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investment banking job interview questions and answers
Errata: Known Errors in "Heard on The Street"
7th Edition (Edition 2001) ISBN: 0-9700552-0-X
Statistics Answer 5.1.18 on page 235.
Cdf is not F(p|H)=p^3/[p^3 + (1-p^2)(1-p)], but rather
simply F(p|H)=p^2. This is corrected in the 8th Edition.
I state incorrectly in two places that
S(t)N(d1)=exp[-r(T-t)]E[S(T)|S(T)>X],
where S(T) is stock price at option expiration, X is
strike, and t is now. In fact, assuming
dS(t)=rS(t)dt+sigmaS(t)dw in
the risk neutral world, then the result should be
S(t)N(d1)=exp[-r(T-t)]E[S(T)|S(T)>X]N(d2).
The N(d2) is missing.
Norbert Wiener's name is mis-spelled in several places
as Weiner.
The disks on poles question. Answer 2.29.
Last three digits of V(64) should be 615, not 616.
8th Edition (Edition 2003) ISBN: 0-9700552-1-8
I state incorrectly in two places that
S(t)N(d1)=exp[-r(T-t)]E[S(T)|S(T)>X],
where S(T) is stock price at option expiration, X is
strike, and t is now. In fact, assuming
dS(t)=rS(t)dt+sigmaS(t)dw in
the risk neutral world, then the result should be
S(t)N(d1)=exp[-r(T-t)]E[S(T)|S(T)>X]N(d2).
The N(d2) is missing.
Norbert Wiener's name is mis-spelled in several places
as Weiner.
The disks on poles question. Answer 2.29.
Last three digits of V(64) should be 615, not 616.
9th Edition (Edition 2004) ISBN: 0-9700552-3-4.
These are all minor, and do not affect the answers.
p5, line 19, "...so that I can you ask..." should be "...so that I can ask you..."
p6, line 11, "Macaulay duration, and the curvature of the plot cannot"
should be "negative Macaulay duration, and the curvature of the plot cannot".
p10, line 9 ...Sincerely, YT. should be ...Sincerely, YT."
p88, where I say four choose 2 is 4!/(2! x (4-2!))=12, it should say 4x3=12. There are thus
12x4x2=96 different solutions different solutions.
p75, Answer 1.14, Where I say "Place 10 potentially heavy coins and
10 potentially light coins on each side of the scales, while holding out
10 potentially heavy coins and 10 potentially light coins"
I should say "Set aside 10 potentially heavy coins and 10 potentially
light coins. Take the remaining 20 potentially heavy and 20 potentially
light coins and swap 10 of them from one side of the scales with
10 of them from the other side of the scales, keeping track of which
were swapped and which stayed put"
p79, Footnote 12, should read "|f'(z_0)|<=1/r sup..." rather than 1/r^2.
p93, let me emphasize that P(X=0)=(1-(1/N))^N requires that N be
infinitely large; it is an approximation only for N less than infinity.
This is implicitly stated, but I should have been more explicit.
p95, last three digits of V(64) should be 615, not 616.
p162, m=(r+(1/2)sigma^2)*(alpha-1) should read m=(r+(alpha/2)sigma^2)*(alpha-1).
Same problem a couple of pages later.
pp218-219, my solution to the sports betting problem assumes a very simple
problem statement. Most firms are asking a more complicated question
that requires you to bet a positive amount on every game (until
someone wins the series) and lock
in either a sure doubling of your money or a sure loss of 100%. This solution
can be worked out by backward induction to arrive at an initial bet of $31.25 on your team.
Minor typos: p3 "questions-." should be "questions.";
p9 "fuacet" should be "faucet"; p138 "Wirh" should be "With";
p10 ending quote marks missing after "Sincerely, YT.";
p247 citation 114 (Lorenz), Jou-rnal should be Journal.
eBook version of 9th Edition (Edition 2004) Parent ISBN: 0-9700552-3-4, eBook
ISBN: B0001DLM26.
Most are minor, and do not affect the answers.
p5, line 19, "...so that I can you ask..." should be "...so that I can ask you..."
p6, line 11, "Macaulay duration, and the curvature of the plot cannot"
should be "negative Macaulay duration, and the curvature of the plot cannot".
p10, line 9 ...Sincerely, YT. should be ...Sincerely, YT."
Question 1.17 need slight clarification. It should say "Each of the lily pads
is one square foot in area."
p69, Answer 1.14, Where I say "Place 10 potentially heavy coins and
10 potentially light coins on each side of the scales, while holding out
10 potentially heavy coins and 10 potentially light coins"
I should say "Set aside 10 potentially heavy coins and 10 potentially
light coins. Take the remaining 20 potentially heavy and 20 potentially
light coins and swap 10 of them from one side of the scales with
10 of them from the other side of the scales, keeping track of which
were swapped and which stayed put"
Appendix A, Footnote 12, should read "|f'(z_0)|<=1/r sup..." rather than 1/r^2.
p88, where I say four choose 2 is 4!/(2! x (4-2!))=12, it should say 4x3=12. There are thus
12x4x2=96 different solutions different solutions.
p87, let me emphasize that P(X=0)=(1-(1/N))^N requires that N be
infinitely large; it is an approximation only for N less than infinity.
This is implicitly stated, but I should have been more explicit.
p89, last three digits of V(64) should be 615, not 616.
Answer 2.18 has a simple error in it (which has been there for 10 years and 10 editions!).
This answer discusses the plots in Figure B.4. The plots are
correct, but the text that discusses the middle plot (c(t) versus F) is not correct. You can see
in the middle plot that the dashed line rising from the strike price has slope slightly less than 1 (compare it
to the top plot to see this). The accompanying text incorrectly says that it has slope=1. In fact,
the slope is exp[-r(T-t)] which gets close to one as maturity approaches, but which is strictly less
than one (as in the plot) at any prior time. There are different ways to see this, but I like to look
at (partial c/partial F) using the chain rule. It is (partial c/partial S)*(partial S/partial F). The first
is the delta which goes to 1 when F and S are large; the second is just exp[-r(T-t)].
p159, m=(r+(1/2)sigma^2)*(alpha-1) should read m=(r+(alpha/2)sigma^2)*(alpha-1).
Same problem a couple of pages later.
pp212-213, my solution to the sports betting problem assumes a
very simple problem statement. Most firms are asking a more complicated question
that requires you to bet a positive amount on every game (until
someone wins the series) and lock
in either a sure doubling of your money or a sure loss of 100%. This solution
can be worked out by backward induction to arrive at an initial bet of $31.25 on your team.
Minor typos: p3 "questions-." should be "questions.";
p9 "fuacet" should be "faucet"; p132 "Wirh" should be "With";
p10 ending quote marks missing after "Sincerely, YT.";
p241 citation 114 (Lorenz), Jou-rnal should be Journal.
10th Edition (Edition 2007) ISBN: 0-9700552-5-0.
Typo: Page 8, 10 lines from bottom, "women" should be "woman"
Question 1.17 needs slight clarification. It should say "Each of the lily pads
is one square foot in area."
p85, Answer 1.24 where I say four choose 2 is 4!/(2! x (4-2!))=6, it should say 4x3=12. There are thus
12x4x2=96 different solutions different solutions not 6x4x2=48.
Answer 1.54 has a typo in it. The denominator "(10-5!)" should read "(10-5)!"
Answer 1.59 is numerically correct, but the logic is wrong. Yes, the probability equals 0.25,
but no, the distribution of L is not uniform. Here is one example of how you might work it out:
Duke Math Web Page, but
I have a simpler argument that I will give for the next edition of the book. Note that one candidate
was asked the expected length of the shortest portion of stick in a recent interview (the answer is
1/9, but you need to derive the density to see that; the density for the shortest portion of stick is f_S(s)=-18s+6 for
0<=s<=(1/3)).
Typo, p103, in Answer 1.37. Eight lines down from top of page, "n>=3" should be "n>=75"
Answer 2.18 has a simple error in it (which has been there for 10 years and 10 editions!).
This answer discusses the plots in Figure B.4. The plots are
correct, but the text that discusses the middle plot (c(t) versus F) is not correct. You can see
in the middle plot that the dashed line rising from the strike price has slope slightly less than 1 (compare it
to the top plot to see this). The accompanying text incorrectly says that it has slope=1. In fact,
the slope is exp[-r(T-t)] which gets close to one as maturity approaches, but which is strictly less
than one (as in the plot) at any prior time. There are different ways to see this, but I like to look
at (partial c/partial F) using the chain rule. It is (partial c/partial S)*(partial S/partial F). The first
is the delta which goes to 1 when F and S are large; the second is just exp[-r(T-t)].
Answer 3.19. p212 of hard copy. Last line. A factor of 2 is missing. The second derivative is actually 2 times what I state, and
thus it is 0.14, not 0.07. The qualitative answer (it is a minimum) remains unchanged.
Typo, p227, top of page. 0x0.50 should read 0x0.30. This does not change the final anwser.
Answer 4.24 (20,000 placed into five funds) is not correct. Many of my "different" outcomes are indistinguishable from
each other. In fact, they are not different and the answer is wrong. Correct answer is the same as the number of
distinct outcomes (i.e., the size of the sample space) in a multinomial
distribution with 20 trials and five types of outcomes (remember the multinomial is a
generalization of the binomial). This is (N+k-1)CHOOSE(k-1), that is
(N+k-1)!/((k-1)!*N!). It gives 10,626 in case N=20, k=5. In the simple case N=20, k=2, it gives 21, which makes sense.
Alternatively, use a stars and bars approach to come to the same answer. This *****|********||*******| has
20 stars and four bars to indicate an allocation of 5,8,0,7,0. So, the answer must be (24)choose(4).
Question 4.25 (the cookie dough question). The given solution is an extremely good approximation to the
correct answer when there are many cookies and many chips (it is out by at most 1 chip).
I assume independence between the event that one cookie has at least one chip and the event
that another cookie has at least one chip. In fact, there is very small dependence. There are
two exact solutions: one uses a recursion algorithmm, the other uses the inclusion-exclusion formula and
some combinatorics.
11th Edition (Edition 2008) ISBN: 0-9700552-6-9.
p83, Answer 1.24 where I say four choose 2 is 4!/(2! x (4-2!))=6, it should say 4x3=12. There are thus
12x4x2=96 different solutions different solutions not 6x4x2=48.
Typo, p101, in Answer 1.37. Eight lines down from top of page, "n>=3" should be "n>=75"
Typo, p115, middle of page. A minus sign is missing from in front of the "a" on the LEFT hand side of the equality.
Typo, p225, top of page. 0x0.50 should read 0x0.30. This does not change the final anwser.
Answer 4.24 (20,000 placed into five funds) Alternatively, use a stars and bars approach to come to the same
answer. This *****|********||*******| has 20 stars and four bars to indicate an allocation of 5,8,0,7,0. So, the answer must be (24)choose(4).
Question 4.25 (the cookie dough question). The given solution is an extremely good approximation to the
correct answer when there are many cookies and many chips (it is out by at most 1 chip).
I assume independence between the event that one cookie has at least one chip and the event
that another cookie has at least one chip. In fact, there is very small dependence. There are
two exact solutions: one uses a recursion algorithmm, the other uses the inclusion-exclusion formula and
some combinatorics.
12th Edition (July 2009) ISBN: 0-9700552-7-7.
Posted Sept 3, 2009: Question 1.16 about the ants is mis-stated.
In the original question the ants can only walk parallel to the edge of the ruler.
As such, they can only ever meet head on. With this amendment, the given answer is correct.
Thank you to Craig Smith for finding this.
Posted May 20, 2010: Answer 4.5 on page 191 says that 8/3 is the same as 2 1/3. Obviously
8/3 = 2 2/3.