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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 60 "Solutions to Background K
nowledge Probe, Due August 29, 2003" }}}{EXCHG {PARA 18 "" 0 "" {TEXT
-1 67 "Math 496/889, Stochastic Processs and Advanced Mathematical Fin
ance" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Steven R. Dunbar" }}
{PARA 0 "" 0 "" {TEXT -1 40 "Department of Mathematics and Statistics
" }}{PARA 0 "" 0 "" {TEXT -1 30 "University of Nebraska-Lincoln" }}
{PARA 0 "" 0 "" {TEXT -1 19 "Lincoln, NE 68588, " }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "sdunbar@unl.edu" }}{PARA
0 "" 0 "" {TEXT -1 25 "www.math.unl.edu/~sdunbar" }}}{EXCHG {PARA 0 ">
" 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 9 "Probl
em 1" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 629 "\\item % Present value ca
lculations\n %adapted from Problem 6, page 41, \{\\it Principles \+
of Corporate\n %Finance\}, Brealy and Myers, 1984, McGraw-Hill\n \+
As winner of the lottery, you can choose one of the following\n priz
es:\n \\begin\{enumerate\}\n \\item \\$100,000 now\n \\item \\$
180,000 at the end of 5 years\n \\item \\$11,400 a year forever\n \+
\\item \\$19,000 for each of the of the next ten years\n \\item \+
\\$6500 next year and increasing thereafter by 5\\% a year thereafter.
\n \\end\{enumerate\}\nWhich is the most valuable prize in terms of p
resent value?\nAssume the interest rate is 12\\% and is compounded con
tinuously." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(finance)
;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7.%-amortizationG%(annuityG%-blac
kscholesG%*cashflowsG%.effectiverateG%,futurevalueG%/growingannuityG%2
growingperpetuityG%,levelcouponG%+perpetuityG%-presentvalueG%0yieldtom
aturityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "(a) Has a present val
ue of $100,000" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "apr := ef
fectiverate(0.12, infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$apr
G$\"*_o\\F\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "apr_dir
ect := exp(0.12)-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+apr_directG$
\"*_o\\F\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "PVb := pr
esentvalue( 180000, apr, 5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$PVb
G$\"+I%4'y)*!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "PVb_dir
ect := 180000/exp(0.12*5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+PVb_d
irectG$\"+_%4'y)*!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "PV
c := perpetuity(11400, apr);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$PVc
G$\"+NsRT*)!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "PVc_dire
ct := sum( 11400/exp(0.12)^n, n = 1..infinity);" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#>%+PVc_directG$\"+NsRT*)!\"&" }}}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 31 "PVd := annuity(19000, apr, 10);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$PVdG$\"+fLQT5!\"%" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 49 "PVd_direct := sum( 19000/exp(0.12*n), n = 1..10);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+PVd_directG$\"+gLQT5!\"%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "payments_e_100 := [seq(6500*
(1.05)^(n-1), n=1..100)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
55 "Pve_approximate_100 := cashflows( payments_e_100, apr);" }}{PARA
11 "" 1 "" {XPPMATH 20 "6#>%$PveG$\"+b1m!Q)!\"&" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 42 "PVe := growingperpetuity(6500, apr, 0.05);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$PVeG$\"+muV(Q)!\"&" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "PVe_direct := sum( 6500*(1.05)^(n-1
)/(1 +apr)^n, n = 1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$
PVeG$\"+muV(Q)!\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "Take (d), \+
$19,000 a year for each of the next 10 years, it offers the best prese
nt value of $104,138.33" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "
" 0 "" {TEXT -1 9 "Problem 2" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 323 "
\\item %Interest calculations\n %adapted from Problem 6, page 41,
\{\\it Principles of Corporate\n %Finance\}, Brealy and Myers, 1
984, McGraw-Hill\nA investment of \\$232 will be worth \\$312.18 in 2 \+
years. What is the\nannual interest rate assuming quarterly compoundi
ng? Assuming\n continuously compounded interest?" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "apr := fsolve( 312.18 = 232.0*(1 + \+
r/4)^8, r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$aprG6$$!+*G47:)!\"*$
\"+CF47:!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "apr := fsolv
e( 312.18 = 232.00*exp(2*r), r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
$aprG$\"+kG@%[\"!#5" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 9 "Problem 3" }}{EXCHG {PARA 0 "" 0
"" {TEXT -1 492 "\\item %Calculating binomial probabilities\n % A
dapted from Problem 3.6.2, page 145, \{\\it An Introduction to\n \+
% Probability Theory and Its Applications\}, Larsen and Marx,\n %
Prentice-Hall, 1985, \nEach day a stock moves up one point or down on
e point with\n probabilities $1/4$ and $3/4$ respectively. What \+
is the\n probability that after 4 days, the stock will have retur
ned to\n its original price. Assume the daily price fluctuations
are\n independent events." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1
228 "The stock can return to its original price if and only in the fou
r days, it has increased twice,a nd decreased twice. That is, out of \+
four independent rials we have 2 successes (up days) and consequently \+
2 failures (down days)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "
prob := stats[statevalf,pf,binomiald[4,1/4]](2);" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#>%%probG$\"++]P4@!#5" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 45 "prob_direct := binomial(4,2)*(1/4)^2*(3/4)^2;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,prob_directG#\"#F\"$G\"" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#$\"++]P4@!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA
3 "" 0 "" {TEXT -1 9 "Problem 4" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1
483 "\\item % Calculating binomial and geometric probabilities\n \+
% Adapted from Problem 31, page 118, \{\\it A First Course in\n %
Probability\} by S Ross, Macmillan, 1976.\nConsider a roulette wheel \+
consisting of 38 numbers, $1$ through $36$,\n$0$ and double $0$. If B
ond always bets that the outcome will be one\nof the numbers $1$ throu
gh $12$, what is the probability that Bond\nwill lose his first $5$ be
ts? What is the probability that his first\nwin will occur on his fou
rth bet?\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "prob_5_losses
:= ((38-12)/38)^5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.prob_5_losse
sG#\"'$Hr$\"(*4wC" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+&*y]*\\\"!#5" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "prob_win_on_4 := ((38-12)/38)^3*(12
/38);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.prob_win_on_4G#\"&#=8\"'@.
8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$\"+nB]65!#5" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 9 "Problem 5" }}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 394 "\\item % Calculating normal probabilities\n % Ada
pted from Problem 10, page 146, \{\\it A First Course in\n % Prob
ability\} by S Ross, Macmillan, 1976.\nIf $X$ is a normal random varia
ble with parameters $\\mu = 10$ and\n$\\sigma^2 = 36$, compute\n\\begi
n\{enumerate\}\n \\item $\\Pr[ X > 5]$\n \\item $\\Pr[4 < X < 16]$\n
\\item $\\Pr[X < 8]$\n \\item $\\Pr[X < 20]$\n \\item $\\Pr[X > 16
]$\n\\end\{enumerate\}\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50
"Pra := 1 - stats[statevalf,cdf,normald[10, 6]](5);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$PraG$\"+!>;n(z!#5" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 88 "Prb := stats[statevalf,cdf,normald[10, 6]](16) - stat
s[statevalf,cdf,normald[10, 6]](4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%$PrbG$\"+A\\*o#o!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "P
rc := stats[statevalf,cdf,normald[10, 6]](8);" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#>%$PrcG$\"+-MT%p$!#5" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 47 "Prd := stats[statevalf,cdf,normald[10, 6]](20);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$PrdG$\"+xk4A&*!#5" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 51 "Pre := 1 - stats[statevalf,cdf,normald[10
, 6]](16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$PreG$\"+RDb'e\"!#5" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Problem \+
6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 ""
0 "" {TEXT -1 308 "\\item % Using normal approximation to the binomial
\n % Adapted from Problem 10, page 146, \{\\it A First Course in
\n % Probability\} by S Ross, Macmillan, 1976.\nIn $10,000$ indep
endent tosses of a coin, the coin landed heads $5800$\n times. I
s it reasonable to assume the coin is not fair? Explain." }{MPLTEXT 1
0 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "mu := 10000/2; s
igma := sqrt(10000*(1/2)*(1/2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
#muG\"%+]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sigmaG\"#]" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Z := (5800 - mu)/sigma;" }}{PARA
11 "" 1 "" {XPPMATH 20 "6#>%\"ZG\"#;" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 59 "significance := 1 - stats[statevalf,cdf,normald[0, 1]
](16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-significanceG$\"\"!F&" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 213 "It is reasonable to believe the \+
coin is not fair. The coin came up heads more than 16 standard deviat
ions away from the mean, an event that is virtually impossible under t
he null hypothesis that the coin is fiar." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 9 "Problem 7"
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 305 "\\item %Calculating moments and
variances.\n %Adapted from Problem 40, page 78 of S. Ross, \{\\i
t Introduction\n % to Probability Models\}, S. Ross, Academic Pre
ss, 1985. \nIf $X$ is a uniformly distributed random variable over $(0
,1)$, then\n calculate $E[X^n]$ and $\\Var[X^n]$ for $n=1,2,3,\\d
ots$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "assume(n, posint);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "nth_moment := int( x^n,
x = 0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+nth_momentG*&\"\"\"F
&,&%#n|irGF&F&F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "nt
h_variance := int( (x^n - nth_moment)^2, x = 0..1);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%-nth_varianceG*&%#n|irG\"\"#,**&F'\"\"\")F&\"\"$F*F
**&\"\"&F*)F&F'F*F**&\"\"%F*F&F*F*F*F*!\"\"" }}}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 27 "subs( n = 1, nth_variance);" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6##\"\"\"\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA
3 "" 0 "" {TEXT -1 9 "Problem 8" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1
387 "\\item % Conditional probabilities\n % Adapted from Problem \+
10, page 79, \{\\it A First Course in\n % Probability\} by S Ross
, Macmillan, 1976.\nThree cards are randomly selected, without replace
ment, from an\n ordinary deck of 52 playing cards. Compute the c
onditional\n probability that the first card selected is a spade,
given the\n second and third cards are spades." }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 33 "PrSSS := (13/52)*(12/51)*(11/50);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&PrSSSG#\"#6\"$])" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 34 "PrNSSS := (39/52)*(13/51)*(12/50);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'PrNSSSG#\"#R\"$])" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 43 "conditional_prob := PrSSS/(PrSSS + PrNSSS
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1conditional_probG#\"#6\"#]" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1
"" {XPPMATH 20 "6#$\"+++++A!#5" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 9 "Problem 9" }}{EXCHG
{PARA 0 "" 0 "" {TEXT -1 471 "\\item % Conditional probabilities\n \+
% Adapted from Problem 10, page 79, \{\\it A First Course in\n \+
% Probability\} by S Ross, Macmillan, 1976.\nIf two fair dice are roll
ed, what is the conditional probability that\n the first one land
s on $6$, given that the sum of the dice is\n $k$, $k=2,3,\\dots \+
12$. Compute for all values of $k$. What is\n the probability t
hat at least one of the fair dices lands on $6$\n given that the \+
sum of the dice is $k$?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 117 "The probability that at least one of the
dice lands on 6, given that the sum of the dice is k is 0 for k = 2,3
,4,5,6:" }}{PARA 0 "" 0 "" {TEXT -1 45 " Pr( 6 | sum = k) = 0, for
k = 2,3,4,5, 6" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 32 " Pr(6 | sum = 7) = 2/6 = 1/3" }}{PARA 0 "" 0 ""
{TEXT -1 26 " Pr(6 | sum = 8) = 2/5" }}{PARA 0 "" 0 "" {TEXT -1
32 " Pr(6 | sum = 9) = 2/4 = 1/2" }}{PARA 0 "" 0 "" {TEXT -1 27 " \+
Pr(6 | sum = 10) = 2/3" }}{PARA 0 "" 0 "" {TEXT -1 25 " Pr(6 |
sum = 11) = 1" }}{PARA 0 "" 0 "" {TEXT -1 24 " Pr(6 | sum =12) = \+
1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}{MARK "3 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1
1 1 }{PAGENUMBERS 0 1 2 33 1 1 }