Exam FM/2 Interest Theory Formulas
by (/iropracy
This is a collaboration of formulas for the interest theory section of the SOA Exam FM / CAS Exam 2.
This study sheet is a free non-copyrighted document for students taking Exam FM/2.
The author of this study sheet is using some notation that is unique so that no designation will repeat. Each
designation has only one meaning throughout the sheet.
Fundamentals of Interest Theory and Time Value of Money
FV = PV (1 + i )
n
i
(1 + i )
1
v=
(1 + i )
d=
a (t )
A(t )
PV =
FV
(1 + i )n
d = 1− v
i − d = id
v =1− d
d = iv
≡
The amount an initial investment of 1 grows to by time t
≡
The amount an initial investment of A(0 ) grows to by time t
a(t ) = (1 + i ) = et⋅ln (1+i )
A(t ) = A(0)(1 + i ) = A(0 ) e t⋅ln (1+i )
δ = ln(1 + i )
a(t ) = e δ⋅t
t
t
v n = (1 + i ) = e − δn
−n
δ (t ) =
a′(t )
a(t )
t
e ∫0
δ (u ) du
t
A(0 ) e ∫0
= a(t )
δ (u ) du
= A(t )
Effective interest rate with nominal rate i (m ) convertible m-thly
m
⎛ i (m ) ⎞
⎟⎟ − 1
i = ⎜⎜1 +
m
⎝
⎠
Effective discount rate with nominal rate d ( p ) convertible p-thly
⎛ d ( p) ⎞
⎟⎟
1 − d = ⎜⎜1 −
p
⎝
⎠
p
Nominal Rate Equivalence
m
⎛ i (m ) ⎞
⎛ d ( p) ⎞
1
1
⎟⎟ = ⎜⎜1 −
⎟
1+ i = e = =
= ⎜⎜1 +
v 1− d ⎝
m ⎠
p ⎟⎠
⎝
δ
−p
Effective annual rate it during the t-th year is given by:
it =
amount earned
a(t ) − a(t − 1) A(t ) − A(t − 1)
=
=
beginning amount
a(t − 1)
A(t − 1)
Note that the t-th year is given by the time period [t − 1,t ]
Therefore, the interest earned during the t-th year is given by:
A(t − 1) ⋅ i = A(t ) − A(t − 1)
For equivalent measures of interest we have the following relationship:
d < d ( 2 ) < d (3 ) <
<δ<
< i (3 ) < i ( 2 ) < i
Annuities
Annuity Immediate— payments are made at the end of the period
Annuity Due— payments are made at the beginning of the period
Annuity Immediate
an | i = v + v +
2
1 − vn
+v =
i
n
n −1
n−2
sn | i = (1 + i ) + (1 + i ) +
i
sn | i = (1 + i ) ⋅ an | i
n
an | i = v n ⋅ s n | i
Annuity Due
an | i = 1 + v +
n
(
1 + i) − 1
+1 =
+v
n −1
1 − vn
=
d
s n | i = (1 + i ) + (1 + i )
n
n −1
+
n
(
1 + i) −1
+ (1 + i ) =
d
sn | i = (1 + i ) ⋅ an | i
n
an | i = v n ⋅ sn | i
Identities for Annuity Immediate and Annuity Due
an | =
i
a = (1 + i ) an |
d n|
sn | =
i
s = (1 + i ) sn |
d n|
a n | = 1 + a n −1|
sn | = sn+1| − 1
Perpetuity
a∞ | i = lim an | i = v + v 2 + v 3 +
=
n→∞
1
i
a∞ | i = lim an | i =
n→∞
1
d
Continuous Annuities
an | i
1 − vn i
=
= an | i
δ
δ
n
sn | i
n
(
1 + i) − 1
=
=
δ
n
t
− δ (u ) du
PV = ∫ e ∫0
p(t ) dt
n
FV = ∫ e ∫t
0
δ ( u ) du
n
i
s
δ n|i
p ( t ) dt
a n | i = ∫ v t dt
0
where p (t ) = payment function
0
Increasing Annuities— Payments are 1, 2, … , n
(Ia )n | i =
an | − nv n
(Ia )n | i
i
(Is )n | i = (1 + i )n (Ia )n | i =
(Ia )∞ | i = lim
(Ia )n | i =
n →∞
sn | − n
an | − nv n
i
= (Ia )n | i = (1 + i )(Ia )n | i =
d
d
(Is )n | i = i (Is )n | i = (1 + i )n (Ia )n | i =
d
i
(Ia )∞ | i = lim
(Ia )n | i =
n →∞
1 1 1
= +
di i i 2
sn | − n
d
1
d2
Decreasing Annuities— Payments are n, n-1,…, 2, 1
(Da )n | i =
n − an | i
(Da )n | i = i (Da )n | i = (1 + i )(Da )n | i =
d
i
(Ds )n | i = (1 + i ) (Da )n | i =
n
n(1 + i ) − sn | i
n
i
(Ds )n | i = (1 + i )n (Da )n | i
Present Value of the annuity with terms X , X + Y , X + 2Y , … , X + (n − 1)Y
⎛ an | − nv n ⎞
⎟
X ⋅ an | i + Y ⎜
⎟
⎜
i
⎠
⎝
Present Value of the perpetuity with terms X , X + Y , X + 2Y , …
X Y
+
i i2
n − an | i
d
Annuities with Terms in Geometric Progression— 1, (1 + q ), (1 + q ) , … , (1 + q )
n −1
2
Present Value is V (0 ) = 1 ⋅ v + (1 + q ) ⋅ v 2 + (1 + q ) ⋅ v 3 +
+ (1 + q )
n −1
2
1 − (1 + q ) v n
i−q
n
⋅ vn =
Useful Identities
(
an+ k | = an | + v n ak |
v n − v m = i am | − an |
a2 n |
1 = v n + i an |
an |
=
a2 n |
an |
=
1 − v 2n
= 1 + vn
1 − vn
)
(Da )n | + (Ia )n | = (n + 1) an |
s2 n |
sn |
=
s2n |
sn |
=
(1 + i )2 n − 1 = (1 + i )n + 1
(1 + i )n − 1
If the interest rate varies:
an | =
1
1
+
+
a (1) a(2)
+
1
a (n )
sn | =
a(n ) a(n )
+
+
a(1) a(2 )
+
a(n )
a(n )
If the compounding frequency of the interest exceeds the payment frequency of k years—
Use an equivalent interest rate over k years: j = (1 + i ) − 1
k
If the payment frequency exceeds the compounding frequency of the interest—
(1) Use an m-thly annuity
an(m| ) =
i
i
(m )
an |
sn(m| ) =
i
i
(m )
sn |
an(m| ) =
d
a
d (m ) n |
sn(m| ) =
d
s
d (m ) n |
(2) Use an equivalent interest rate effective over the payment period: j = (1 + i )
1m
an(m| i) = an | j
sn(m| i) = sn | j
an(m| i) = an | j
−1
sn(m| i ) = sn | j
an | i − nv n
1 2
n
(m )
If the payments are , ,… , , then the present value is (Ia )n | i =
i (m )
m m
m
an(m| i) − nv n
1 2
n
(m ) (m )
If the payments are 2 , 2 ,… , 2 , then the present value is (I a )n | i =
i (m )
m m
m
Loan Repayment— Amortization
Amortization Method— when a payment is made, it must be first applied to pay interest
due and then any remaining part of the payment is applied to pay principle
Notation
L
≡
amount of the loan
n
≡
number of payment periods
PA
≡
amount of level payment at the end of the period (amortized payment)
P(k )
≡
loan payment at time k
i
≡
effective interest rate per payment period
Bk
≡
balance at time k, balance after k-th payment.
Pk
≡
principle paid in payment P(k )
Ik
≡
interest paid in payment P(k )
Note that B0 = L
Useful Equations for Level Payments
L
an | i
L = PA ⋅ an | i
PA =
Prospective Method
Bk = PA ⋅ an−k | i
Retrospective Method
Bk = L(1 + i ) − PA ⋅ s k | i
k
Bk +t = Bk (1 + i ) and Pk +t = Pk (1 + i )
t
PA = Pk + I k
t
(
I k = i ⋅ Bk −1 = PA 1 − v n−k +1
)
Pk = PA − I k = PA ⋅ v n−k +1
Useful Equations for Non-Level Payments
L = P(1)v + P(2 )v 2 +
I k = i ⋅ Bk −1
+ P(n )v n
Bk = P(k +1)v + P(k +2 )v 2 +
Pk = P(k ) − I k = Bk −1 − Bk
+ P(n )v n−k = Bk −1 (1 + i ) − P(k )
Loan Repayment— Sinking Fund
Sinking Fund Loan (SFL)— accumulate money in a separate fund by making a payment,
in addition to the regular interest payment, every period.
Notation
L
≡
amount of the loan
n
≡
number of payment periods
i
≡
effective interest rate per payment period by the borrower to the lender
j
≡
effective interest rate earned by the borrower in the sinking fund
DS
≡
periodic sinking fund deposit (SFD), assumed to be level
PS
≡
periodic outlay by the borrower = interest payment to lender + SFD
Sk
≡
sinking fund balance after k-th deposit
Lk
≡
net loan balance at time k
Useful Equations
L = DS ⋅ s n | j
S k = DS ⋅ s k | j = L
DS =
sk | j
sn | j
L
sn | j
PS = Li + DS = Li +
L
sn | j
Lk = L − DS ⋅ sk | j
Net Principal Paid
S k − S k −1 = DS ⋅ sk | j − DS ⋅ sk −1| j = DS (1 + j )
Net Interest Paid
Li − jS k −1 = Li − jDS ⋅ sk −1| j
k −1
Notes on Loans
Amortized Loan— over time interest paid decreases and principal paid increases
SFL— for each outlay interest paid to lender is constant
Installment Loan— over time interest paid decreases while the principal paid is constant
Bonds
Bonds— interest bearing securities; basically loans from lenders perspective
Callable Bond— a bond that can be paid off (called) before maturity
Notation
F
≡
par value
r
≡
coupon rate (interest rate of bond)
Fr
≡
coupon amount (payment to lender)
C
≡
redemption value (usually = F )
n
≡
number of coupon periods to maturity
P
≡
market price of the bond
BVk
≡
book value of the bond (bond amortized balance after k-th payment)
i
≡
yield per period to investor at price P
vi =
1
1+ i
K = Cvin
g=
Fr
C
≡
Present value of the redemption value
≡
modified coupon rate
Premium— If i > r then the bond is priced at a premium. P > C , and P − C is the
amount of the premium.
Premium ≡ P − C = (Fr − iC ) an | i
(
P − C = Pk v k −1 + v k −2 +
+ v + 1 + (1 + i ) +
+ (1 + i )
n−k
) = P (a
k
k −1 | i
+ s n−k +1| i
)
Discount— If i < r then the bond is priced at a discount. P < C , and C − P is the
amount of the discount
Discount ≡ C − P = (iC − Fr ) an | i
Par— If i = r the bond is selling at the price P = C we say that it sells at par.
Price and Premium-Discount Formula
P = Fran | i + K
(
)
P = C 1 + (g − i ) an | i
if F = C , then P = F 1 + (r − i ) an | i
(
)
BVk = BVm (1 + i )
Fr = I k + Pk
Bond Amortized
k −m
BVk = Fran−k | i + Cvin−k
(
)
I k = i ⋅ BVk −1 = Fr 1 − v n−k +1 + iC v n−k +1
If F = C , then
− Fr ⋅ sk −m | i
Pk = Fr v n−k +1 − iC v n−k +1
Pk +t
t
= (1 + i )
Pk
Write-Up during the first k years (Discount) ≡ BVk − P
Write-Down during the first k years (Premium) ≡ P − BVk
Write-Up/Write-Down in general during time m to time k, (k > m ) ≡ BVk − BVm
WDk = ( Fr − iC ) v n − k +1
Makeham’s Formula
g
P = K + (C − K )
i
WU k = ( iC − Fr ) v n − k +1
if F = C , then P = K +
r
(F − K )
i
Maturity to use in Pricing a Callable Bond
Type of Bond
Premium Bond
Discount Bond
Take N using…
Earliest Possible Redemption Date
Latest Possible Redemption Date
Price Between Payment Dates
t=
number of days from last coupon date to settlement date
number of days in the bond period
Price Plus Accrued ≡ P0 (1 + i )
t
Accrued Interest ≡ t (Fr )
P ≡ Price Plus Accrued − Accrued Interest = P0 (1 + i ) − t (Fr )
t
Yield Rate of an Investment
Internal Rate of Return (IRR)— the rate of interest at which the present value of all
amounts invested is equal to the present value of all the amounts paid back to the investor
Internal Rate of Return (IRR)
Given investment cash flows C0 , C1 , C2 , … , Cn , the IRR is a solution for i of the equation
C0 +
Cn
C1
C2
+
+…+
=0
2
(1 + i ) (1 + i )
(1 + i )n
or
C0 + C1v + C2 v 2 + … + Cn v n = 0
Time Weighted Rates of Interest (TWR)
Ck′
≡
Contribution at time t k
Bk′
≡
Fund value at time t k before the contribution Ck′ is made
jk
≡
Effective rate over [t k −1 , t k ]
1 + jk =
Bk′
Bk′ −1 + C k′ −1
TWR
→
1 + i = (1 + j1 )(1 + j2 ) ⋅
Dollar Weighted Rates of Interest (DWR)
A
≡
Initial fund balance
B
≡
Final fund balance
I
≡
Interest earned
Ct
≡
Contribution or withdrawal at time t (cash flows)
C Net
≡
Net contribution
C Net = ∑ Ct
DWR
→
B = A + C Net + I
i=
I
A + ∑ Ct (1 − t )
⇒
I = B − A − C Net
(1 + jm )
Term Structure of Interest Rates
Spot Rates
Denoted by sn ≡ the n-year spot rate
(1) The annual interest rate on the n-year Treasury STRIP is called the n-year
spot rate, and the series of spot rates over time is called the yield curve.
(2) To value a bond, take the present value of each payment at the appropriate
yield curve rate and sum the present values.
P=
P(1)
+
P( 2)
(1 + s1 ) (1 + s2 )2
+
+
P( n )
(1 + sn )
n
=
P(1)
P( 2)
+
(1 + f1 ) (1 + f1 )(1 + f 2 )
+
+
P( n )
n
∏ (1 + f )
i =1
i
(3) Once we have found the price of a bond using the yield curve we can find the
yield to maturity as the constant yield on the bond at that price.
For example—
Purchasing a bond with coupons has cash flows given by − P, P(1) , P(2 ) , … , P(n )
If payments P(1) , P(2 ) , … , P(n−1) are not level
Using the BA-II Plus—
1. → CF Worksheet → Set CFo = − P, C01 = P(1) , C02 = P(2 ) , … , CN = P(n )
2. → IRR
→ CPT
⇒
Constant Yield on the Bond = IRR
If payments P(1) , P(2 ) , … , P(n−1) are level
Using the BA-II Plus
1. → TVM Worksheet → Set PV = − P, N = n, PMT = P(n−1) , FV = C
2. → I/Y
→ CPT
⇒
Constant Yield on the Bond = I/Y
Forward Rates
Denoted by f n ≡ the n year forward rate
The rate agreed upon today for a one-year loan to be made n years in the future
(1 + sn )
1 + fn =
n −1
(1 + sn−1 )
n
⇒
(1 + sn )
n
= (1 + f n )(1 + sn −1 )
n −1
Duration
Duration— a measure of sensitivity of a financial asset to changes in interest rates
Investment Cash Flows
C1 , C2 , … , Cn
Investment Price
P(i ) = vC1 + v 2C2 +
wt =
Weights for Macaulay Duration
+ v n C n = ∑ v t Ct
v t Ct
v t Ct
=
P (i ) vC1 + v 2 C 2 +
t >0
+ v nCn
=
v t Ct
∑ v t Ct
t >0
∑ tv C
=
∑v C
t
DM = 1 ⋅ w1 + 2 ⋅ w2 +
Macaulay Duration
+ n ⋅ wn
t >0
t
t >0
Modified Duration
D=−
Duration of a Level Payment Investment
D=
t
t
d
1
P(i ) =
DM
di
1+ i
(Ia )n | i
an | i
Macaulay Duration of a coupon bond with face value F and coupon Fr for n periods and
redemption value C
Fr (Ia )n | i + nCv n
DM =
Fran | i + Cv n
The Duration of a Zero-Coupon Bond payable in n periods is n
Modified Duration of a Portfolio with Investments X k
DTot = W1 D1 + W2 D2 +
+ Wn Dn
where
Wk =
Xk
X1 + X 2 +
+ Xn
Convexity and Approximations in Change of Price
Convexity and Estimate
Change in Price
C=
1 d 2 P (i )
P(i − Δi ) − 2 P(i ) + P(i + Δi )
≈
2
2
P (i ) di
P(i ) ⋅ (Δi )
P′(i )
P (i )Δi = − D ⋅ P(i )Δi
P(i )
P(i ) 2
P′′(i ) 2
Δi
ΔP ≈ P′(i )Δi +
Δi = − D ⋅ P(i )Δi +C ⋅
2
2
ΔP = P(i + Δi ) − P(i ) ≈ P′(i )Δi =
Immunization
Notation
A(i )
≡
Present Value of Assets
At
≡
Asset Amount at time t
L(i )
≡
Present Value of Liabilities
Lt
≡
Liability Amount at time t
S (i )
≡
Surplus
S (i ) = A(i ) − L(i )
Conditions for Immunization
To achieve immunization we must have S (io ) = 0 , S ′(io ) = 0 , and S ′′(io ) > 0
Immunization in terms of duration and convexity we need…
(1) PV Matching
A(io ) = L(io )
(2) Duration Matching
d
A(i )
di
(3) Greater Convexity for Assets
d
A(i )
di 2
=
i0
2
d
L(i )
di
>
i0
i0
d
L(i )
di 2
2
Immunization in terms of the asset and liability amounts at time t…
(1) PV Matching
∑ Av
= ∑ Lt vit0
∑ tA v
= ∑ tLt vit0
t >0
(2) Duration Matching
t
t i0
t
t i0
t >0
(3) Greater Convexity for Assets
∑t
t >0
2
t >0
t >0
At vit0 = ∑ t 2 Lt vit0
t >0
i0
Special Cases
Yield Rate Reinvestments
Notation
y
n
k
i
j
≡
≡
≡
≡
≡
annual yield of total investment (IRR)
number of years
number of payments
k effective interest in fund X
k effective interest in fund Y
General Case—
Suppose you make an initial investment of C0 . The yield rate y is the actual rate
of return you are receiving on the investment. AV is the accumulated value of your
investment.
n
C0 (1 + y ) = AV
• Suppose you are investing payments into a fund X at the end of each k period
• …and reinvesting the interest accrued each k period into fund Y
an y (1 + y )
= sn y = k + i ( Is )k j
n
AV of initial investment
AV of reinvestment
• Suppose you make an initial investment of C0 into fund X at t = 0
• You reinvest interest accrued in fund X after each k period into fund Y starting at t = 1
• You reinvest interest accrued in fund Y after each k period into fund Z starting at t = 2
C0 (1 + y ) = C0 + k ( iX C0 ) + iY iX C0 ( Is )k −1 i
n
Z
Sum of principal and interest
after k periods
⇒
(1 + y )
n
= 1 + kiX + iY iX ( Is )k −1 i
Z
Bond Reinvestments
This refers to the case where we have bought a bond for a price of P = Fran | i + K and we
reinvest the coupon payments Fr into a separate account at the time they are received.
Notation
y
n
k
≡
≡
≡
annual yield of total investment
number of years
number of payments the bond pays
Frsk | i + C is the AV of the account and the price P is the initial investment.
P (1 + y )
n
⎛
y (m ) ⎞
⎟
= P ⎜⎜1 +
m ⎟⎠
⎝
m⋅n
= Frsk | i + C
⇒
(1 + y )n =
Frs k | i + C
Frak | i + K
Of course we can have more than one bond involved. If that is the case we just need to
combine prices and coupon payments accordingly.
Matching Liabilities Using Bonds
We are going to cover the case that liability frequency matches the coupon frequency.
(e.g. We would not have a liabilities at year 1 and year 2 with coupons semiannually).
Let F1 , r1 and C1 denote the par value, coupon rate and redemption value, respectively,
for the bond with the longest duration. Denote F2 , r2 and C2 for the bond with the next
longest duration, and so on.
C1
of the bond. This is a percentage.
F1r1 + C1
Step1- Purchase
Step2- This gives F1r1 ⋅
C1
, a fractional amount of the coupon the period before.
F1r1 + C1
Step 3- Determine the amount left we need to match. C 2 − F1r1 ⋅
C1
F1r1 + C1
C1
F1r1 + C1
of the bond.
F2 r2 + C 2
C 2 − F1r1 ⋅
Step 4- Purchase
C1
F1r1 + C1
C1
P1 . This matches
P2 +
F2 r2 + C2
F1r1 + C1
C2 − F1r1 ⋅
Price of the bond to match liabilities is:
liabilities at time 1 and time 2.
Stupid Yield Curve Stuff
(2) To value a bond, take the present value of each payment at the appropriate yield
curve rate and sum the present values.
P=
P(1)
+
P( 2)
(1 + s1 ) (1 + s2 )2
+
+
P( n )
(1 + sn )
n
=
P(1)
+
P( 2)
(1 + f1 ) (1 + f1 )(1 + f 2 )
+
+
P( n )
n
∏ (1 + f )
i =1
i