Exam FM
Formula & Review Sheet
ACTEX Learning
(updated 06/29/2023)
TIME VALUE OF MONEY
Accumulation and Amount Functions:
A(t) = Ka(t),
Effective Interest Rate:
it =
Simple Interest:
a(t) = 1 + it
Compound Interest:
a(t) = (1 + i)t
Effective Discount Rate:
dt =
A(0) = K,
a(0) = 1
a(t) − a(t − 1)
A(t) − A(t − 1)
=
a(t − 1)
A(t − 1)
it =
i
1 + i(t − 1)
it = i
a(t) − a(t − 1)
A(t) − A(t − 1)
=
a(t)
A(t)
i
1 1
d=
= 1 − v = iv
v = (1 + i)−1
− =1
1+i
d
i
m
m
i(m)
d(m)
1+
=1+i 1−
=1−d
m
m
h
i
h
i
1
1
i(m) = m (1 + i) m − 1 d(m) = m 1 − (1 − d) m
Discount Rate:
Nominal Rates:
a0 (t)
A0 (t)
d
=
=
ln a(t)
a(t)
A(t)
dt
Force of Interest:
δt =
Constant Force of Interest:
eδ = 1 + i
PV of $1 Due in t Years:
PV =
AV at time t of $1 invested at time 0:
AV at time t2 for $1 Invested at time t1 :
Rt
a(t) = e 0 δr dr
δ = ln(1 + i)
1
= (1 + i)−t = v t = e−δt = (1 − d)t
a(t)
−mt mt
i(m)
d(m)
= 1+
= 1−
m
m
mt −mt
i(m)
d(m)
AV = a(t) = (1+i)t = eδt = (1−d)−t = 1 +
= 1−
m
m
R
t
2
a(t2 )
AV =
= e t1 δt dt
a(t1 )
ANNUITIES
Annuity-Immediate:
(PV one period before first payment, AV at time of last payment)
P V = an = v + v2 + · · · + vn =
1 − vn
i
AV = s n = 1 + (1 + i) + · · · + (1 + i)n−1 =
Annuity-Due:
(1 + i)n − 1
= a n (1 + i)n
i
(PV at time of first payment, AV one period after last payment)
1 − vn
i
P V = ä n = 1 + v + · · · + v n−1 =
= (1 + i)a n =
a n = 1 + a n−1
d
d
(1 + i)n − 1
AV = s̈ n = (1 + i) + (1 + i)2 + · · · + (1 + i)n =
d
i
= (1 + i)s n =
s n = s n+1 − 1
d
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ACTEX Learning
Exam FM
Page 2
Rn
Rn
1 − vn
i
=
a n = 0 v t dt = 0 e−δt dt
δ
δ
Rn
Rn
(1 + i)n − 1
i
AV = s̄ n =
=
s n = 0 (1 + i)n−t dt = 0 eδ(n−t) dt
δ
δ
Continuous Annuity:
P V = ā n =
Deferred Annuity:
m |a n = m+1 |ä n = v
Perpetuity:
a∞ =
1
i
m
a n = a m+n − a m
ä ∞ =
1
d
ā ∞ =
1
δ
Increasing Annuity - Payments in Arithmetic Progression:
ä n − nv n
i
ä n − nv n
(Iä) n =
d
(Ia) n =
s n+1 − (n + 1)
s̈ n − n
=
i
i
s n+1 − (n + 1)
s̈ n − n
(I s̈) n =
=
d
d
(Is) n =
Decreasing Annuity - Payments in Arithmetic Progression:
n(1 + i)n − s n
i
n(1 + i)n − s n
(Ds̈) n =
d
n − an
i
n − an
(Dä) n =
d
(Da) n =
(Ds) n =
Increasing Perpetuity - Payments in Arithmetic Progression:
1
1
1
1
+ 2 =
(Iä) ∞ = 2
i
i
id
d
1 − vn
1 − vn
(m)
(m)
a n = (m)
ä n = (m)
i
d
(m)
n
ä n − nv n
ä
−
nv
n
(m)
(m) (m)
(Ia) n =
(I
a)
=
n
i(m)
i(m)
n
R
R
¯ n = ā n − nv = n tv t dt = n te−δt dt
(Iā)
0
0
δ
Rn
s̄ n − n R n
n−t
¯
(I s̄) n =
= 0 t(1 + i) dt = 0 teδ(n−t) dt
δ
(Ia) ∞ =
m-thly Annuity:
Continuously Increasing
Annuity:
Annuity with Varying Payments and Varying Force of Interest:
PV =
Rn
AV =
Rn
0
0
f (t)e−δt dt
If force of interest varies:
PV =
Rn
f (t)eδ(n−t) dt
If force of interest varies:
FV =
Rn
0
0
Rt
f (t)e− 0 δr dr dt
Rn
f (t)e t δr dr dt
Geometric Annuity-Immediate: 1st Payment is 1. Subsequent payments increase by a factor of (1 + k).
n
1+k
1−
(1 + i)n − (1 + k)n
1+i
PV =
FV =
i−k
i−k
Geometric Annuity-Due:
Sum of a Geometric Series:
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1st Payment is 1. Subsequent payments increase by a factor of (1 + k).
n
1+k
1−
(1 + i)n − (1 + k)n
1+i
PV =
FV =
d − kv
d − kv
n
1
−
r
a + ar + ar2 + · · · + arn−1 = a
1−r
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ACTEX Learning
Exam FM
Page 3
LOANS
Amortization Method: Level payment P , Outstanding Balance Bt , Principal Repayment P rint
L = B0 = P a n
P = P rint + It
It = iBt−1
P rint = P − It = P v n−t+1
P rint+s = P rint (1 + i)s
It = P − P rint = P (1 − v n−t+1 )
n
P
n
P
It = nP − L
Bt = Bt−1 − P rint
P rint = L
t=1
t=1
Bt = P a n−t (Prospective)
Bt = L(1 + i)t − P s t (Retrospective)
BONDS
Price Formulas:
Number of coupon payments n, Coupon rate r, Face amount F , Maturity value C
P = F ra n + Cv n
P = C + (F r − Ci)a n
g
Fr
P = K + (C − K), where K = Cv n and g =
i
C
Callable Bonds:
To calculate appropriate price:
If Bond is sold at a Premium, assume Earliest Redemption date
If Bond is sold at a Discount, assume Latest Redemption date
Exception: If a premium bond has a call premium, calculate price both ways
(using Earliest Redemption date and using Latest Redemption date).
Use the smaller of the 2 calculated values.
If g > i, then P > C, and Premium = P − C = (F r − Ci)a n = (Cg − Ci)a n
If g < i, then P < C, and Discount = C − P = (Ci − F r)a n = (Ci − Cg)a n
Book Value at time t:
Bt = F ra n−t + Cv n−t
Interest Earned:
It = iBt−1
If g > i, Premium Amortized at time t = F r − It = Bt−1 − Bt = (F r − Ci)v n−t+1
If g < i, Discount Accumulated/Accrued at time t = It − F r = Bt − Bt−1 = (Ci − F r)v n−t+1
GENERAL CASH FLOW
Reinvestment Rates:
Interest rate i, Reinvestment rate i0
A single deposit of $1, AV = 1 + is n i0
Deposits of $1 at beginning of each year, AV = n + i(Is) n i0
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Spot Rates:
Forward Rates:
(Annual Effective)
Inflation Rate:
Exam FM
Effective annual rate on investment for t years:
rt
Price of a t-year zero-coupon Bond:
Pt = (1 + rt )−t
(1 + rt+1 )t+1
Pt
−1=
−1
(1 + rt )t
Pt+1
Forward rate for the period (t, t + 1):
f[t,t+1] =
Forward rate for the period (t, t + m):
(1 + rt )t (1 + f[t,t+m] )m = (1 + rt+m )t+m
1
(1 + rt+1 )t+m m
f[t,t+m] =
−1
(1 + rt )t
Real rate of interest i0 , Inflation rate r
1 + i = (1 + i0 )(1 + r)
Duration:
Page 4
i = i0 + r + i0 r
d
P
P
tAt v t
dδ
Dmac (i) = P
=−
t
At v
P
1+i
1
Perpetuity: Dmac =
=
i
d
Bond: Dmac =
d
P
P
tAt v t+1
Dmac
di
Dmod (i) = P
=−
=
t
At v
P
1+i
(Ia) n
Mortgage or Level Annuity: Dmac =
an
F r(Ia) n + nCv n
P
Bond Sold at Par: Dmac = ä n
Convexity:
P 2
d2
t At v t
dδ 2 P
Cmac = P
=
At v t
P
Duration of a Portfolio:
Dt and Pt are duration and price of tth components of Portfolio
D(Portfolio) =
P
Cmod =
2
d
t(t + 1)At v t+2
Cmac + Dmac
2P
P
= di
=
t
At v
P
(1 + i)2
D1 P1 + D2 P2 + · · · + Dn Pn
P1 + P2 + · · · + Pn
First-Order Modified Price Approximation:
P (i) ≈ P (i0 ) − Dmod (i0 )P (i0 )(i − i0 )
Dmac (i0 )
1 + i0
P (i) ≈ P (i0 )
1+i
First-Order Macaulay Price Approximation:
Second-Order Modified Price Approximation:
P (i) ≈ P (i0 )−Dmod (i0 )P (i0 )(i−i0 )+Cmod (i0 )P (i0 )
(i − i0 )2
2
IMMUNIZATION
Requirements for Redington Immunization:
If same interest rate applies to all CF’s
(i)
PV(Assets) = PV(Liabilities)
(i)
PA = PL
(i)
P
At v t =
(ii)
Dmod (Assets) = Dmod (Liabilities)
(ii)
PA0 = PL0
(ii)
P
tAt v t =
(iii)
Cmod (Assets) > Cmod (Liabilities)
(iii)
PA00 > PL00
(iii)
P 2
P
t At v t > t 2 L t v t
Full Immunization:
P
Lt v t
P
tLt v t
(i) and (ii) as above, (iii) There is a single liability CF that is matched (in PV and
Dmod ) with 2 or more asset CFs, including at least one that occurs before and at least
one that occurs after the liability CF.
Exact Matching or Dedication: Match both the amount and the time of Asset Cash Flows and Liability Cash Flows.
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