Formulae

ACTS 4308
FORMULA SUMMARY
Section 1: Calculus review and effective rates of interest and discount.
1. Some useful finite and infinite series:
(i) sum of the first n positive integers:
1 + 2 + ··· + n =
n(n + 1)
2
(ii) finite geometric series:
1 − rk+1
r − rk+1
; r + r2 + · · · + rk =
1−r
1−r
(iii) infinite geometric series: if |r| < 1 then
r
1
; r + r2 + · · · =
1 + r + r2 + · · · =
1−r
1−r
(iv) increasing geometric series:
1
1 + 2r + 3r2 + · · · =
(1 − r)2
1 + r + r2 + · · · + rk =
(v) exponential and natural log series:
ex = 1 + x +
x2 x3
+
+ ···
2!
3!
x2 x3
+
− ···
2
3
Simple interest i: An amount P deposited for the time t accumulates to P (1 + it).
Compound interest i: An amount P deposited for the time t accumulates to P (1 + i)t .
Cash flow principle: when using compound interest, for a series of deposits and withdrawals
that occur at various points in time, the balance in the account at any given time point is the
accumulated values of all deposits minus the accumulated values of all withdrawals to that time
point.
Present value: the present value of 1 due in one year is the amount required now to accumulate
to 1 in one year.
1
v=
1+i
Present value of 1 due in t years for simple interest
1
1 + it
Present value of 1 due in t years for compound interest
1
= vt
(1 + i)t
for |x| < 1 ln(1 + x) = x −
2.
3.
4.
5.
6.
7.
8. d = annual effective rate of discount =
9.
10.
11.
12.
13.
A(1)−A(0)
A(1)
i
d = 1+i
v = 1 − d is the present value of 1 due in 1 year
d
i = 1−d
; id = i − d; d1 = 1i + 1
Compound discount: (1 + i)n = v −n = (1 − d)−n
Simple discount: present value of 1 due in t years is 1 − dt
2
Section 2: Nominal Rates of Interest and Discount
1. i(m) =
rate of interest compounded (or convertible) m times per year
nominal annual
= m (1 + i)1/m − 1
1
i(m)
2. The effective rate of interest for each -year is
.
m
m
!m
i(m)
3. 1 +
=1+i
m
4. d(m) = nominal annual
rate of discount compounded (or convertible) m times per year
= m 1 − (1 − d)1/m
1
d(m)
5. The effective rate of discount for each -year is
.
m
m
!m
d(m)
=1−d
6. 1 −
m
!−p
!m
d(p)
i(m)
−1
7. 1 −
= (1 − d) = 1 + i = 1 +
p
m
3
Section 3: Force of Interest, Inflation
1. i(∞) = d(∞) = δt =
A0 (t)
A(t)
=
d
dt
[ln A(t)]
(m) m
(p) −p
2. For a constant force of interest: eδ = 1 + i = 1 + i m
= 1 − dp
Z n
1 dA(t)
δt dt
3. For a non-constant force of interest: δt =
⇒ ln A(n) =
A(t) dt
0
A0 (t)
i
4. Simple interest: δt =
=
A(t)
1 + it
i−r
5. Real interest rate - interest rate adjusted for inflation: iREAL = 1+r
Pn
Pn
6. The equation of value for cash flow X = k=1 Xk : k=1 Xk v tk = Xv t
7. Exact solution
P
P
n
n
Xk tk
Xk tk
ln
− ln
k=1 X v
k=1 X v
t=
=
.
ln v
δ P
n
X t
8. Method of equated time: uses the approximate solution t̄ = k=1X k k
4
Section 4: Annuity-immediate and annuity-due.
1. The present value of an annuity-immediate with n level payments of 1 is
1 − vn
an i =
i
2. The accumulated value at time n of an annuity-immediate with n level payments of 1 is
(1 + i)n − 1
sn i =
i
3. The present value of an annuity-due with n level payments of 1 is
1 − vn
än i =
d
4. The accumulated value at time n of an annuity-due with n level payments of 1 is
(1 + i)n − 1
s̈n i =
d
5. sn i = (1 + i)n an i ; s̈n i = (1 + i)n än i
6. sn+1 i = 1 + (1 + i)sn i ; s̈n+1 i = 1 + (1 + i)s̈n i
7. A loan of 1 can be repaid by level payments throughout the n year period of the loan, or by paying
interest every year, and making payments that accumulate to the loan amount at the end of the
n year period of the loan:
1
= s 1 + i; ä 1 = s̈ 1 + d
a
n
i
n
i
n
i
n
i
8. än = (1 + i)an ; s̈n = (1 + i)sn
9. än = 1 + an−1 ; s̈n = sn+1 − 1
10. The present value for a perpetuity-immediate is
1
a∞ i = .
i
11. The present value for a perpetuity-due is
ä∞ i =
1
.
d
5
Section 5: Annuity valuation at any time point.
1. Suppose this is the balance on an account at time n and there are no more deposits into the
account, but the account continues to earn interest at an annual effective compound rate of i after
time n. Then, m periods after the final deposit into the account, the balance of the account is
sn i (1 + i)m = sn+m i − sm
i
2. The present value of a deferred annuity is
v m an i = an+m i − am
i
3. Suppose that deposits of 1 are made at the end of each period into an account earning interest
at effective rate i until the end of the nth period. Then, just after the nth deposit the effective
interest rate changes to j and deposits are made at the end of each period for m more periods.
(a) The accumulated value of this annuity at time n + m is
sn i (1 + j)m + sm
j
(b) The present value of this annuity is
an i + vin am
j
= an i + (1 + i)−n am
j
6
Section 6: Annuities with differing interest and payment period.
1. General Rule
When payment frequency and interest compounding are different, use the payment frequency
as your basis for calculation and adjust compounding to your payment frequency by finding the
effective interest rate per payment period.
For example, if payments are monthly, but compounding is semi-annual with annual nominal
(12)
rate of i(2) , the period is one month and the effective interest rate per period is j = i 12 . To find
j, use the equivalence principle between the nominal rates:
!12
!2
!1
i(12)
i(2)
i(12)
i(2) 6
1+
= 1+
⇒j=
= 1+
−1
12
2
12
2
2. Continuous Annuities
• Basic continuous annuity: payment of 1 per year is paid uniformly and continuously for
an n-year period
• The present
value is
Z n
1 − vn
t
v dt =
.
ān =
δ
0
• The accumulated
value at time
n is
Rn
n
s̄n = 0 (1 + i)n−t dt = (1+i)δ −1 .
• ān = lim a(m)
; s̄n = lim s(m)
m→∞ n
m→∞ n
• General continuous annuity: continuous rate of payment of ct at time t for an n-year
period
Rn
Rn
• The present value is 0 ct v t dt = 0 ct e−δt dt
Rn
Rn
• The accumulated value at time n is 0 ct (1 + i)n−t dt = 0 ct eδ(n−t) dt.
7
Section 7: Annuities whose payments follow a geometric progression.
1. Consider an annuity-immediate with payments which follow a geometric progression; i.e., payments
of K, K(1 + r), K(1 + r)2 , . . . , K(1 + r)n−1 are made at times 1, 2, 3, . . . , n.
2. If i 6= r, then the present value is
n 

1 − 1+r
1+i

PV = K 
i−r
3. If i 6= r, then the accumulated value at time n is
(1 + i)n − (1 + r)n
.
FV = K
i−r
4. If i = r, then the present value is
P V = nKv
and the accumulated value at time t = n is
F V = nK(1 + i)n−1 .
5. Suppose a particular stock pays dividends at the end of each year forever (perpetuity with a
geometric progression).
6. Dividend payments of K, K(1 + r), K(1 + r)2 , . . . are made at times 1, 2, 3, . . . where r is the
growth rate for the dividends.
7. If i > r, then the theoretical price of the stock (the present value of the dividend payments) is
K
S=
.
i−r
8
Section 8: Annuities whose payments follow an arithmetic progression.
1. Increasing annuity-immediate: payments of 1, 2, 3, . . . , n are made at the end of the 1st, 2nd,
3rd, . . ., nth period, respectively.
2. The present value of an increasing annuity-immediate is
ä − nv n
(Ia)n = n
.
i
3. The accumulated value of an increasing annuity-immediate at time n is denoted by
− (n + 1)
s
s̈ − n
(Is)n = n
= n+1
.
i
i
4. Increasing annuity-due: payments of 1, 2, 3, . . . , n are made at the beginning of the 1st, 2nd, 3rd,
. . . , nth period, respectively; i.e., payments are made at times 0, 1, . . . , n − 1
ä − nv n
5. (Iä)n = n
d
s̈n − n
6. (I s̈)n =
d
1
1
1
7. Increasing perpetuities: (Ia)∞ = + 2 and (Iä)∞ = 2 , if i > 0
i
i
d
8. Decreasing annuity-immediate: payments of n, n − 1, n − 2, . . . , 1 are made at the end of the 1st,
2nd, 3rd, . . . , nth period, respectively
9. The present value of a decreasing annuity-immediate is
n − an
(Da)n =
.
i
10. The accumulated value of a decreasing annuity-immediate at time n is
n(1 + i)n − sn
.
(Ds)n =
i
11. Decreasing annuity-due: payments of n, n − 1, n − 2, . . . , 1 are made at the beginning of the 1st,
2nd, 3rd, . . . , nth period, respectively; i.e., payments are made at times 0, 1, . . . , n − 1
n − an
12. (Dä)n =
d
n(1 + i)n − sn
13. (Ds̈)n =
d
14. Reinvestment rate for a single payment: Suppose interest is reinvested at a rate j on an investment
of 1 made at time 0. Then the accumulated value of the principal and interest at time n is
1 + isn j .
15. Reinvestment rate for a series of level payments: Suppose interest is reinvested at a rate j on
payments of 1 at the end of each period for n periods. Then the accumulated value of the principal
and interest at time n is
n + i (Is)
.
n−1
j
9
Section 9: Amortization of a loan.
1.
L
.
an i
payment (valued at time t)
L = Kan i =⇒ K =
2. OBt = outstanding balance after the tth
= Kan−t i (prospective formula)
= L(1 + i)t − Kst i (retrospective formula)
3. The balance goes up between payments because the loan is accruing interest according to the
formula
OBt+s = (1 + i)s × OBt , 0 < s < 1.
4. It = interest paid in the tth payment = i × OBt−1 = K(1 − v n−t+1 )
5. P Rt = principal paid in the tth payment = K − It = Kv n−t+1
6. So the outstanding balance after the tth payment
OBt = OBt−1 − P Rt = (1 + i)OBt−1 − K.
7. The principle portion of consecutive payments increases by the factor (1 + i):
P Rt = (1 + i)P Rt−1 .
8. The interest portion of consecutive payments decreases as follows:
It = It−1 − i(K − It−1 ).
9. For non-level payments: L = K1 v + K2 v 2 + . . . + Kn v n .
10.
It = i × OBt−1
P Rt = Kt − It
OBt = OBt−1 − P Rt = (1 + i)OBt−1 − Kt
10
Section 10: The sinking fund method of loan repayment.
1. i = effective interest rate per payment period paid by the borrower to the lender
2. j = effective interest rate earned by the borrower on the sinking fund
L
3. D = periodic sinking fund deposit =
sn j
4. Li = interest payment to the lender
5. K = periodic outlay by the borrower = Li + D
11
Section 11: Bond valuation.
1. “Frank” formula: P = F ran j + K where K = Cvjn is the present value of the redemption value
2. Premium-Discount formula: P = C + (F r − Cj)an j
3. There are different ways of computing bond prices at fractional points in the coupon periods.
BVt = F ran−t j + Cvjn−t
Purchase Price
Compound
Compound
Simple
Simple
Accrued Coupon
Compound
Simple
Compound
Simple
BVt+u
BVt (1 + j)u
BVt (1 + j)u
BVt (1 + ju)
BVt (1 + ju)
M Vt+u
u
BVt (1 + j)u − F r (1+j)j −1
BVt (1 + j)u − F ru
u
BVt (1 + ju) − F r (1+j)j −1
BVt (1 + ju) − F ru
12
Section 12: Bond amortization, callable bonds.
1. BVt = Book value (or Amortized value): value at time t of the remaining payments of a bond
after the tth coupon payment has been made
2. BV0 = P, BVn = C
3. Kt = tth coupon payment
4. BVt+1 = (1 + j)BVt − Kt
5. It = Interest portion of the tth coupon payment
6. It = j × BVt−1
7. P Rt = Amount of amortization of the premium with the tth coupon payment
8. P Rt = BVt−1 − BVt = Kt − It
9. If BVt > BVt−1 (equivalently P Rt < 0), then the bond is written up. If BVt < BVt−1 (equivalently P Rt > 0), then the bond is written down.
10. BVt = F ran−t j + Cvjn−t = C + (F r − Cj)an−t j
11. BVt+1 = (1 + j)BVt − F r ⇒
BVt+k = (1 + j)k BVt − F rs
for k = 1, . . . , n − t
k
j
12. P Rt = (F r − Cj)v n−t+1
13. P Rt+1 = (1 + j)P Rt ⇒
P Rt+k = (1 + j)k P Rt for k = 1, . . . , n − t
14. If a bond is bought at a premium (P > C or equivalently, F r > Cj), then the bond is written
up with each coupon payment.
15. If a bond is bought at a discount (P < C or equivalently, F r < Cj), then the bond is written
down with each coupon payment.
16. Basic rules regarding callable bonds:
If the bond is bought at premium (P > C), then for a given yield j, calculating the price based
on the earliest redemption date is the maximum price that guarantees the yield will be at least j.
If the bond bought at discount (P < C), then for a given yield j, calculating the price based on
the latest redemption date is the maximum price that guarantees the yield will be at least j.
13
Section 13: Measures of the rate of return on a fund.
I
1. Dollar-weighted rate of return: iD =
, where B = A + C + I
n
X
A+
Ck (1 − tk )
k=1
2. Time-weighted rate of return = iT = (1 + j1 )(1 + j2 ) · · · (1 + jn+1 ) − 1, where Bk = account
balance just before investment Ck and 1 + jk = “effective rate of return between two consecutive
Bk
(C0 = 0, Bn+1 = B)
contributions Ck−1 and Ck ” =
Bk−1 + Ck−1
3. Fund accumulation methods:
Portfolio method: All investors are pooled together in the same overall portfolio, and every
investor gets the same return.
Investment year method: Segregate the money of all individuals who started in a given year
and give them all the return on that segregated fund that is unique to them.
14
Section 14: The term structure of interest rates, forward rates of interest and duration.
1. Let sn be the spot rate for n years to maturity for n = 1, 2, . . ..
2. Let fj be the forward rate from time j until time j + 1 for j = 1, 2, . . ..
3. The spot rates can be determined from the forward rates using
(1 + sn )n = (1 + f0 )(1 + f1 ) · · · (1 + fn−1 ).
4. The forward rates can be determined from the spot rates using
(1 + sn+1 )n+1
,
f0 = s1 .
(1 + sn )n
5. Duration: measure of sensitivity to changes in the interest rate
6. At time 0, suppose a series of future cashflows will occur at times t1 , . . . , tn of amounts A1 , . . . , An ,
respectively.
7. The present value as a function of the force of interest δ is
n
X
P =
e−δtk Ak .
1 + fn =
k=1
P]
= − d[ln
8. (Macaulay) Duration = D = −dP/dδ
P
dδ
n
n
X
X
dP
−δtk
9. −
=
tk e
Ak =
tk v tk Ak
dδ
k=1
Pnk=1 t
n
X
kA
t
v
v tk Ak
k
k
=
t
w
where
w
=
10. D = Pk=1
k
k
k
n
tk
P
k=1 v Ak
k=1
11. Modified duration (volatility) = DM = −dP/di
= − d[lndiP ]
P
dP dδ
1 dP
dP
dP
12. −
=−
=−
= −v
since δ = ln (1 + i)
di
dδ di
1 + i dδ
dδ
D
13. DM =
= vD
1+i
14. Duration of a portfolio: Suppose there are m cashflow streams with respective present values
P1 , P2 , . . . , Pm . Then the duration (either Macaulay or modified) of the combined set of cashflows
is
P1
P2
Pm
× D1 +
D2 + . . . +
Dm
D=
P
P
P
where Dk is the duration of the kth cashflow stream and P = P1 + P2 + . . . + Pm .
15. The Macaulay duration of an annuity immediate with n payments is
Pn
(Ia)n
kv k
D = Pk=1
=
.
n
k
an
k=1 v
16. The Macaulay duration of a coupon bond with face value F and coupon rate r for n periods and
with redemption value C is
P
F r(Ia)n + nK
F r nk=1 kv k + nCv n
Pn
D=
.
=
k
n
F ran + K
F r k=1 v + Cv
17. Cash flow matching: the present value of assets PB and the present value of liabilities PA are
matched at interest rate i0
18. The liabilities are immunized by the assets if the present value of assets is greater than or equal
to the present value of liabilities for any small change in interest rates.
15
Section 15: Introduction to financial derivatives, forward and futures contracts.
P =S .
1. If an asset pays no dividends, then the prepaid forward price is F0,T
0
2. If an asset pays dividends with amounts D1 , . . . , Dn at times t1 , . . . , tn , then the prepaid forward
price is
n
X
P
F0,T = S0 −
Di e−rti
i=1
3. If an asset pays dividends as a percentage of the stock price at a continuous (lease) rate δ, then
P = S e−δT .
the prepaid forward price is F0,T
0
4. Cost of carry = r − δ
P erT
5. F0,T = F0,T
6. Implied fair price: the implied value of S0 when it is unknown based on an equation relating S0
to F0,T
7. Implied repo rate: implied value of r based
on
the price of a stock and a forward
F0,T
1
8. Annualized forward premium = T ln S0
16
Section 16: Introduction to options.
1. Assets
(a) S0 = asset price at time 0; it is assumed the purchaser of the stock borrows S0 and must repay
the loan at the end of period T
(b) Payoff(Purchased Asset)= ST
(c) Profit(Purchased Asset)= ST − S0 erT
(d) Payoff(Short Asset)= −ST
(e) Profit(Short Asset)= −ST + S0 erT
2. Calls
(a) C0 = premium for call option at time 0; it is assumed the purchaser of the call borrows C0
and must repay the loan when the option expires
(b) Payoff(Purchased Call)= max {0, ST − K}
(c) Profit(Purchased Call)= max {0, ST − K} − C0 erT
(d) Payoff(Written Call)= − max {0, ST − K}
(e) Profit(Written Call)= − max {0, ST − K} + C0 erT
3. Puts
(a) P0 = premium for put option at time 0; it is assumed the purchaser of the put borrows P0
and must repay the loan when the option expires
(b) Payoff(Purchased Put)= max {0, K − ST }
(c) Profit(Purchased Put)= max {0, K − ST } − P0 erT
(d) Payoff(Written Put)= − max {0, K − ST }
(e) Profit(Written Put)= − max {0, K − ST } + P0 erT
17
Section 17: Option Strategies (Part I).
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Floor: buying an asset with a purchased put option
Cap: short selling an asset with a purchased call option
Covered put (short floor): short selling an asset with a written put option
Covered call (short cap): buying an asset with a written call option
Synthetic Forward: combination of puts and calls that acts like a forward; purchasing a call
with strike price K and expiration date T and selling a put with K and T guarantees a purchase
of the asset for a price of K at time T
Short synthetic forward: purchasing a put option with strike price K and expiration date T
and writing a call option with strike price K and expiration date T
No Arbitrage Principle: If two different investments generate the same payoff, they must have
the same cost.
C(K, T ) = price of a call with an expiration date T and strike price K
P (K, T ) = price of a put with an expiration date T and strike price K
Put-Call parity: C0 − P0 = C(K, T ) − P (K, T ) = (F0,T − K)v T
If the asset pays no dividends, then the no-arbitrage forward price is F0,T = S0 erT ⇒ S0 = F0,T v T .
C(K, T ) + Kv T = P K, T ) + S0
18
Section 18: Option Strategies (Part II).
1. Spread: position consisting of only calls or only puts, in which some options are purchased and
some written
2. Bull: investor betting on increase in market value of an asset
3. Bull spread: purchased call with lower strike price K1 and written call with higher strike price
K2
4. A bull spread can also be created by a purchased put with strike price K1 and a written put with
strike price K2 .
5. Bear: an investor betting on a decrease in market value of an asset
6. Bear spread: written call with lower strike price K1 and purchased call with higher strike price
K2
7. Ratio spread: purchasing m calls at one strike price K1 and writing n calls at another strike
price K2
8. Box spread: using options to create a synthetic long forward at one price and a synthetic short
forward at a different price
9. Purchased collar: purchased put option with lower strike price K1 and written call option with
higher strike price K2
10. Collar width: difference between the call and put strikes
11. Written collar: writing a put option with lower strike price K1 and purchasing a call option with
higher strike price K2
12. Zero-cost collar: choosing the strike prices so that the cost is 0
13. Purchased straddle: purchased call and put with the same strike price K
14. Written straddle: written call and put with the same strike price K
15. Purchased strangle: purchased put with lower strike price K1 and purchased call with a higher
strike price K2
16. Written strangle: written put with lower strike price K1 and written call with a higher strike
price K2
17. Butterfly spread: a written straddle with a purchased strangle
18. Asymmetric butterfly spread: purchasing λ calls with strike price K1 , purchasing 1 − λ calls
K3 − K2
and K1 < K2 < K3
with strike price K3 , and writing 1 call with strike price K2 where λ =
K3 − K1
19
Section 19: Swaps.
1. sn = n-year spot rate as defined in Section 14
2. r0 (n, n + 1) = n-year forward rate as defined in Section 14:
1 + r0 (n, n + 1) =
(1 + sn+1 )n+1
(1 + sn )n
3. P (t1 , t2 ) = price of a zero-coupon bond that is purchased at time t1 and pays 1 at time t2
P (0, n − 1)
1
=
4. P (0, n) =
n
(1 + sn )
1 + r0 (n − 1, n)
5. R = Swap rate is the rate for which the present value of level interest payments at the end of
each period equals the present value of interest payments at the forward rates at the end of each
period
n
n
X
X
1 − P (0, n)
6. R
P (0, k) =
r0 (k − 1, k)P (0, k) ⇒ R = Pn
k=1 P (0, k)
k=1
k=1
7. Deferred swap: a swap that starts at some time in the future but the swap rate is agreed upon
today
8. For a deferred swap with the first loan payment in m periods on an n-period loan, the swap rate
is
P (0, m − 1) − P (0, m − 1 + n)
Pn
R=
k=1 P (0, m − 1 + k)

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