets are very competitive; one of the implications of this fact is thatriches do not come easily.High expected returns require bearing some risk, andobvious bargains are few and far between.Odds arethatthe only one getting rich fromthe book is its author.17.Financialassets provide for a means to acquire real assets as well as an expansionof these real assets.Financial assets provide a measure of liquidity to real assetsand allow for investors tomore effectivelyreduce risk through diversification.18.Allowing traders to share in the profits increases the traders’ willingness toassumerisk.Traders will share in the upside potential directlyin the form ofhigher compensationbut only in the downside indirectlyin the form of potentialjob loss if performance is bad enough.This scenario creates a form of agencyconflict known as moral hazard, in which the owners of the financial institutionshare in both the total profits and losses, while the traders will tend to share moreof the gains than the losses.

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CHAPTER 1: THE INVESTMENT ENVIRONMENT1-619.Answers may vary, however, students should touch on the following: increasedtransparency, regulations to promote capital adequacy by increasing the frequencyof gain or loss settlement, incentives to discourage excessive risk taking, and thepromotion of more accurate and unbiased risk assessment.

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Chapter 2-Asset Classesand FinancialInstruments2-1CHAPTER 2:ASSET CLASSES ANDFINANCIALINSTRUMENTSPROBLEM SETS1.Preferred stock is like long-term debt in that it typically promises a fixed paymenteach year.In this way, it is a perpetuity.Preferred stock is also like long-term debtin that it does not give the holder voting rights in the firm.Preferred stock is like equity in that the firm is under no contractual obligation tomake the preferred stock dividend payments.Failure to make payments does not setoff corporate bankruptcy.With respect to the priority of claims to the assets of thefirm in the event of corporate bankruptcy, preferred stock has a higher priority thancommon equity but a lower priority than bonds.2.Money market securities are calledcash equivalentsbecause of theirhigh levelofliquidity.The prices of money market securities are very stable, and they canbe converted to cash (i.e., sold) on very short notice and with very lowtransaction costs.Examples of money market securities include Treasury bills,commercial paper, and banker's acceptances, each of which is highly marketableand traded in the secondary market.3.(a)A repurchase agreement is an agreement whereby the seller of a securityagrees to “repurchase” it from the buyer on an agreed upon date at an agreedupon price.Repos are typically used by securities dealers as a means forobtaining funds to purchase securities.4.Spreads between risky commercial paper and risk-free government securitieswill widen.Deterioration of the economy increasesthe likelihood of defaultoncommercial paper, making them more risky.Investors will demand a greaterpremium onall riskydebt securities, not just commercial paper.5.Corp. BondsPreferred StockCommon StockVotingrights (typically)YescontractualobligationYesPerpetualpaymentsYesYesAccumulateddividendsYesFixedpayments (typically)YesYesPaymentpreferenceFirstSecondThird

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Chapter 2-Asset Classesand FinancialInstruments2-26.Municipalbond interest is tax-exemptat the federal level and possibly at thestate level as well.When facing higher marginal tax rates, a high-incomeinvestor would be more inclined toinvest intax-exempt securities.7.a.You would have to pay the askprice of:101.9297% of parvalue of $1,000= $1,019.297b.The coupon rate is3.000%;implying coupon payments of $30.00annuallyor, more precisely $15.00(semiannually).c.The yield to maturity on a fixed income security is also known as its requiredreturn and is reported byThe Wall Street Journaland others in the financialpress as the ask yield. In this case, the yield to maturity is 2.902%. An investorbuying this security today and holding it until it matures will earn an annualreturn of 2.902%.Students will learn in a later chapter how to compute boththe price and the yield to maturity with a financial calculator.8.Treasury bills are discount securities that mature for $10,000. Therefore, a specific T-bill price is simply the maturity value divided by one plus the semi-annual return:P= $10,000/1.02 = $9,803.929.The total before-tax income is $4.After the50% exclusion for preferred stockdividends, the taxable income is: 0.50$4 = $2.00Therefore, taxes are: 0.30$2.00 = $0.60After-tax income is: $4.00–$0.60= $3.40Rate of return is: $3.40/$40.00 =8.50%10.a.You could buy: $5,000/$57.94=86.30shares. Since it is not possible to tradein fractions of shares, you could buy86shares ofHerbalife.b.Your annual dividend income would be:86$1.20=$103.20c.The price-to-earnings ratio is47.75and the price is $57.94.Therefore:$57.94/Earnings per share =47.75Earnings per share =$1.21d.Herbalifeclosed today at $57.94, which was $1.39lowerthan yesterday’spriceof$59.33.

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Chapter 2-Asset Classesand FinancialInstruments2-311.a.Att= 0, the value of the index is: (90 + 50 + 100)/3 = 80Att= 1, the value of the index is: (95 + 45 + 110)/3 = 83.333The rate ofreturn is: (83.333/80)−1 = 4.17%b.In the absence of a split, Stock C would sell for 110, so the value of theindex would be:(95+45+110)/3 =250/3 = 83.333with a divisor of 3.After the split,stock C sells for 55.Therefore, we need to find the divisor(d) such that:83.333 = (95 + 45 + 55)/dd = 2.340. The divisor fell,which is always the case after one of the firms in an index splits itsshares.c.The return is zero.The index remains unchanged because the return foreach stock separately equals zero.12.a.Total market value att= 0 is: ($9,000 + $10,000 + $20,000) = $39,000Total market value att= 1 is: ($9,500 + $9,000 + $22,000) = $40,500Rate ofreturn = ($40,500/$39,000)–1 = 3.85%b.The return on each stock is as follows:rA= (95/90)–1 = 0.0556rB= (45/50)–1 =–0.10rC= (110/100)–1 = 0.10The equallyweighted average is:[0.0556 + (-0.10) + 0.10]/3 = 0.0185 = 1.85%13.The after-tax yield on the corporate bonds is: 0.09(1–0.30) = 0.063 = 6.30%Therefore, municipals must offera yield to maturity ofat least 6.30%.14.Equation (2.2) shows that the equivalent taxable yield is:r=rm/(1–t), so simplysubstitute each tax rate in the denominator to obtain the following:a.4.00%b.4.44%c.5.00%d.5.71%

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Chapter 2-Asset Classesand FinancialInstruments2-415.In an equallyweighted index fund, each stock is given equal weight regardless of itsmarket capitalization.Smaller cap stocks willhavethe same weight as larger capstocks.The challenges are as follows:•Givenequalweights placed to smaller cap and larger cap, equal-weighted indices (EWI)willtend to be more volatile than their market-capitalizationcounterparts;•It follows thatEWIsare not good reflectorsof the broad marketthattheyrepresent;EWIs underplaythe economic importance of largercompanies.•Turnover rates will tend to be higher, asanEWI must be rebalancedback to its original target.By design, many of the transactions would beamong the smaller, less-liquid stocks.16.a.The ten-year Treasury bond with thehigher couponrate will sell for a higherprice because its bondholder receives higher interest payments.b.The calloptionwith the lower exercise pricehas more value than one with ahigher exercise price.c.The putoption writtenon the lower priced stockhas more value than onewritten on a higher priced stock.17.a.You bought the contract when the futures price was$3.96(seeTable2.8).The contract closes at a price of$4.06, which is$0.10morethan theoriginal futures price.The contract multiplier is5000.Therefore, thegainwillbe:$0.085000 = $400.0018.a.The call option gives you the right, but not the obligation to buy at $100;thestock is trading in the secondary market at $103.Since the stock priceexceeds the exercise price, you exercise thecall.The payoff on the option will be:$103-$100= $3Thecostwasoriginally$3.81, so the profit is:$3-$3.81=-$.81b.Since the stock price is greater than the exercise price, you will exercise the call.The payoff on the option will be: $103-$95= $8The option originally cost $7.65,so the profit is $8-$7.65=$.35.c.Owning the put option gives you the right, but not the obligation, to sell at $105, butyou could sell in the secondary market for $03if you exercise the call the payoff onthe option will be: $105-$103= $2.The option originally cost $4.79, so the profit is $2.00-$4.79=-$2.79.

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Chapter 2-Asset Classesand FinancialInstruments2-519.There is always a possibility that the option will be in-the-money at some time prior toexpiration.Investors will pay something for this possibility of a positive payoff.20.Value ofCall atExpirationInitial CostProfita.04-4b.04-4c.04-4d.541e.1046Value ofPut atExpirationInitial CostProfita.1064b.56-1c.06-6d.06-6e.06-621.A put option conveys therightto sell the underlying asset at the exercise price.Ashort position in a futures contract carries anobligationto sell the underlying assetat the futures price.Both positions, however, benefit if the price of the underlyingasset falls.22.A call option conveys therightto buy the underlying asset at the exercise price.Along position in a futures contract carries anobligationto buy the underlying assetat the futures price.Both positions, however, benefit if the price of the underlyingasset rises.CFAPROBLEMS1.(d)There are tax advantages for corporations that own preferred shares.2.The equivalent taxable yield is: 6.75%/(1−0.34) = 10.23%3.(a)Writing a call entails unlimited potential losses as the stock price rises.4.a.The taxable bond.With a zero tax bracket, the after-tax yield for the

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Chapter 2-Asset Classesand FinancialInstruments2-6taxable bond is the same as the before-tax yield (5%), which is greater thanthe yield on the municipal bond.b.The taxable bond.The after-tax yield for the taxable bond is:0.05(1–0.10) = 4.5%c.You are indifferent.The after-tax yield for the taxable bond is:0.05(1–0.20) = 4.0%The after-tax yield is the same as that of the municipal bond.d.The municipal bond offers the higher after-tax yield for investors in taxbrackets above 20%.5.If the after-tax yields are equal, then: 0.056 = 0.08×(1–t)This implies thatt= 0.30 =30%.

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CHAPTER 3: HOW SECURITIES ARE TRADED3-1CHAPTER 3:HOWSECURITIESARE TRADEDPROBLEM SETS1.Limit buy order: an order that purchases stock if the pricefalls below apredetermined level. Limit sell order:sells stock when the price rises above apredetermined level.Limite orders are not guaranteed to execute since the pricemay not reach the trigger point.Market order:either a buy or sell order that isexecuted immediately at the current market price2.In response to the potential negative reaction to large [block] trades, trades will be splitup into many small trades, effectively hiding the total number of shares bought or sold.3.The use of leverage necessarily magnifies returns to investors.Leveragingborrowed money allows for greater return on investment if the stock price increases.However, if the stock price declines, theinvestor must repay the loan, regardless ofhow far the stock price drops,and incur a negative rate of return.For example, if aninvestor buys an asset at $100 and the price rises to $110, the investor earns 10%.If an investor takes out a $40 loan at 5% and buys the same stock, the return will be13.3%, computed as follows: $10 capital gain minus $2 interest expense divided bythe $60 original investment. Of course, if the stock price falls below $100, thenegative return will be greater for the leveraged account.4.a. False:An investor who wishes to sell shares immediately should ask his or herbroker to enter amarketorder.b. False: The ask price isgreaterthan the bid price. (note: the opposite is true foryields)c. False: An issue of additional shares of stock to the public by Microsoft would becalledseasoned offering.d.True5.(a)A broker market consists of intermediaries who have the discretion to tradefor their clients. A large block trade in an illiquid security would most likelytrade in this market as the brokers would have the best access to clientsinterested in this type of security.The advantage of anelectroniccommunicationnetwork(ECN) is that it canexecute large block orders without affecting the public quote.Since thissecurity is illiquid, large block orders are less likely to occur and thus it wouldnot likely trade through an ECN.

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CHAPTER 3: HOW SECURITIES ARE TRADED3-2Electroniclimit-order markets(ELOM) transact securities with high tradingvolume.This illiquid security is unlikely to be traded on an ELOM.6.a.The stock is purchased for:300$40 = $12,000The amount borrowed is $4,000.Therefore, the investor put up equity, ormargin, of $8,000.b.If the share price falls to $30, then the value of the stock falls to $9,000.Bythe end of the year, the amount of the loan owed to the broker grows to:$4,0001.08 = $4,320Therefore, the remaining margin in the investor’s account is:$9,000−$4,320= $4,680c.The percentage margin is now: $4,680/$9,000= 0.52,or52%> 30%.Therefore, the investor will not receive a margin call.d.Using an end price of $30, the rate of return on the investment over the yearis:(Ending equity in the account−Initial equity)/Initial equity= ($4,680−$8,000)/$8,000 =−0.415,or−41.5%Alternatively, divide the initial equity investments into the change in valueplus the interest payment:($3,000 loss + $320 interest)/$8,000 =-0.415.7.a.The initial margin was: 0.501,000$40 = $20,000As a result of the increase in the stock priceOld Economy Traders loses:$101,000 = $10,000Therefore,margin decreases by $10,000.Moreover, Old Economy Tradersmust pay the dividend of $2 per share to the lender of the shares, so that themargin in the account decreases by an additional $2,000.Therefore, theremaining margin is:$20,000–$10,000–$2,000 = $8,000b.The percentage margin is: $8,000/$50,000 = 0.16,or16%So there will be a margin call.

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CHAPTER 3: HOW SECURITIES ARE TRADED3-3c.The equity in the account decreased from $20,000 to $8,000 in one year, for arate of return of:(−$12,000/$20,000)=−0.60,or−60%8.a.The buy orderfor FinTradewill be filled at the best limit-sell order price:$50.25b.The next market buy order will be filled at the next-best limit-sellorder price: $51.50c.You would want to increase your inventory. There is considerable buyingdemand at prices just below $50, indicating that downside risk is limited.Incontrast, limit sell orders are sparse, indicating that a moderate buy order couldresult in a substantial price increase.9.a.You buy 200 shares of Telecom for $10,000.These shares increase in value by10%, or $1,000.You pay interest of: 0.08$5,000 = $400The rate of return will be:$1, 000$4000.1212%$5, 000−==b.The value of the 200 shares is 200P.Equity is (200P–$5,000).You willreceive a margin call when:PP200000,5$200−= 0.30whenP= $35.71 or lower10.a.Initial margin is 50% of $5,000,or $2,500.b.Total assets are $7,500 ($5,000 from the sale of the stock and $2,500 put up formargin).Liabilities are 100P.Therefore,equityis ($7,500–100P).A margincall will be issued when:PP100100500,7$−= 0.30whenP= $57.69 or higher11.The total cost of the purchase is: $201,000= $20,000You borrow $5,000 from your broker and invest $15,000 of your own funds.Your margin account starts out with equity of $15,000.a.(i)Equity increases to: ($221,000)–$5,000 = $17,000

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CHAPTER 3: HOW SECURITIES ARE TRADED3-4Percentage gain = $2,000/$15,000 = 0.1333,or13.33%(ii)With price unchanged, equity is unchanged.Percentage gain = zero(iii)Equity falls to ($181,000)–$5,000 = $13,000Percentage gain = (–$2,000/$15,000) =–0.1333,or–13.33%The relationship between the percentage return and the percentage change inthe price of the stock is given by:% return = % change in priceequityinitialsInvestor'investmentTotal= % change in price1.333For example, when the stock price rises from $20to $22, the percentage change inprice is 10%, while the percentage gain for the investor is:% return = 10%000,15$000,20$= 13.33%b.The value of the1,000shares is1,000P.Equity is (1,000P–$5,000).You willreceive a margin call when:PP000,1000,5$000,1−= 0.25whenP= $6.67or lowerc.The value of the1,000 shares is1,000P.But now you have borrowed $10,000instead of $5,000.Therefore, equity is (1,000P–$10,000).You will receive amargin call when:PP000,1000,10$000,1−= 0.25whenP= $13.33or lowerWith less equity in the account, you are far more vulnerable to a margin call.e.By the end of the year, the amount of the loan owed to the broker grows to:$5,0001.08 = $5,400The equity in your account is (1,000P–$5,400).Initial equity was $15,000.Therefore, your rate of return after one year is as follows:(i)000,15$000,15$400,5$)22$000,1(−−= 0.1067,or10.67%(ii)000,15$000,15$400,5$)20$000,1(−−=–0.0267,or–2.67%(iii)000,15$000,15$400,5$)18$000,1(−−=–0.1600,or–16.00%

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CHAPTER 3: HOW SECURITIES ARE TRADED3-5The relationship between the percentage return and the percentage change inthe price ofXtelis given by:% return =equityinitialsInvestor'investmentTotalpriceinchange%−equityinitialsInvestor'borrowedFunds%8For example, when the stock price rises from $40 to $44, the percentagechange in price is 10%, while the percentage gain for the investor is:000,15$000,20$%10−000,15$000,5$%8=10.67%e.The value of the1000shares is1,000P.Equity is (1,000P–$5,400).You willreceive a margin call when:PP000,1400,5$000,1−= 0.25when P = $7.20or lower12.a.The gain or loss on the short position is: (–1,000ΔP)Invested funds = $15,000Therefore: rate of return = (–1,000ΔP)/15,000The rate of return in each of the three scenarios is:(i)Rate of return = (–1,000$2)/$15,000 =–0.1333,or–13.33%(ii)Rate of return = (–1,000$0)/$15,000 = 0%(iii)Rate of return = [–1,000(–$2)]/$15,000 = +0.1333,or+13.33%b.Totalassets in the margin account equal:$20,000 (from the sale of the stock) + $15,000 (the initial margin) = $35,000Liabilities are 500P.You will receive a margin call when:PP000,1000,1000,35$−= 0.25whenP= $28or higherc.With a $1 dividend, the short position must now pay on the borrowed shares:($1/share1000shares) = $1000.Rate of return is now:[(–1,000ΔP)–1,000]/15,000(i)Rate of return = [(–1,000$2)–$1,000]/$15,000 =–0.2000,or–20.00%(ii)Rate of return = [(–1,000$0)–$1,000]/$15,000 =–0.0667,or–6.67%

Solution Manual For Investments, 12th Edition - Page 19 preview image

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CHAPTER 3: HOW SECURITIES ARE TRADED3-6(iii)Rate of return = [(–1,000)(–$2)–$1,000]/$15,000 = +0.067,or+6.67%Totalassets are $35,000, and liabilities are (1,000P+1,000).A margin call willbe issued when:PP000,1000,1000,1000,35−−= 0.25when P = $27.2or higher13.The broker is instructed to attempt to sell yourMarabel, Inc.stock as soon as theMarabel, Inc.stock trades at a bid price of $70or less.Here, the broker will attemptto execute but may not be able to sell at $70, since the bid price is now $69.95.Theprice at which you sell may be more or less than $70because the stop-loss becomesa market order to sell at current market prices.14.a.$55.50b.$55.25c.The trade will not be executed because the bid price is lower than the pricespecified in thelimit-sell order.d.The trade will not be executed because the asked price is greater than the pricespecified in thelimit-buy order.15.a.You will not receive a margin call.You borrowed $20,000 and with another$20,000 of your own equity you bought 1,000 shares ofIxnayat $40 pershare.At $35 per share, the market value of the stock is $35,000, your equityis $15,000, and the percentage margin is: $15,000/$35,000 = 42.9%Your percentage margin exceeds the required maintenance margin.b.You will receive a margin call when:PP000,1000,20$000,1−= 0.35whenP= $30.77 or lower16.The proceeds from the short sale (net of commission) were: ($21100)–$50 = $2,050A dividend payment of $200 was withdrawn from the account.Covering the short sale at $15per share costs(with commission):$1,500 + $50 =$1,550Therefore, the value of your account is equal to the net profit on the transaction:$2,050–$200–$1,550 = $300

Solution Manual For Investments, 12th Edition - Page 20 preview image

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CHAPTER 3: HOW SECURITIES ARE TRADED3-7te that your profit ($300) equals (100 sharesprofit per share of $3). Your netproceeds per share were:$21selling price of stock–$15repurchase price of stock–$ 2dividend per share–$ 12 trades$0.50 commission per share$ 3CFAPROBLEMS1.a.In addition to the explicit fees of $70,000, FBN appears to have paid animplicit price in underpricing of the IPO.The underpricing is $3 per share, ora total of $300,000, implying total costs of $370,000.b.No.The underwriters do not capture the part of the costs correspondingto the underpricing.The underpricing may be a rational marketingstrategy.Without it, the underwriters would need to spend more resourcesin order to place the issue with the public.The underwriters would thenneed to charge higher explicit fees to the issuing firm.The issuing firmmay be just as well off paying the implicit issuance cost represented bythe underpricing.2.(d)The broker will sell, at current market price, after the first transaction at$55 or less.

Solution Manual For Investments, 12th Edition - Page 21 preview image

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CHAPTER 4: MUTUAL FUNDS ANDOTHER INVESTMENT COMPANIES4-1CHAPTER 4: MUTUAL FUNDS ANDOTHER INVESTMENT COMPANIESPROBLEM SETS1.The unit investment trust should have lower operating expenses.Because theinvestment trust portfolio is fixed once the trust is established, it does not have topay portfolio managers to constantly monitor and rebalance the portfolio asperceived needs or opportunities change.Because the portfolio is fixed, the unitinvestment trust also incurs virtually no trading costs.2.a.Unit investment trusts:Diversification from large-scale investing, lowertransaction costs associated with large-scale trading, low management fees,predictable portfolio composition, guaranteed low portfolio turnover rate.b.Open-end mutualfunds:Diversification from large-scale investing, lowertransaction costs associated with large-scale trading, professional managementthat may be able to take advantage of buy or sell opportunities as they arise,record keeping.c.Individual stocks and bonds: No management fee;ability to coordinaterealization of capital gains or losses with investors’ personal tax situations;capability of designingportfolio to investor’s specific riskand returnprofile.3.Open-end funds are obligated to redeem investor's shares at net asset value and thusmust keep cash or cash-equivalent securities on hand in order to meet potentialredemptions.Closed-end funds do not need the cash reserves because there are noredemptions for closed-end funds.Investors in closed-end funds sell their shareswhen they wish to cash out.4.Balanced funds keep relatively stable proportions of funds invested in each assetclass.They are meant as convenient instruments to provide participation in a rangeof asset classes.Life-cycle funds are balanced funds whose asset mix generallydepends on the age of the investor.Aggressive life-cycle funds, with largerinvestments in equities, are marketed to younger investors, while conservative life-cycle funds, with larger investments in fixed-income securities, are designed forolder investors.Asset allocation funds, in contrast, may vary the proportionsinvested in each asset class by large amounts as predictions of relative performanceacross classes vary.Asset allocation funds therefore engage in more aggressivemarket timing.

Solution Manual For Investments, 12th Edition - Page 22 preview image

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CHAPTER 4: MUTUAL FUNDS ANDOTHER INVESTMENT COMPANIES4-25.Unlike an open-end fund, in which underlying shares are redeemed when the fundis redeemed, a closed-end fund trades as a security in the market.Thus, their pricesmay differ from the NAV.6.Advantages of an ETF over a mutual fund:•ETFs are continuously traded and can be sold or purchased on margin.•There are nocapitalgainstax triggers when an ETF is sold (shares are justsold from one investor to another).•Investors buy frombrokers, thus eliminating the cost of direct marketingto individual small investors.This implies lower management fees.Disadvantagesof an ETFover a mutual fund:•Prices can depart from NAV (unlike an open-end fund).•There isabroker fee when buying and selling (unlike a no-load fund).7.The offering price includes a 6% front-end load, or sales commission, meaning thatevery dollar paid results in only $0.94 going toward purchase of shares.Therefore:Offering price =06.0170.10$Load1NAV−=−= $11.388.NAV =Offering price(1–Load) = $12.30.95 = $11.699.StockValueHeld byFundA$ 7,000,000B12,000,000C8,000,000D15,000,000Total$42,000,000Net asset value =000,000,4000,30$000,000,42$−= $10.4910.Value of stocks sold and replaced = $15,000,000Turnover rate =000,000,42$000,000,15$= 0.357,or35.7%

Solution Manual For Investments, 12th Edition - Page 23 preview image

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CHAPTER 4: MUTUAL FUNDS ANDOTHER INVESTMENT COMPANIES4-311.a.40.39$000,000,5000,000,3$000,000,200$NAV=−=b.Premium (or discount) =NAVNAVicePr−=40.39$40.39$36$−=–0.086,or-8.6%The fund sells at an 8.6% discount from NAV.12.100NAVNAVDistributions$12.10$12.50$1.500.088, or 8.8%NAV$12.50−+−+==13.a.Start-of-year price:P0= $12.00×1.02 = $12.24End-of-year price:P1= $12.10×0.93 = $11.25Although NAV increased by $0.10, the price of the fund decreased by$0.99.Rate of return =100Distributions$11.25$12.24$1.500.042, or 4.2%$12.24PPP−+−+==b.An investor holding the samesecuritiesas thefundmanager would haveearned a rate of return based on the increase in the NAV of the portfolio:100NAVNAVDistributions$12.10$12.00$1.500.133, or 13.3%NAV$12.00−+−+==14.a.Empirical research indicates that past performance of mutual funds is nothighly predictive of future performance, especially for better-performingfunds.While theremaybe some tendency for the fund to be an above averageperformer next year, it is unlikely to once again be a top 10% performer.b.On the other hand, the evidence is more suggestive of a tendency for poorperformance to persist.This tendency is probably related to fund costs andturnover rates.Thus if the fund is among the poorest performers, investorsshould be concerned that the poor performance will persist.15.NAV0= $200,000,000/10,000,000 = $20Dividends per share = $2,000,000/10,000,000 = $0.20NAV1is based on the 8% price gain, less the 1% 12b-1 fee:NAV1= $201.08(1–0.01) = $21.384Rate of return =20$20.0$20$384.21$+−= 0.0792,or7.92%

Solution Manual For Investments, 12th Edition - Page 24 preview image

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CHAPTER 4: MUTUAL FUNDS ANDOTHER INVESTMENT COMPANIES4-416.The excess of purchases over sales must be due to new inflows into the fund.Therefore, $400 million of stock previously held by the fund was replaced by newholdings.So turnover is: $400/$2,200 = 0.182,or18.2%.17.Fees paid to investment managers were: 0.007$2.2 billion = $15.4 millionSince the total expense ratio was 1.1% and the management fee was 0.7%, weconclude that 0.4% must be for other expenses.Therefore, other administrativeexpenses were: 0.004$2.2 billion = $8.8 million.18.As an initial approximation, your return equals the return on the shares minus thetotal of the expense ratio and purchase costs: 12%−1.2%−4% = 6.8%.But the precise return is less than this because the 4% load is paid up front, not atthe end of the year.To purchase the shares, you would have had to invest: $20,000/(1−0.04) = $20,833.The shares increase in value from $20,000 to: $20,000(1.12−0.012) = $22,160.The rate of return is: ($22,160−$20,833)/$20,833 = 6.37%.19.Assume $1,000 investmentLoaded-Up FundEconomy FundYearlygrowth(ris 6%)(1.01.0075)r+−−(.98)(1.0025)r+−t=1year$1,042.50$1,036.35t=3years$1,133.00$1,158.96t=10years$1,516.21$1,714.0820.a.$450,000,000$10,000000$1044,000,000−=b. The redemption of 1 million shares will most likely trigger capital gains taxeswhich will lower the remaining portfolio by an amount greater than $10,000,000(implying a remaining total value less than $440,000,000).The outstanding sharesfall to 43 million andthe NAV drops to below $10.21.Suppose you have $1,000 to invest.The initial investment in Class A shares is$940( = $1000 × [1-.06])net of the front-end load.After four years, your portfolio willbe worth:$940(1.10)4= $1,376.25Class B shares allow you to invest the full $1,000, but your investment performancenet of 12b-1 fees will be only 9.5%, and you will pay a 1% back-end load fee if yousell after four years.Your portfolio value after four years will be:$1,000(1.095)4= $1,437.66

Solution Manual For Investments, 12th Edition - Page 25 preview image

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CHAPTER 4: MUTUAL FUNDS ANDOTHER INVESTMENT COMPANIES4-5After paying the back-end load fee, your portfolio value will be:$1,437.66.99 = $1,423.28Class B shares are the better choice if your horizon is four years.With a15-year horizon, the Class A shares will be worth:$940(1.10)15= $3,926.61For the Class B shares, there is no back-end load in this case since the horizon isgreater than five years.Therefore, the value of the Class B shares will be:$1,000(1.095)15= $3,901.32At this longer horizon, Class B shares are no longer the better choice.The effect ofClass B's 0.5% 12b-1 fees accumulates over time and finally overwhelms the 6%load charged to Class A investors.22.a.After two years, each dollar invested in a fund with a 4% load and a portfolioreturn equal torwill grow to: $0.96(1 +r–0.005)2.Each dollar invested in the bank CD will grow to: $11.062.If the mutual fund is to be the better investment, then the portfolio return (r)must satisfy:0.96(1 +r–0.005)2> 1.0620.96(1 +r–0.005)2> 1.1236(1 +r–0.005)2> 1.17041 +r–0.005 > 1.08191 +r> 1.0869Therefore:r> 0.0869 = 8.69%b.If you invest for six years, then the portfolio return must satisfy:0.96(1 +r–0.005)6> 1.066= 1.4185(1 +r–0.005)6> 1.47761 +r–0.005 > 1.0672r> 7.22%The cutoff rate of return is lower for the six-year investment because the“fixed cost” (the one-time front-end load) is spread over a greater number ofyears.

Solution Manual For Investments, 12th Edition - Page 26 preview image

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CHAPTER 4: MUTUAL FUNDS ANDOTHER INVESTMENT COMPANIES4-6c.With a 12b-1 fee instead of a front-end load, the portfolio must earn a rate ofreturn (r) that satisfies:1 +r–0.005–0.0075 > 1.06In this case,rmust exceed 7.25% regardless of the investment horizon.23.The turnover rate is 50%.This means that, on average, 50% of the portfolio is soldand replaced with other securities each year.Trading costs on the sell orders are0.4% and the buy orders to replace those securities entail another 0.4% in tradingcosts.Total trading costs will reduce portfolio returns by: 20.4%0.50 = 0.4%24.For the bond fund, the fraction of portfolio income given up to fees is:%0.4%6.0= 0.150,or15.0%For the equity fund, the fraction of investment earnings given up to fees is:%0.12%6.0= 0.050,or5.0%Fees are a much higher fraction of expected earnings for the bond fund andtherefore may be a more important factor in selecting the bond fund.This may help to explain why unmanaged unit investment trusts are concentrated inthe fixed income market.The advantages of unit investment trusts are low turnover,low trading costs,andlowmanagement fees.This is a more important concern tobond-market investors.25.Suppose that finishing in the top half of all portfolio managers is purely luck, andthat the probability of doing so in any year is exactly ½.Then the probability thatany particular manager would finish in the top half of the sample five years in a rowis (½)5= 1/32.We would then expect to find that [350(1/32)] = 11 managersfinish in the top half for each of the five consecutive years.This is precisely whatwe found.Thus, we should not conclude that the consistent performance after fiveyears is proof of skill.We would expect to find11managers exhibiting preciselythis level of "consistency" even if performance is due solely to luck.

Solution Manual For Investments, 12th Edition - Page 27 preview image

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CHAPTER 5: INTRODUCTION TO RISK, RETURN, ANDTHE HISTORICAL RECORD5-1CHAPTER 5: INTRODUCTION TO RISK, RETURN, ANDTHE HISTORICAL RECORDPROBLEM SETS1.The Fisher equation predicts that the nominal rate will equalthe equilibriumreal rate plus theexpected inflationrate.Hence, ifthe inflation rate increasesfrom 3% to 5%whilethere is no change in the real rate,thenthe nominal ratewill increase by 2%.On the other hand, it is possible that an increaseintheexpected inflationratewouldbe accompanied by a change in thereal rateofinterest.While it is conceivable that the nominal interest rate could remainconstant as the inflation rate increased,implying that the real rate decreasedas inflation increased, this is not a likely scenario.2.Ifweassumethatthe distribution of returnsremainsreasonably stable overthe entire history, then a longer sample period(i.e., a larger sample)increasesthe precisionof the estimate of the expected rate of return; this is aconsequence of the fact thatthe standard errordecreasesas the sample sizeincreases.However,if weassumethatthe mean of the distributionof returnsis changingover timebutwe arenot in a position to determinethe nature ofthis change, then theexpected returnmust be estimated from a more recentpart of thehistorical period.In this scenario,we must determinehow farback, historically,togo in selecting the relevant sample.Here, it is likely tobe disadvantageous to use the entire dataset back to 1880.3.NominalRealReal11.4511.115411.54%11.30rrri+++===→=++4.For the money market fund, your holding-period return for the next yeardepends on the level of 30-day interest rates each month when the fund rollsover maturing securities.The one-year savings deposit offers a5% holdingperiod return for the year.If you forecast that the rate on money marketinstruments will increase significantly above the current3% yield, then themoney market fund might result in a higher HPR than the savings deposit.The 20-year Treasury bond offers a yield tomaturity of5% per year, which is100 basis points higher than the rate on the one-year savings deposit;however, you could earn a one-year HPR much less than4% on the bond iflong-term interest rates increase during the year.If Treasury bond yields riseabove5%, then the price of the bond will fall, and the resulting capital losswill wipe out some or all of the5% return you would have earned if bondyields had remained unchanged over the course of the year.

Solution Manual For Investments, 12th Edition - Page 28 preview image

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CHAPTER 5: INTRODUCTION TO RISK, RETURN, ANDTHE HISTORICAL RECORD5-25.a.If businesses reduce their capital spending, then they are likely todecrease their demand for funds.This will shift the demand curve inFigure 5.1 to the left and reduce the equilibrium real rate of interest.b.Increased household saving will shift the supply of funds curve to theright and cause real interest rates to fall.c.Open market purchases of U.S. Treasury securities by the FederalReserve Boardareequivalent to an increase in the supply of funds (ashift of the supply curve to the right).The FED buys treasuries withcash from its own account or it issues certificates which trade likecash. As a result, there is an increase in the money supply, and theequilibrium real rate of interest will fall.6.a.The “Inflation-Plus” CD is the safer investment because it guarantees thepurchasing power of the investment.Using the approximation that the realrate equals the nominal rate minus the inflation rate, the CD provides a realrate of1.5% regardless of the inflation rate.b.The expected return depends on the expected rate of inflation over the nextyear.If the expected rate of inflation is less than3.5% then the conventionalCD offers a higher real return than theinflation-plus CD; if the expected rateof inflation is greater than 3.5%, then the opposite is true.c.If you expect the rate of inflation to be 3% over the next year, then theconventional CD offers you an expected real rate of return of2%, which is0.5% higher than the real rate on the inflation-protected CD.But unless youknow that inflation will be 3% with certainty, the conventional CD is alsoriskier.The question of which is the better investment then depends on yourattitude towards risk versus return.You might choose to diversify and investpart of your funds in each.d.No.We cannot assume that the entire difference between the risk-freenominal rate (on conventional CDs) of5% and the real risk-free rate (oninflation-protected CDs) of1.5% is the expected rate of inflation.Part of thedifference is probably a risk premium associated with the uncertaintysurrounding the real rate of return on the conventional CDs.This impliesthat the expected rate of inflation is less than 3.5% per year.7.E(r) = [0.35×44.5%] + [0.30×14.0%] + [0.35×(–16.5%)] = 14%2= [0.35×(44.5–14)2] +[0.30×(14–14)2] + [0.35×(–16.5–14)2] = 651.175= 25.52%The mean is unchanged, but the standard deviation has increased, as theprobabilities of the high and low returns have increased.

Solution Manual For Investments, 12th Edition - Page 29 preview image

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CHAPTER 5: INTRODUCTION TO RISK, RETURN, ANDTHE HISTORICAL RECORD5-38.Probability distribution of price and one-year holding period return for a 30-year U.S. Treasury bond (which will have 29 years to maturity atyear-end):EconomyProbabilityYTMPriceCapitalGainCouponInterestHPRBoom0.2011.0%$ 74.05−$25.95$8.00−17.95%Normalgrowth0.508.0100.000.008.008.00Recession0.307.0112.2812.288.0020.289.E(q) = (0 × 0.25) + (1×0.25)+(2×0.50)= 1.25σq=[0.25×(0–1.25)2+ 0.25×(1–1.25)2+ 0.50×(2–1.25)2]1/2= 0.829210.(a) With probability 0.9544,the value of a normally distributedvariable willfallwithin2standard deviations of the mean; that is,between–40% and 80%.Simply add and subtract 2 standarddeviations to and from the mean.11.From Table 5.4, the average risk premiumBig/Valuefor the period1927-2018was:11.69%per year.Adding11.69% to the 3% risk-free interest rate, the expected annual HPR fortheBig/Value portfoliois: 3.00% +11.69% =14.69%.12.(01/1930-6/1974)SmallBigLow2HighLow2HighAverage0.99%1.17%1.48%0.76%0.81%1.19%SD8.29%8.38%10.17%5.70%6.72%8.89%Skew1.301.632.350.171.751.77Kurtosis9.7413.1017.697.0617.8014.64(07/1974-12/2018)SmallBigLow2HighLow2HighAverage1.00%1.35%1.45%0.99%1.05%1.13%SD6.69%5.28%5.49%4.70%4.35%4.90%Skew-0.43-0.55-0.47-0.33-0.43-0.54Kurtosis2.083.604.301.992.572.96

Solution Manual For Investments, 12th Edition - Page 30 preview image

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CHAPTER 5: INTRODUCTION TO RISK, RETURN, ANDTHE HISTORICAL RECORD5-4. The distributions from (01/1930–06/1974) and (07/1974–12/2018) periodshave distinct characteristics due to systematic shocks to the economy andsubsequent government intervention. While the returns from the two periods do notdiffer greatly, their respective distributions tell a different story. The standarddeviation for all six portfolios is larger in the first period. Skew is also positive, butnegative in the second, showing a greater likelihood of higher-than-normal returnsin the right tail. Kurtosis is also markedly larger in the first period.13.anominalnominalreal10.800.7010.0588,5.88%111.70rrirorii+−−=−===++b.nominal.80.70.10realrir−=−=Clearly, the approximation gives a real HPR that is too high.14.From Table 5.3, the average real rate on T-bills has been0.46%.a.T-bills: 0.46% real rate + 3% inflation = 3.46%b.Expected return onBig/Value:3.46% T-bill rate +11.69% historical risk premium =15.15%c.The risk premium on stocks remains unchanged.A premium, thedifference between two rates, is a real value, unaffected by inflation.15.Real interest rates are expected to rise.The investment activity will shiftthe demand for funds curve (in Figure 5.1) to the right.Therefore theequilibrium real interest rate will increase.16.a.Probabilitydistribution of the HPR on thestockmarket andput:STOCKPUTState of theEconomyProbabilityEnding Price+ DividendHPREndingValueHPRExcellent0.25$131.0031.00%$0.00−100%Good0.45114.0014.00$0.00−100Poor0.2593.25−6.75$20.2568.75Crash0.0548.00−52.00$64.00433.33Remember that the cost of the index fund is $100 per share, and the costof the put option is $12.

Solution Manual For Investments, 12th Edition - Page 31 preview image

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CHAPTER 5: INTRODUCTION TO RISK, RETURN, ANDTHE HISTORICAL RECORD5-5b.The cost of one share of the index fund plus a put option is $112.Theprobability distribution of the HPR on the portfolio is:State of theEconomyProbabilityEnding Price+ Put +DividendHPRExcellent0.25$131.0017.0%= (131−112)/112Good0.45114.001.8= (114−112)/112Poor0.25113.501.3= (113.50−112)/112Crash0.05112.000.0= (112−112)/112c.Buying the put option guarantees the investor a minimum HPR of0.0%regardless of what happens to the stock's price.Thus, it offers insuranceagainst a price decline.17.The probability distribution of the dollar return on CD plus call option is:State of theEconomyProbabilityEnding Valueof CDEnding Valueof CallCombinedValueExcellent0.25$114.00$16.50$130.50Good0.45114.000.00114.00Poor0.25114.000.00114.00Crash0.05114.000.00114.0018.a.Total return of the bond is (100/84.49)-1 = 0.1836. Witht= 10, the annualrate on the real bond is (1 + EAR) == 1.69%.b.With a perquarter yield of 2%, the annual yield is= 1.0824,or8.24%. The equivalent continuously compounding (cc) rate is ln(1+.0824)= .0792,or 7.92%. The risk-free rate is 3.55%with a cc rate of ln(1+.0355)= .0349,or 3.49%. The cc risk premium will equal .0792-.0349 = .0443,or4.433%.c.The appropriate formulais,where.Using solver or goal seek, setting thetarget cell to theknowneffective cc rate by changing theunknown variance(cc)rate, the equivalentstandard deviation(cc)is18.03%(excel mayyield slightly different solutions).d.The expected value of the excess return will grow by 120 months (12months over a 10-year horizon). Therefore theexcess return will be 120×4.433% = 531.9%. The expected SD grows by the square root of time

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