Sample Mid-Term Examination Fall Some Useful Formulas

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15.407 Sample

Mid-Term

Examination

–

Fall

2008

Please check to be certainthat your copy of this examinationcontains 18pages (including thisone). WriteyournameandMITIDnumberoneverypage. Youareallowedone81₂×11 sheet of notes and onenon-programmable non-PDA calculator. Unlike the group projects, youaretoanswertheseexaminationquestionswithout consulting anyone. Noscratchpaper is allowed—doall your workon theseexaminationpages.

For simplicity, assumethat thecoupon payments for all coupon bonds occur annually,i.e., once ayear, notsemi-annually asis theconvention inpractice.

You have eighty (80) minutes to complete this examination. Credit for each question is proportionaltotheamountoftimeyououghttospendonit. Therefore,do notagonizeover a 10-pointquestion withouthavingtackled a 40-pointquestion. Goodluck!

Some

Useful

Formulas

n

� C C 1

NPV = = 1 − (Annuity) (1)

(1+r)k r (1+r)n

k=1

∞

� C C

NPV = = (Perpetuity) (2)

(1+r)k r

k=1

∞

� D D

Price = = (DDM) (3)

(1+r)k r

k=1

∞

� D(1+g)k−1 D

Price = = (DDM withgrowth) (4)

(1+r)k r−g

k=1

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15.407 Mid-Term

Examination

2008

Grade

Sheet

1. / 10

2. / 10

3. / 20

4. / 30

5. / 20

6. / 30

120

Total /

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Question 1 (10points):

If the annualinterest rate is 10percent, how long wouldyou havetowait before a $17,500 investment atleastdoubles invalue? Givethe minimumnumberof years required.

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Yourcardealeroﬀersyoualoanforpartof thepurchasepriceofanewcar,citing anannual percentagerate (APR)of8.5%. Whatistheeﬀectiveannual rateof suchaloan (recallthat an auto loantypically requires monthlypayments)?

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Company ABC has just paid a dividend of 50 cents per share. Because of its growth po tential, itsdividend is forecastedto grow ata rate of 7 percent per year indeﬁnitely. If the company’s appropriate costof capital(given its risk) is 11percent, what was ABC’s share price immediately before it paid its 50 cent dividend, i.e the stock price right before the ex-dividend date?

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Extra SpaceFor Question 3:

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The current market priceof a two-year 25 percent couponbond with a $1,000face value is $1,219.71 (recall that such a bond pays coupons of $250 at the end of years 1 and 2, and the principal of $1,000 at the end of year 2). The current market price of a one-year pure discount bondwith a $50face valueis $44.64.

(a) (15 points) Whatmust the price of a two-year pure discount bond witha $2,500 face value bein ordertoavoid arbitrage?

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Question 4 Continued:

(b) (15 points) Suppose the price of the two-year pure discount bond with a $2,500 face valueisonly$1,900. Isthereanarbitrageopportunity? Isyes,howwouldyoustructure a trade that has zero cash-ﬂowin years 1 and2 and a positivecash-ﬂow onlyin year 0 (i.e. now).

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Your friendis celebrating her35th birthday today andwants to start savingfor her antici patedretirementatage65(shewillretireonher65thbirthday). Shewouldliketobeableto withdraw $80,000 from her savingsaccounton each birthday foratleast 20years following her retirement (the ﬁrst withdrawal will be on her 66th birthday). Your friend intends to invest her money in the local savings bank which oﬀers 4 percent per year. She wants to makeequalannual depositson eachbirthday inanew savingsaccountshe willestablishfor herretirement fund.

If she starts making these deposits on her 36th birthday and continues to make deposits untilsheis65(thelastdepositwillbeon her65thbirthday),whatamountmustshedeposit annually tobe ableto makethedesiredwithdrawals upon retirement?

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Thecurrentlevelof theS&P500is1040. The risk-freeinterestrate peryearis2%. Assume negligibledividends. The 6month futures contractis trading at1060.

(a)(15 points) Is therean arbitrage opportunity? Brieﬂy explain.

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Question 7 Continued:

(b)(15points) Ifthereisanarbitrageopportunity,whatstrategywouldyouusetoexploit it withoutusingany fundsof your own?

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Extra SpaceFor Question 6(b):

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Extra SpaceFor Answers– Please SpecifyQuestion Number:

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15.401 Finance Theory I

Fall 2008