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A Mathematician's Take on FM
Part I
Interest Theory (65 80%)1 Interest Rates and Time Value of
Money
1. Easy formulas (most of these are denitions/easy to
derive):
d =i
1 + i,
v =1
1 + i,
v + d = 1,
i d = id,
d = iv,
and
d < d(m) < < i(m) < i.
2. When given nominal rates, instead of calculating the eective
interest rate using ICONV and then calculate (1 + i)n
where n is the number of years, it is much much easier to use
the nominal rate directly, i.e.,(
1 + i(m)
m
)nmor(
1 d(m)m)nm.
3. It is important to understand these three formulas,
1 + i =
(1 +
i(m)
m
)m,
1 d =(
1 d(m)
m
)m,
and (1 +
i(m)
m
)m=
(1 d
(m)
m
)m.
Use intuition, not brute force.
4. Continuous interest: = ln (1 + i). 5% compounded continuously
means = 0.05, not i = 0.05! Equivalently, we
can also understand it as a nominal rate of 5% convertible
continuously/innitely. Formula to memorize:
a (t) = a (0) et. (1)
Here is always a constant, for example, 0.05 mentioned above. It
is not to be confused with the instantaneous
force of interest below:
5. Force of interest: (t) = a(t)a(t) is a function of t, and
astart amount
eend time
start time
(u)du = aend amount
. (2)
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Some a (t) are easy to get given (t), for example, when (t) =
mt+n , a (t) = (t+ n)m. (m and n are positive
integers). Also, when a (t) is of the form (1), then (t) happens
to be a constantso the previous bullet point is
really just a special case where the force of interest is always
a number, for example, 0.05, it is basically a constant
function (of t).
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2 Annuities
1. Make the best out of STO and RCL features! Trust me, this is
the dierence maker! I am surprised that ACTEX
doesn't mention this at all. Practice with your calculator! You
calculator is your best friend!
2. Make sure you know how to use the calculator to solve the
CF-related questions.
3. Always remember to CLR TVM. CPT N if you are not sure if you
did, if Error 1, then you are good to go. Or just
simply CLR TVM again. CLR TVM does not clear CF data, you need
to CLR WORK within the CF interface to
do that.
4. For problems that do not have both present and future values
involved, there is no need to worry about plus/minus
signs if you are not calculating the interest. When the payment
is plus (minus), then the present value or future
value is minus (plus). Edit: after a while, especially after
later chapters, it becomes a second nature to get the signs
right, so I guess it does not hurt to train yourself early.
5. Check if it's beginning or end on calculator. Default is end
(does not show BGN). My friend, who is an accountant,
said that I should never bother switching between beginning and
end, just do everything as if it is end, then multiply
the result by (1 + i) if it is beginning.
anq =1 vni(3)
3 is all you need to memorize for level payments, here is
why
Immediate = end = denominator is i = normal: 3.
Due = beginning = denominator is d = two dots: anq = 1vn
d .
Continuous = denominator = bar: anq = 1vn
.
Again, given the same amount of payment, then the present value
of due = (1 + i) present value of immediate,
because x =(
x1+i
)(1 + i), think about it.
6. For future values, just multiply dierent versions of 3 by (1
+ i)naccordingly.
7. All you need for arithmetic payment problems is this
all-inclusive formula:
Panq +Q(anq nvn
i
).
Write it down on scrap paper at the beginning of the exam.
1
Remember, when it is Ia, P = Q = 1, when it is Da,
P = n, Q = 1. When Q 6= 1, make sure you multiply the nvn term
with Q too when you break the bracket.
8. We will show that it is intuitively easy to understand
1(
1+g1+i
)ni g . (4)
But you need to rst memorize the perpetuity formula (which is
easy to memorize)
1
i g . (5)
Since 5 is positive and has i g at the bottom, we assume that i
> g (actually when g > i, the perpetuity divergesand
therefore the formula is meaningless). In order for the fraction
part of the numerator of 4,
(1+g1+i
)n, to go
to zero when n goes to innity, its numerator should be smaller
than the denominator, thus 1 + g is on top, and
1
See 11 for a complete short list of important formulas that you
should write down at the beginning of the exam.
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therefore, 1 + i is at the bottom. The power is n also makes
sense because before the rst payment, i.e., when n = 0,
the numerator should zero, so it checks out.
9. Just to emphasize, 5 and etc. is for immediates, it
calculates the present value of the total payments 1 period
before
the rst payment. Also keep in mind that the nal result is
rst payment1
(1+g1+i
)ni g ,
where the rst payment could be of the form x (1 + g)
already.
10. The future value of A at time t1 is twice the future value
of B at time t2 does not mean the current value (present
value today) of A is twice the current value of B. It is
extremely important that you are always crystal clear about
the time t when you talk about present values and/or future
values.
11. ALWAYS distinguish between the interest rate and the actual
interest value, do NOT confuse between K% and K,
the calculator uses 5 for 5%, make sure you use 0.05 when you do
calculations!
12. If money is deposited constantly at c (t), and force of
interest is (t) compounded continuously, then the nal amount
is end
begin
c (t) eend
t(u)dudt. (6)
6 is pretty complicated on the surface, yet it is actually a
straight forward formula. It tests your understanding of
1 and 2 greatly. (Courtesy of question 8)
13. If you need x retirement income and you are told that every
thousand in the fund will generate k each period, then
you need
xk1000 in the fund. (Courtesy of question 12)
14. Keep in mind that the end of the 10th year is the beginning
of the 11th year. Don't just stop at the beginning of the
10th year! When you are drawing a cash ow chart, always go one
extra year at the end, just in case. Also, keep in
mind that, on the time axis, n always indicates the end of the
nth year, because you start with 0!
15. When a geometrically increasing payment decreases by the
same percentage, you have to calculate the new payment
manually, for example, at 10% increase and decrease, we have
100, 110, 1100.9 = 99, not 100, 110, 100. In generalit is x (1 +
i)
n, x (1 + i)
n+1, x (1 + i)
n+1(1 i), not x (1 + i)n, x (1 + i)n+1, x (1 + i)n! (Courtesy of
question 24)
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3 Loan Repayment
1. Loan payment at time k: Pk. Notation.
2. Balance after Pk is made: Bk. Notation.
3. Loan amount: B0. Obvious since 0 payment has been made.
4. Interest paid in Pk+1: Ik+1 = i (Bk). Your new payment always
pays the interest of the old balance o rst. Key
concept!
5. Principal paid in Pk+1: Rk+1 = Pk+1 Ik+1. Obvious!
6. Balance after Pk+1 is made:
Bk+1 = Bk Rk+1 = Bk (1 + i) Pk+1 (7)
There are two ways to look at the balance after Pk+1 is
made:
(a) Old balance subtract the new principal paid, this is
straight forward.
(b) The second is a mathematical result, it is counter-intuitive
but let us rst look at an example: say we borrow
10 today and we repay 1 year from now, we obviously need to
repay 10 (1 + i). The new balance after our
repayment is 0 regardless if we calculate 10 10 (rst way) or 10
(1 + i) 10 (1 + i) (second way). The newbalance is the future value
of the loan one period from now, so if we do not pay anything one
period from now,
the new balance would simply be Bk (1 + i), but we are going to
pay Pk+1, thus Bk (1 + i) Pk+1.
Some usual ways to understand equations involving indices n, k,
etc.:
(a) Look at simple examples.
(b) Consider extreme cases like when the index is 0 or .
7. Prospective method: the loan balance after Pk is the present
value of the rest payments at the time of Pk. If we
calculated the present value of all the future payments at the
time we borrow the money, it would have to equal to
the loan amount, otherwise it's not a fair trade! So then it is
simple: consider the new loan balance after payment
Pk a brand new loan that you borrow right after payment Pk!
8. Retrospective balance:
Bk = B0 (1 + i)k
k1
Pj (1 + i)kj
. (8)
ACTEX does a good job explaining this one: if we do not pay
anything until payment Pk, then the loan would
grow to B0 (1 + i)k, so to calculate the balance after Pk, when
we know that we in fact did make payments, we
would calculate the present value of all the payments at time
when Pk is made, so at time k. The rst term of the
summation is P1 (1 + i)k1, because the rst payment grows to P1
(1 + i)
k1at time k. For example, when k = 2,
it would be P1 (1 + i)21
= P1 (1 + i), not P1 (1 + i)2! The last term of the summation is
Pk (1 + i)
kk= Pk which
is obviously true, the current value (present value today) of $1
paid today is simply $1. If you accept this analogy,
then the second part of 7 also becomes intuitively true because
it would simply be a special case.
9. For Level Payment x, the principal paid in payment t is:
xvnt+1. (9)
Remember, with each payment you make, you pay more and more
principal. So clearly, there should be an (n t)somewhere in the
power of v, now to remember the extra +1, you just have to realize
that you NEVER make a
payment that is 100% principal, so if it was not there, you
would be paying xvnn = x principal, so 100% principal,in the last
payment.
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10. For Level Payment x, the interest paid in payment t is:
xxvnt+1 = x (1 vnt+1). You do not need to memorizethis one, it is
trivial given 9!
11. For Level Payment x, the prospective balance at time k,
xankq. Same as above, look at it this way: you have anew loan with
n k payments of x left, what is the loan amount?
12. For Level Payment x, the retrospective balance is Bk = B0 (1
+ i)k xskq. It is nothing but a special case of 8.
13. Sinking Fund Loan, Loan interest rate: i, Sinking fund
deposit rate: j.
14. Sinking Fund Deposit: x = Lsnqj . Simple, because you want
xsnqj = L. This can usually be done with a calculator.
15. Term interest payable to the lender: y = Li. Trivial.
16. Total sinking fund payment: z = x+ y = Lsnqj + Li = L(i+
1snqj
). Trivial.
17. Balance in the fund after payment k: Bk = xskqj . Trivial,
but notice that when k = n, you get L which is expected.
18. Principal paid in payment t is x (1 + j)t1
= xv1tj . This is similar to 9, except, there is no n. So
memorize it bybrute force, keep in mind the v is based on j not i,
because i is only related to the interest!
19. Net interest in payment t: z xv1tj . Obvious, you take the
whole payment, subtract the principal (which wememorized by brute
force), and you are left with the interest.
20. Another way to calculate net interest in payment t: yBt1j.
This is actually easy to understand: y itself is 100%interest, but
you make some interest back from your fund in this period too, the
interest that you make is Bt1j, sosubtract and ta-da!
21. Do not confuse between the nth year and the nth payment! If
the rst payment is made at time 0, then payment
n+ 1 is actually made at the spot where you write n in the time
axis.
22. When it is monthly payment but the question asks in k years,
make sure you input N as 12k, I made this silly
mistake!
23. Just to emphasize, the loan balance at the end of one year
is the present value of all future payments at the end of
one year, not today! Again, present value does not necessarily
mean today!
24. APR means annual percentage rate, it is the true interest
rate. You calculate the payment with the original loan,
then CPT PMT, then change the original loan to the actual amount
(usually something like original (1 2.5%)),CPT I/Y.
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4 Bonds
1. A bond is basically a series of xed level payments (returns),
to the investor x, called coupons, with a huge last
payment, x+R, coupon + redemption.
2. The price of a bond, P , is the present value of all of
those
P = xanq +Rvn. (10)
3. r = xF is called the coupon rate, where F , xed, is called
the face value, since x and F are both xed, r is xed too.
4. Sometimes the interest rate demanded by investors go up (or
down), compared to the xed r, i.e., they will settle
with a higher (or lower) yield per period rate i, which is the
interest rate used in 10. The only way to achieve those
while 10 remains equal is by increasing (decreasing) R or
decreasing (increasing) P . Usually R is the same as F and
does not change, so we are left with one option: decrease
(increase) P . Do not worry if you feel lost here, we will
show that this makes sense intuitively. For example, there is a
bond with a face value $5, the buyer of the bond gets
$1 each year for the next 10 years and $5 at the end of 10
years. If you want to impress your boss, then you need
a higher yield per period rate, so paying $4 to get the package
would be much better (higher yield per period rate
than paying $5) than paying $6 to get the exact same package
(lower yield per period rate than paying $5).
5. The previous bullet point also makes sense mathematically:
under the assumption that R = F , 10 is of the form
P = xanq +Rvn
= x1
(1
1+i
)ni
+ F
(1
1 + i
)n
= x1
(1
1+i
)ni
+ Fi
(1
1+i
)ni
= (x Fi)1
(1
1+i
)ni
+ F= (Fr Fi)
1(
11+i
)ni
+ F= F (r i) anq + F. (11)
(a) If i = r, then P stays the same, P = F .
(b) If i > r, i.e., when the demanded yield per period rate
goes up (you want to impress your boss, remember?),
the new P is the sum of F and a negative number, so P goes down,
P < F .
(c) If i < r, i.e., when the demanded yield per period rate
goes down (there is another person that wants to buy it,
so you are willing to settle for less), the new P is the sum of
F and a positive number, so P goes up, P > F .
F (r i) anq is called the premium (if positive) or discount (if
negative). It is worth mentioning that, in real life,premium bonds
usually come with a higher coupon rate than discount bounds, so
people are willing to pay for more
than face value just for the big coupons.
6. Callable bonds are bonds that can be redeemed early by THE
BOND ISSUER (no more coupons chaps, here is
your redemption and get out of here). It is intuitively true
that a callable premium bond gives the minimum yield
to the investor when called at the earliest possible date,
because of an early capital lossyou did pay more than
the face value. Would you be happy if I sell you a $10-par bond
for $11 (therefore a premium bond) and call it
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immediately? No, because you just lost a dollar in one second.
Another intuitive way to see it is, remember what
we said earlier, in real life, you buy premium bonds for the big
coupons, so the more coupons you get, the better.
Although this is not the case in FMwe often see premium and
discount bonds that oer the same couponsthis
is because you are only learning the concepts so it is
convenient for the books/guides to use the same numbers
(especially coupon rate) over and over. Just think: if things
were like this in real life, then who on earth would buy
a premium coupon when they can buy a discount coupon that oer
the exact same amount of returns (dollar-wise)?
Similarly, a callable discount bond gives the minimum yield to
the investor when called at the latest possible date,
because of a late capital gainyou did pay less than the face
value. Would you be happy if I sell you a $10-par
bond for $9 (therefore a discount bond) and call it immediately?
Yes, because you just made a dollar in one second.
7. Callable bonds are priced as low as possible, because
investors are all prudentthey are afraid that the bond issuer
might call the bond at the worst time (minimum yield rate to the
investor), so they want to pay the smallest P !
A callable premium bond is priced as if it will be redeemed at
the earliest possible time (if n gets smaller in 11,
then the premium, a positive number, gets less positive, and so
P gets smaller). Similarly, a callable discount bond
is priced as if it will be redeemed at the latest possible time.
(if n gets bigger in 11, then the discount, a negative
number, gets more negative, and so P gets smaller). If they are
not priced in this way, prudent investors will simply
not buy these bonds.
8. The original formula 10 and the premium-discount formula 11
are two ways to calculate the price of a bond. There
is a third formula, the Makeham's Formula, it is in a way
similar to 11instead of the dierence between r and i,
we use the ratio of r and i. Here is the derivation, keep in
mind that all of our v and anq are based on i, not r, and
also that R = F is assumed as usual:
P = xanq +Rvn
= x1
(1
1+i
)ni
+K
= Fr1
(1
1+i
)ni
+K
=r
i
(F F
(1
1 + i
)n)+K
=r
i(F K) +K
where K = Fvn = F(
11+i
)n. Also, note that this formula is basically 10 except we write
xanq in a fancy way.
So you do not really have to memorize this formula, you can just
derive it during the exam, assuming you are
comfortable with calculations. But I personally nd this formula
cute and decided to memorize it.
9. Assuming R = F , principal of premium or discount paid in
period t is F (r i) vnt+1. But wait a minute, this looksjust like
9, except we use F (r i), which is a small number. But it makes
sense, because the premium or discount isa (relatively speaking)
small number to begin with! Also, notice that, when r > i, F (r
i) is positive, because thepremium is positive, and when r < i,
F (r i) is negative, because the discount is negative! It is nice
when thingscheck out! Lastly, just like 9 and many other
t-in-the-power formulas, when given the value of F (r i) vnt1+1,
wecan calculate a dierent F (r i) vnt2+1 by F (r i) vnt2+1 = F (r
i) vnt1+1 vt1t2 .
10. Make sure you know how to use calculator to calculate date
dierence. Price of a bond sold between two coupons:
P0 (1 + i)t tx,
where t = settlement datelast coupon date# of days in this
period and P0 is the price immediately after the last coupon date.
If you sell thebond to another person immediately after the last
coupon date, it would be a brand new bond to that person, the
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price would be P0 which is easy to compute. However, when the
settlement date is in between two coupon dates,
we have two problems:
(a) The bond price (P0) has increased to P0 (1 + i)t(or P0 (1 +
ti) if you are asked to use simple interest), because
some time has passed since the bond was worth exactly P0. This
part is called the dirty price.
(b) The dirty price uctuates too much, so for appearance
purposes, all quote prices take away tx from the dirty
price to obtain the clean price P0 (1 + i)t tx, the clean price
is what we use for FM, it is NOT the actualtrading price.
Make sure you know how to use the BOND worksheet on your
calculator! The BOND worksheet gives you the
clean price (not using the simple interest), so when the
question asks you to nd the purchase price, make sure you
add the accrued interest back on. Also, when the question asks
you to use simple interest specically, you have to
calculate P0 (1 + ti) manually and not rely on the BOND
worksheet.
11. I am surprised that these did not came up in the rst few
chapters, but when the question gives you the value of
vn, say vn = k, yet the bond has semi-annual coupons, make sure
you use k2 when you calculate things, because I
am pretty sure that is what you actually need! Also, when the
question asks you for the number of yeas a coupon
is held for, always divide your N from the calculator, most
likely half years or quarters, by 2 or 4, respectively.
Tricky bastards! My friend says always write down Question #:
annually/semi-annually/quarterly(4)/monthly on
the scrap paper and circle it. So something like this,
Question 21, quarterly (4) , should keep those pesky
bastards
from stealing points from you.
12. (2005 Exam FM Sample Questions #30) As of 12/31/03, an
insurance company has a known obligation to pay
1, 000, 000 on 12/31/2007. To fund this liability, the company
immediately purchases 4-year 5% annual coupon
bonds totaling $822, 703 of par value. The company anticipates
reinvestment interest rates to remain constant at
5% through 12/31/07. The maturity value of the bond equals the
par value. Under the following reinvestment
interest rate movement scenarios eective 1/1/2004, what best
describes the insurance company's prot or (loss) as
of 12/31/2007 after the liability is paid?
Insight from this question:
(a) 4-year 5% annual coupon bonds totaling $822, 703 of par
value means the coupon is an annual coupon that pays
822703 0.05 = $41135.15 per year for 4 years, this sentence does
not tell you about the redemption value yet.Them fancy words!
(b) When you reinvest the coupon payments at the same rate as
your bond, and when the redemption rate is par,
then your money is petty much growing at i, like here, 822703
(1.05)4 = 1, 000, 000 which is desired. This ismathematically true,
because xsnq + F = xanq (1 + i)
n+ F = (xanq + Fvn) (1 + i)
n= F (1 + i)
n.
(c) But then you might ask, why do I have to reinvest? The yield
per period rate i is already at 5%! Because
the coupons that you get do not magically grow according to the
bond's yield per period rate i if you do not
reinvestthe x you get from the rst coupon will still be x at the
time of redemption if you just let it sit on
your desk!
i. Andrew reinvests at the bond's yield per period rate i, so
his original investment is growing at i, so he ends
up with an even over-all yield per period iA = i by reinvesting
at i.
ii. Bill reinvests at a rate higher than the bond's yield per
period rate i , he ends up with more money than
Andrew in the end, so he achieves a higher over-all yield per
period rate iB > iA = i.
iii. Chris reinvests at a lower rate than the bond's yield per
period rate i , he ends up with less money than
Andrew in the end, so he achieves a lower over-all yield per
period rate iC < iA = i.
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5 Yield Rate of an Investment
1. The internal rate of return formula is just a special case of
the net present value formula:
NPV = C0 + C1v + C2v2 + Cnvn (12)
When you calculate the net present value, i is given so that the
left hand side can be calculated. When you calculate
the internal rate of return, the left hand side of 12 is 0 to
begin with, you are required to nd v such that the
equation holds. It is nothing but an nth degree polynomial. We
are however, only interested in roots that are close
to 1 because v = 11+i is close to 1. Once we solve for v, we can
solve for i. The nthdegree polynomial that you are
going to get will be an easy one, it helps to know the quadratic
formula and some other simple techniques. But at
the end of the day, you could always nd the answer by trial and
error, not recommended but can be a last resort.
2. Know how to use the CF worksheet. When the money at time t1
is reinvested until time t2, the Ct1 entry becomes
0, but the Ct2entry now increases by whatever the total amount
of the total payo of the reinvestment is at time t2.
This is very important, you should NEVER keep the Ct1 entry and
just add the earned interest to Ct2 . For example,
3. Eective (means actual interest rate +1) rate of time weighted
rate of interest is simply the product of eective rate
of interest from equal-length sub-periods,
1 + i = (1 + i1) (1 + i1) (1 + in) . (13)
Notice that in actual exam questions, it may tell you the
balance at the end of January 31st, November 30th,
December 31st only, the sub-periods might seem unequal at rst
glance, but you can just assume the monthly
interest rate from February 1st to October 30th to be 0, then
calculate (1 + i1) 1 1 (1 + i11) (1 + i12).Basically, unless you
have negative interest periods and positive interest periods that
multiply out to be exactly 1,
like when 0.80 1.25 = 1, the time weighted rate of interest is
rarely 0.
4. Dollar weighted Rate of Interest is a little bit more tricky.
First, we give the formula for net cashow. Here we do
not care about interest/time value of money at all, it is just
elementary addition and subtraction,
Net cashow = End balance Initial balanceNet contribution.
Remember contributions are only deposits (positive
contributions) and withdraws (negative contributions) , when
the fund simply drops or increases on its own, we do not count
that towards the contribution.
Then, we give the formula for weighted contributions, notice how
this is dierent from net contribution because now
we actually care about when these contributions are made,
Weighted contribution =
Ct (1 t) ,
where Ct is the contribution at time t (for example, when the
rst contribution is made at the end of the rst month,
when the total period is 1 year, then t = 112 ). Finally, dollar
weighted rate of interest given by
Net cashow
Initial balance+Weighted contributions(14)
Keep in mind that in the numerator you subtract the (net)
contributions, in the denominator you add the (weighted)
contributions. Do lots of practice because it is very easy to
mess up the signs. Word of advice: it is easier to keep
the contributions in a bracket of its own, that way, you do not
have to worry about signs too much.
5. Keep in mind that, 13 gives the eective interest rate (1 +
i), but 14 gives the actual interest rate.
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6. Portfolio method and investment year method are pretty
straight forward. Just practice with a table until you are
absolutely comfortable with it.
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6 Term Structure of Interest Rates
1. A zero coupon bond is a bond that does not give any coupons.
The nyear maturity quote of a Treasury STRIPzero-coupon bond is the
spot rate sn. So 1 dollar today would be worth, n years from now,
(1 + sn)
n.
2. Let us look at one simple example,
Year Spot Rate
1 s1 = 2%
2 s2 = 3%
3 s3 = 3.5%
4 s4 = 4%
5 s5 = 4.5%
The price of a four year $1000 par bond with 3% coupon rate,
based on the above table, is 301.02 +30
1.032 +30
1.0353 +10301.044 =
965.20.
3. Eective forward rate from time n to time n+ 1 (in, n+1 is
called the n-year forward rate) is
1 + in, n+1 =(1 + sn+1)
n+1
(1 + sn)n (15)
The index and power in the numerator are both n+ 1, the index
and power in the denominator are both n, so this
should be an easy to remember formula. Notice that a direct
result of 15 is
(1 + sn)n
= (1 + i0, 1) (1 + i1, 2) (1 + in2, n1) (1 + in1, n) (16)
where i0, 1 is dened to be s1. 16 makes perfect sense
intuitively if you stare at it long enough. It also helps to
realize that the n-year forward rate today is simply the 1year
spot rate in year n!
4. One year forward rate for year two means i1, 2. The for year
two part is redundant and meant to trick you into
calculating i2, 3 instead. Remember the n-year forward rate is
simply in, n+1.
5. One year eective rate during year n is simply in1, n. This is
just a dierent way of saying (n 1)-year forwardrate.
6. (Nov 05 #15) You are given the following term structure of
spot interest rates:
Term (in years) Spot Rate
1 s1 = 5%
2 s2 = 5.75%
3 s3 = 6.25%
4 s4 = 6.5%
A three-year annuity-immediate will be issued a year from now
with annual payments of 5000. Using the forward
rates, calculate the present value of this annuity a year from
now.
Insight from this question:
(a) A three-annuity-immediate issued one year from now means the
rst of the three payments do not come in until
2 on the time axis.
(b) The question itself is easy to solve, we can simply
calculate the current value of the payments and then multiply
by 1.05:(
50001.05752 +
50001.6253 +
50001.0654
) 1.05(c) However, this question and its solution provided by
ACTEX raises an interesting issue: how do we calculate
spot rate and forward rate at a future time? The 1-year spot
rate n year from now, s (n)1 is simply the
-
n-year forward rate today, in, n+1 =(1+sn+1)
n+1
(1+sn)n . The 2-year spot rate n year from now, s (n)2, must
satisfy
(1 + s (n)2)2
= (1 + in, n+1) (1 + in+1, n+2) =(1+sn+1)
n+1
(1+sn)n
(1+sn+2)n+2
(1+sn+1)n+1 =
(1+sn+2)n+2
(1+sn)n . Similarly, (1 + s (n)3)
3=
(1+sn+3)n+3
(1+sn)n . So in general, we have, the k-year spot rate n year
from now is
(1 + s (n)k)k
=(1 + sn+k)
n+k
(1 + sn)n . (17)
17 makes perfect sense if we move the denominator to the left
hand side and obtain
(1 + s (n)k)k
(1 + sn)n
= (1 + sn+k)n+k
.
Also notice that, when n = 0, that is, when we talk about today,
17 gives (1 + sk)k
= (1 + sk)kwhich is
trivially true!
-
7 Term Structure of Interest Rates
1. Macaulay duration, D (i) = D, is a function of i.2 This is an
important concept, you have to remember that you
are dealing with a very complicated function here, not just a
number.
D (i) =vC1 + 2v
2C2 + . . .+ nvnCn
vC1 + v2C2 + . . .+ vnCn
=
(1
1+i
)C1 + 2
(1
1+i
)2C2 + . . .+ n
(1
1+i
)nCn(
11+i
)C1 +
(1
1+i
)2C2 + . . .+
(1
1+i
)nCn
, (18)
where Ck's are investment cash ows at time k. It is worth
mentioning that, when Ck's are non-negative, 1 D nas shown below,
let
(1
1+i
)kCk = ak. Since ak's are positive, we have
1 =a1 + a2 + . . .+ ana1 + a2 + . . .+ an
a1 + 2a2 + . . .+ nana1 + a2 + . . .+ an
= D na1 + na2 + . . .+ nana1 + a2 + . . .+ an
= n.
Usually we write vC1 + v2C2 + . . . + v
nCn = P (i) = P because it is the net present value (the price)
of the
investments.
2. Modied Duration,
M (i) = M = vD = P
P. (19)
This is how I remember M = vD, like in sports they have MVPmost
valuable player, MVDmost valuable
duration.
3. If you clearly understand 18, then you do NOT need to
memorize any of the level payment/bond/zero-coupon bond
duration formulas. They are all easily followed if you write out
18. I will do it for you here, but make sure you do
it on your own.
(a) Level payments with payments x, the duration is
D =vC1 + 2v
2C2 + . . .+ nvnCn
vC1 + v2C2 + . . .+ vnCn=vx+ 2v2x+ . . .+ nvnx
vx+ v2x+ . . .+ vnx=v + 2v2 + . . .+ nvn
v + v2 + . . .+ vn=
(Ia)nqanq
.
It helps that you remember the denominator of D is the price/net
present value of the cashow, so the above
formula makes perfect sense.
(b) Bonds with payments x and redemption payment R, the duration
is
D =vx+ 2v2x+ . . .+ nvn (R+ x)
vx+ v2x+ . . .+ vn (R+ x)=x (Ia)nq + nv
nR
xanq + vnR.
Again, it helps that you remember the denominator of D is the
price/net present value of the cashow, so the
denominator is just the price of the bond! It is very important
that during the exam, you must not forget to
multiply R with n.
(c) Bonds with payments x and redemption payment xi , notice
that the most common bond satisfying this condition
2
We write D (i) = D to mean that we will use D for Macaulay
duration sometimes for convenience, even though it is a function of
i really.
-
is just an r = i par bond, the duration is
D =x (Ia)nq + nv
n xi
xanq + vn xi
=(Ia)nq +
nvn
i
anq + vn
i
=
(anq +
(anqnvn
i
))+ nv
n
i
1vni +
vn
i
=anq +
(anqi
)1i
= anq
(1 +
1
i
)i
= anq (1 + i) = anq.
(d) Zero-coupon bonds
D =vC1 + 2v
2C2 + . . .+ nvnCn
vC1 + v2C2 + . . .+ vnCn=nvnR
vnC= n.
Notice that is is also the maximum of the Macaulay duration
function, it is achieved precisely when there is
only 1 nal payment. Can you spot when the minimum is
achieved?3
4. When you write FM, many duration questions can be solved by
simply calculating M = P P once you gure outwhat P is (do NOT plug
in i yet), M is usually a nice function because P and P often have
common factors andsimplify beautifully. Then once you have M , plug
in the given i to get a number, then do D = M (1 + i). The key
point here is you do NOT plug in i in the early steps.
5. It helps to know the Taylor series, it is such a widely-used
formula, you might as well memorize it.
f (x) = f (a) +f (a)
1!(x a) + f
(a)2!
(x a)2 + f(3) (a)
3!(x a)3 + . (20)
So we let x = x+ x and let a = x in 20, we obtain
f (x+ x) f (x) = f (x)1!
x+f (x)
2!(x)
2+f (3) (x)
3!(x)
3+ . (21)
And luckily, for FM, we only need the rst two terms of the right
hand side of 21
f (x+ x) f (x) = f (x) x+ f (x) (x)2
2. (22)
22 makes sense intuitively, the change of the value of f after
shifting x by a tiny x is directly aected by two
things:
(a) The change in x ( means change), but we need to multiply the
change in x, x, by the rst derivative, which
is a measure of how a function changes as its input changes.
(b) Half of the square of the change in x, that is, (x)2
2 , but we also need to multiply that by something, seeing
it
has a second power, we need to multiply it by the second
derivative.
Sometimes, we only need one term.
f (x+ x) f (x) = f (x) x. (23)
Now to apply the above knowledge. Recall that M = P P , so P =
MP . Using 19 and 23, we have,
P (i+ i) P (i) = P = MPi = vDPi. (24)3
The minimum is also achieved precisely when there is only 1 nal
payment that arrives at time 1, that is, when n = 1.
-
Next, the convexity of a P is dened to be4
=P
P. (25)
So P = P . Using 19 and 22, we have,
P (i+ i) P (i) = P = MPi+ P (i)2
2= vDPi+ P (i)
2
2. (26)
I understand that 24 and 26 are both hard to memorize, but if
you go through the derivation above a few times,
and memorize the small formulas (they are the building blocks of
the big formulas!) like 19, 22 and 25, it becomes
a lot easier.
6. The duration of a portfolio, in which all investments have
the same interest rate shift, is the weighted average of the
durations of the investments. You really do not have the need to
memorize this formula, but here it is anyways,
D =k
wkDk
and
M =k
wkMk,
where the wk's are the weights based on the investment amount,
so bigger investment = bigger weight, smaller
investment = smaller weight.
wk =PkP
7. Immunization is where you have to keep reminding yourself
that P = P (i). Take any random function of x,
f = f (x), as long as it is not too complicated, you should be
able to dierentiate it with respect to x to obtain
f = f (x), and then dierentiate it again to obtain f = f (x).
Immunization is pretty much just this. Let Astand for asset and L
stand for liability, i0 some interest, in order to achieve immunity
near i0, we need three criteria,
PA (i0) = PL (i0) ,
P A (i0) = PL (i0) ,
and
P A (i0) > PL (i0) .
It is important to realize that all P , P and P are functions of
i, so with the bracket (i0) attached to them, it meansto evaluate
these functions at i0. Just like if we have f (x) = f = x
2, then f (x) = f = 2x and f (x) = f = 2, wecan evaluate these
functions of x at some point, say x0 = 3, f (x0) = f (3) = 3
2 = 9, f (x0) = f (3) = 2 3 = 6and f (x0) = f (3) = 2. Again,
these P , P and P are NOT functions of v, we simply use v because
it isconvenientthese functions are functions of i. So remember,
always dierentiate rst, then evaluate, because if you
evaluate rst, then you get a xed number and the derivative is
automatically 0!
8. We can dierentiate with respect to i but keep the result in
v. In fact, if f = vk =(
11+i
)k, then
f =(vk)
=d
di
[(1
1 + i
)k]=
d
di
[(1 + i)
k]
= k (1 + i)k1 = k(
1
1 + i
)k+1= kvk+1 (27)
and
f =(vk)
=d
di
(kvk+1) = k [ ddi
(vk+1
)]= k
( (k + 1) v(k+1)+1
)= k (k + 1) vk+2. (28)
4
Pronounced Gamma, it is the standard letter for convexity.
-
notice how these useful results are completely dierent from f =
kvk1 and f = k (k 1) vk2 which are completelywrong and probably
what you would have expected. This can be explained by realizing
that v is a function of i, soddi (f) =
ddv
dvdi (f) =
ddv (f)
dvdi =
ddv (f)
(v2) which utilizes the chain rule.9. If we are given
PA =
Atvt
and
PL =
Ltvt,
then to achieve immunity, we need
(a) Present value matching:
PA (i0) = PL (i0)
Given
Atvt (i0) =
Ltv
t (i0)
Denition of v
At
(1
1 + i
)t(i0) =
Lt
(1
1 + i
)t(i0)
We evaluate these functions of i at i0 and "match" them
At
(1
i0 + 1
)t=
Lt
(1
i0 + 1
)t.
We dene a new vi0 for easier notation
Atvti0 =
Ltv
ti0 (29)
You do not need to memorize 29 as it is intuitively true and
easy to set up. Again, just to emphasize, all
of these make sense only because we are given an interest rate
i0. We cannot match two functions without
an input. We can only match two evaluated functions. We will
look another example. We cannot compare
f (x) = x and g (x) = x2 in general. We can only compare them if
we know what input we are using. When
x = x0 = 0.1, we have 0.1 = f (x0) = f (0.1) > g (0.1) = g
(x0) = 0.01. When x = x0 = 1, we have
1 = f (x0) = f (1) = g (1) = g (x0) = 1. When x = x0 = 2, we
have 2 = f (x0) = f (2) < g (2) = g (x0) = 4.
-
(b) Duration matching:
MA (i0) = ML (i0)
Denition of M
PA
PA(i0) = P
L
PL(i0)
To evaluate
fg (x0) is (usually, so here) the same as evaluating
f(x0)
g(x0)
PA (i0)
PA (i0)= P
L (i0)
PL (i0)
Because we are already matching present value, so
PA(i0)=PL(i0)
P A (i0) = PL (i0)
See 27
At(tvt+1) (i0) = AL (tvt+1) (i0)
Multiply both sides by 1v , we can do this because t starts at
1
tAtvt (i0) =
tALv
t (i0)
Denition of v
tAt
(1
1 + i
)t(i0) =
tAL
(1
1 + i
)t(i0)
We nally get to evaluate and "match"tAt
(1
1 + i0
)t=
tAL
(1
1 + i0
)tEasier notation
tAtvti0 =
tALv
ti0 (30)
30 is similar to 29 except with an extra t, you need to memorize
it. Also try to understand the reasoning behind
each .
-
(c) Greater convexity for assets:
A (i0) > L (i0)
Denition of
P APA
(i0) >P LPL
(i0)
Same as above
P A (i0) > PL (i0)
See 28
At(t (t+ 1) vt+2
)(i0) >
AL(t (t+ 1) vt+2
)(i0)
Multiply both sides by
1v2, we can do this because t starts at 1
At(t (t+ 1) vt
)(i0) >
AL(t (t+ 1) vt
)(i0)
We are simply breaking the bracket on both sides
t2Atvt (i0) +
Atv
t (i0) >
t2Ltvt (i0) +
Ltv
t (i0)
Denition of P
t2Atvt (i0) + PA (i0) >
t2Ltv
t (i0) + PL (i0)
Again, because PA(i0)=PL(i0)t2Atv
t (i0) >
t2Ltvt (i0)
Denition of v
t2At
(1
1 + i
)t(i0) >
t2Lt
(1
1 + i
)t(i0)
We nally get to evaluate and "match"t2At
(1
1 + i0
)t>
t2Lt
(1
1 + i0
)tEasier notation
t2Atvti0 >
t2Ltv
ti0 (31)
31 is also similar to 30 except with yet another t, you need to
memorize it. But again, try to understand the
reasoning behind each .
10. The price formula for a equal-dividend stock is pretty
trivial, it is just the rst dividend, x, multiplied by 5,
x
i g (32)
Notice that,
(a) When g = 0, it means that the dividend is not increasing and
stays x forever.
(b) When g > i, 32 is meaningless and useless because the
stock price diverges.
11. For the simple liability matching questions, always
calculate the longest investment rst because the earlier
coupons
of it can go towards the earlier liabilities. Never try to
calculate the bond price on paper, use your calculator. The
less calculation you do with pen and paper, the less chance of
making stupid arithmetic mistakes.
-
12. Because 29 and 30 are both equalities, sometimes you are
asked to solve an immunity question using those two. Two
equations means they can only give you two unknowns. What you
need to solve is probably something like thisa = bx+ cyd = ex+
fywhere x and y are unknowns and represent money put into two
investments. If you know that the determinant of a
2 2 matrix(a b
c d
)is ad cb. It is then easy to solve a system of equation like
this using Cramer's rule:
x =
det
a cd f
det
b ce f
= afdcbfec
y =
det
b ae d
det
b ce f
= bdeabfec
The denominator is always the determinant of the coecient
matrix
(b c
e f
), and the numerator is the determinant
of the coecient matrix where the corresponding unknown's
coecients replaced by the column vector
(a
d
). Ignore
this if you did not get A in linear algebra.
13. This really belongs somewhere in the beginning, but again,
make sure you calculate the eective/nominal rate as
requested, do not calculate one when the question is asking for
the other.
14. The full immunization technique is designed to work for any
change in the interest rate. I previously thought for
only small changes. I was wrong.
-
Part II
Financial Economics (20 35%)8 General Derivatives
1. A stock index is an average of the prices of stock in a
specic group.
2. The spot price is the current price of an asset at any
time.
3. Payo is the value of the nancial result at expiration. Prot
or net payo is payo less the future value (at
expiration) of any expenses incurred in setting up the nancial
structure involved. A zero coupon bond's payo is
the redemption value, its prot or net payo is 0.
4. A derivative is an agreement between two people that has a
value determined by the price of the underlying asset
(stock, corn, wine, etc.).
5. Derivatives are used to
(a) Reduce risk (at a cost of reduced prot), called hedging.
(b) Increase risk (in an attempt to prot), called
speculation.
(c) Capitalize on loopholes in regulatory systems in order to
circumvent unfavorable regulation, called regulatory
arbitrage.
6. Bid-ask spread is the amount by which the ask price exceeds
the bid price of the same asset at the same time.
Bid-ask spread is easy if you remember that the investors (who
enter into derivative contracts for the three reasons
listed in (5)) are always screwed by the market
makersintermediaries or traders who will buy/sell derivatives
from
customers who wish to sell/buy: the investors have to buy high
at ask price and sell low at bid price.
7. Over-the-counter market is where buyers and sellers transact
with banks and dealers rather than on an exchange.
So it is dicult to obtain statistics for its volume.
8. The typical, easy to understand buy-today&sell-tomorrow
is called lendingyou pay cash to receive the underlying
asset and late sell it to receive (hopefully more) cash. You are
said to have a long position.
9. Short-selling is the exact opposite of lendingyou borrow the
asset, sell it for cash, then later buy back the asset
and return it. You will most likely have to pay lease rate,
which does not get returned to you, to short sell. You are
said to have a short position.
10. Short-sell can happen for at least three reasons that are
not mutually exclusive,
(a) Speculation, you aim to sell high and then buy low to make a
prot.
(b) Financing, it is a common way to borrow money.
(c) Hedging, you can oset the risk of owning the stock or a
derivative on the stock. Usually done by market-makers.
11. Short-sell has two major issues,
(a) Credit risk, as the short seller, you have an obligation to
return the asset. The lender may hold onto your proceed
you received from selling the asset, and also require you to
hand over an extra amount, called a haircut. When
you return the asset, say after one period,
Collateral (1 + Repo rate)
-
(where Collateral is basically Proceed+Haircut. Repo rate is
also called Short rebate) is returned to you.
Obviously, repo rate/short rebate is lower than the market rate
of interest, or even zero when the asset borrowed
is scarce and hard to replace.
(b) Scarcity risk, again, it might be hard to nd someone willing
to lend you the asset.
12. Mark to Market is a measure of the fair value of assets or
liabilities that can change over time.
13. The margin is a performance bond that can earn interest, NOT
a premium. Both buyers and sellers are required
to post a margin with the broker to protect the counterparty
against your failure to meet your obligations. If a loss
occurs, you can choose to pay it directly; or allow it to be
taken out of the margin balance.
14. Participants are required to maintain the margin at a
minimum level, called the maintenance margin. This is often
set at 70% to 80% of the initial margin level.
15. When the margin balance drops below the maintenance margin,
the broker would make a margin call, requesting
additional margin. Failing to post additional margin would cause
the broker to close the position, and return to the
investor the remaining margin.
-
9 Options
1. Stockholders are typically paid a share of the company's
prots called dividends.
2. A forward contract is a contract to buy or sell a specied
asset at a designated future time. The forward contract
should specify
(a) Underlying asset. The type and quantity of the asset to
deliver.
(b) Expiration date. Time, place and date of delivery.
(c) The sale price.
3. The buyer in the forward contact assumes a long position and
the seller assumes a short position. Remember, the
long position holder always benets if the price of the asset
goes up, just like in (8), in fact they have the same
prot function. Similarly, the short position holder always
benets if the price of the asset goes down, just like in
(9), they have the same prot function.
4. At future exchanges, forward contract can be written for cash
settlement in which a net payment is made for the
dierence between the spot price at expiration and the forward
price.
5. Let t = 0 and t = T denote the beginning and the expiration
of the forward contract. Let S0 and ST denote the
asset prices at t = 0 and t = T . Let F0, T denote the forward
price at t = 0 for a forward contract expiring at t = T .
Let r denote the continuously compounded interest rate for the
forward buyer and seller. We then have
F0, T = S0erT
and
Prot of long forward = ST S0erT = Prot of short forward.
6. Options:
(a) European: can be exercised only on the expiration date.
(b) American: can be exercised any time.
(c) Bermudan: can be exercised only during specied periods.
7. The price of an option is called the premium, it is, of
course, what the option buyer gives to the option seller.
8. Strike price or exercise price is the agreed settlement
price.
9. The option buyer is also called the option purchaser. The
option seller is also called the option writer. I just use
buy and sell so I do not confuse myself.
Call, can buy if protable
With respect to option Buyer Seller
With respect to asset Buyer Seller
Prot/Position Long Short
(33)
This is how I remember this chart: on the left, we have BBB
(long means big), on the right, we have SSS.
Put, can sell if protable
With respect to option Seller Buyer
With respect to asset Buyer Seller
Prot/Position Long Short
(34)
-
This is how I remember this chart: rst row (with respect to
option) is the opposite of the previous chart, the rest
stay the same. When you take the exam, you will write down (33)
and (34) on your scrap paper right after the exam
begins.
5
10. Regarding the payo and prot graph of options. Since the
horizontal axis is the price of the asset at expiration, all
you need to remember is,
(a) When the price of the asset at expiration is low, nothing
will happen with a call, so the left part of the payo
graph should be horizontal for both call buyers and call
sellers.
(b) When the price of the asset at expiration is high, nothing
will happen with a put, so the right part of the payo
graph should be horizontal for both put buyers and put
sellers.
(c) It remains to see that, the right part of the call payo
graph and the left part of the put payo graph go up
or down for long or short positions respectively because as we
mentioned multiple times already: long position
makes money when the price goes up, and short position makes
money when the price goes down.
Once you have the payo graph, the prot graph is obtained by a
simple vertical shift of the pemium amount, it
shifts up for the option seller (who gets the premium) and down
for the option buyer (who pays the premium).
11. An option is in/at/out of the money if the payo would be
positive/0/negative if the option were exercised imme-
diately.
12. To see how homeowner's insurance can be viewed as an
option.
(a) First realize that it is a put option as no one is trying to
buy the underlying asset, the house.
(b) Next realize that, if the value of the house goes down (for
example, a re), then the home owner makes money
by selling it, so he is in the short position, so he is the put
buyer.
(c) Lastly, the strike price is of course the house price
subtract the deductible. It is subtraction, not addition
because in the case that the house suers a small damage, the
house owner does not get to claim it, which
means he does not exercise the put, which means the strike price
should be lower than the house price.
13. Covered writing, option overwriting, or selling a covered
call is the opposite of a cap. Covered put is the opposite
of a oor. We rst use a pair of charts to summarize these four
and memorize them with brute force. It is not too
hard, I will explain.
Prot Equivalent
Cap Short Index & Long Call Short Put
Covered Call Long Index & Short Call Long Put
and
Prot Equivalent
Covered Put Short Index & Long Put Short Call
Floor Long Index & Short Put Long Call
Notice that in these charts, the terms long/short with respect
to options refer back to (33) and (34). So for example,
long call would be the equivalent of buy call, similarly, long
put would the equivalent of sell put.
(a) First Notice how the position with respect to the index (we
use stock index instead of the usual word asset here,
you will see why) is always the opposite of the position with
respect to the option? So even without knowing
anything, you can recreate the middle column of the two
charts:
Prot Equivalent
Short Index & Long Call
Long Index & Short Call
5
See 11 for a complete short list of important formulas that you
should write down at the beginning of the exam.
-
and
Prot Equivalent
Short Index & Long Put
Long Index & Short Put
(b) Next, the equivalent option in the last column is always the
opposite of the option in the middle, with the
position opposite of the position with respect to the option in
the middle (this sentence is long, but it is really
simple and staight forward). So you can ll in some more blanks
without knowing a lot:
Prot Equivalent
Short Index & Long Call Short Put
Long Index & Short Call Long Put
and
Prot Equivalent
Short Index & Long Put Short Call
Long Index & Short Put Long Call
(c) Next, cap, is a hat, so it is made by silc (silk). Therefore
cap goes into the row with Short Index & Long Call.
The other one is then covered call.
Prot Equivalent
Cap Short Index & Long Call Short Put
Covered Call Long Index & Short Call Long Put
(35)
(d) Lastly, oor, has the exact opposite (literal meaning, not
mathematical) prot equivalent of that of capequivalent
to short put, so it goes into the row with long call as prot
equivlant. The other one is then covered put.
Prot Equivalent
Covered Put Short Asset & Long Put Short Call
Floor Long Asset & Short Put Long Call
(36)
This may seem long because I wrote out the entire thinking
process one step at a time. When you take the exam,
you will write down (35) and (36) on your scrap paper right
after the exam begins.
6
14. Now I will try my best to explain the two charts.
(a) A covered call is the opposite of a naked call, which is
when you sell a call without owning the actual asset,
you can easily see why this would be dangerousif the price of
the asset gets too high, you would be broke
because you cannot provide the asset when the call buyer
excersise the option. So to avoid naked call, when
you sell a call, you also buy the asset, then you have a covered
call.
(b) A cap is when you short an asset, but you are afraid that
when the price of the asset gets too high, you would
be broke because you cannot return the asset when it is time to
do so. So to avoid paying too much for the
asset in the future, you buy a call to cover your behind.
(c) Covered put is when you sell a put, but you are afraid that
when the price of the asset gets too low, you
will end up spending a lot of money on something that is not
worth a lot. So to make the (potential) pain
more bearable, you short the asset. This way, when the asset
gets too low, at least you made money from the
short-sale of it.
(d) Floor is just like insurance. You buy an asset, but you are
afraid that it will depreciate, so you buy a put to
cover your behind.
6
See 11 for a complete short list of important formulas that you
should write down at the beginning of the exam.
-
15. Synthetic forward mimics a long (again, long means the party
that makes money when the price of the asset goes
up) forward position on an asset by buying a call and selling a
put, with each option having the same strike price
and time to expiration. Net income from the premiums of a
synthetic forward is Put (K,T )Call (K,T ). Net costof synthetic
forward is the opposite, Call (K,T ) Put (K,T ).
16. Put-call parity is
Put (K,T ) Call (K,T ) = PV (K F0, T ) (37)
(37) makes pefect sense because if you create a synthetic
forward with strike price K, then the present value of the
dierence between K and the forward value of the asset should be
the same as the dierence between your income
from selling the put and expense from buying the call. When K
gets smaller than F0, T , the right hand side becomes
negative. But at the same time, the more you have to pay for the
call because it would seem like a really good deal
to you (bad to the call seller), so the left hand side becomes
negative too. When K gets bigger than F0, T , the right
hand side becomes positive. But at the same time, the more you
get from selling the put because it would seem like
a really good deal to the put buyer (bad to you), so the left
hand side becomes positive too.
-
10 Hedging and Investment Strategies
1. Diversiable risk is unrelated to other risks, such as a
lightning strike. Undiversiable risk is the opposite, such as
a stock market cash. There exist risk-sharing mechanisms that
benet everyone.
2. Catastrophe bonds are bonds that the issuer need not repay if
there is a specied event, such as a large earth quake.
Here is an example of tiered risk-sharing.
Policyholder Insurer Reinsurer Catastrophe bond
Note that, an insurer that purchases reinsurance is also called
the "ceding company" or "cedant".
3. Arbitrage is the simultaneous purchase and sale of an asset
in order to prot from a dierence in the price. It is a
trade that prots by exploiting price dierences of identical or
similar nancial instruments, on dierent markets or
in dierent forms. Arbitrage exists as a result of market
ineciencies; it provides a mechanism to ensure prices do
not deviate substantially from fair value for long periods of
time.
4. An option spread is a position consisting of only calls or
only puts.
(a) Bull spread is what you do when you expect the market value
of an asset to increase. Remember it by realizing
that you simply buy low and sell high (which in a way agrees
with the word increase, just remember that buy
comes before sell). You can do it with both call and put.
(b) Bear spread is what you do when you expect the market value
of an asset to decrease. Remember it by realizing
that you simply buy high and sell low (which in a way agrees
with the word decrease). You can do it with
both call and put.
5. A box spread is simply two synthetic forwards. It builds a
zero coupon bond. Done in an attempt to lower taxes.
6. Buy collar is buy low put and sell high call. Sell collar is
buy high call and sell low put.
(a) Remember this by noting that the sum of the two options is
always the opposite of the position with respect
to the collar. It is always the opposite of what you would
expect.
i. buy low + sell high, you expect the result to be sell because
there is more sell, but it is actually buy low + sell high = buy
collar.ii. buy high + sell low, you expect the result to be buy
because there is more buy, but it is actually buy high + sell low =
sell collar.(b) The high one is always the call because call sounds
similar to collar.
(c) The collar can be used to hedge the price of a purchased
asset.
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Part III
Appendix
11 List of Formulas to Write Down at the Beginning of FM
(t) =a (t)a (t)
astart amount
eend time
start time
(u)du = aend amount
Panq +Q(anq nvn
i
)anq =
1 vni
Call, can buy if protable
With respect to option Buyer Seller
With respect to asset Buyer Seller
Prot/Position Long Short
Put, can sell if protable
With respect to option Seller Buyer
With respect to asset Buyer Seller
Prot/Position Long Short
Prot Equivalent
Cap Short Index & Long Call Short Put
Covered Call Long Index & Short Call Long Put
Prot Equivalent
Covered Put Short Asset & Long Put Short Call
Floor Long Asset & Short Put Long Call
I Interest Theory (65-80%)Interest Rates and Time Value of
MoneyAnnuitiesLoan RepaymentBondsYield Rate of an InvestmentTerm
Structure of Interest RatesTerm Structure of Interest Rates
II Financial Economics (20-35%)General DerivativesOptionsHedging
and Investment Strategies
III AppendixList of Formulas to Write Down at the Beginning of
FM