FM (Actuary Exam) Formulas

 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 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 )

 PV  =

n

i

d  = v=

(1 + i ) 1 (1 + i )

 FV 

(1 + i )n

d  = 1 − v

i − d  = id 

v = 1 − d 

d  = iv

a (t )

≡

The amount an initial investment of 1 grows to by time t 

 A(t )

≡

The amount an initial investment of   A(0) grows to by time t 

a (t ) = (1 + i ) = e t ⋅ln (1+i )

 A(t ) =  A(0 )(1 + i ) =  A(0 ) e t ⋅ln(1+i )

t 

δ

a (t ) = e δ⋅t 

= ln(1 + i ) −n

v n = (1 + i )

δ

t 

(t ) =

a′(t ) a(t )

= e − δn t 

t 

∫ δ (u ) du = a(t ) e0

∫ δ (u ) du =  A(t )  A(0 ) e 0

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 δ ⎟ = ⎜⎜1 − ⎟⎟ = ⎜1 + 1+ i = e = = v 1 − d  ⎜⎝  m  ⎠⎟  p ⎝   ⎠ 1

−p

Effective annual rate it  during the t -th year is given by: it  =

amount  earned  beginning  amount 

=

a (t ) − a (t − 1) a (t − 1)

=

 A(t ) − 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 +  + v = 2

n

1 − vn

n−1

 sn | i = (1 + i )

i

n −2

+ (1 + i )

++1 =

(1 + i )n − 1 i

 sn | i = (1 + i ) ⋅ an | i

an | i = v n ⋅ sn | i

n

Annuity Due n | i = 1 + v +  + v a

n −1

=

1 − vn

n −1

 sn | i = (1 + i ) + (1 + i ) n

d 

n | i = v n ⋅ sn | i a

+  + (1 + i ) =

(1 + i )n − 1 d 

n | i  sn | i = (1 + i ) ⋅ a n

Identities for Annuity Immediate and Annuity Due n | = a

i d 

an | = (1 + i ) an |

 sn | =

i

 s = (1 + i ) sn | d  n |

an | = 1 + a n −1|

 sn | = sn+1| − 1

Perpetuity a∞ | i = lim an | i = v + v 2 + v 3 +  =

1

n→∞

∞ | i = lim a n | i = a n→∞

i

1 d 

Continuous Annuities an | i =

1 − vn δ n

=

i δ

 sn | i =

an | i

δ n

t 

∫ δ (u ) du p(t ) dt   PV  = ∫ e 0 −

(1 + i )n − 1 n

 FV = e∫t 

∫

0

δ

( u ) du

=

i δ

n

∫

t  a n | i = v dt 

sn | i

p ( t ) dt 

0

where  p(t ) = payment function

0

Increasing Annuities— Payments are 1, 2, … , n

( Ia )n | i =

n | − nv n a

n | − nv n a

d 

d 

( I a)n | i = ( Ia )n | i = (1 + i )( Ia )n | i =

i

( Is )n | i = (1 + i )n ( Ia )n | i = ( Ia )∞ | i = lim ( Ia )n | i = n →∞

i

1 di

 sn | − n i

1 1 = + 2 i

i

 sn | − n

d 

d 

( I  s)n | i = ( Is )n | i = (1 + i )n ( I a)n | i = ( I a)∞ | i = lim ( I a)n | i = n →∞

i

1 d 2

Decreasing Annuities— Payments are n, n-1,…, 2, 1

( Da )n | i =

n − an | i

i

n − an | i

d 

d 

( Da)n | i = ( Da )n | i = (1 + i )( Da )n | i =

i n(1 + i ) − sn | i n

( Ds )n | i = (1 + i ) ( Da )n | i = n

( D s)n | i = (1 + i )n ( Da)n | i

i

Present Value of the annuity with terms  X ,  X  + Y ,  X  + 2Y , … ,  X  + (n − 1)Y 

⎛ an | − nv n  ⎞ ⎟  X  ⋅ an | i + Y ⎜ ⎜ ⎟ ⎝  i  ⎠ Present Value of the perpetuity with terms  X ,  X  + Y ,  X  + 2Y , …  X  i

+

Y  i2

n−1

2

Annuities with Terms in Geometric Progression—  1, (1 + q ), (1 + q ) , …, (1 + q ) 2

n −1

Present Value is V (0 ) = 1 ⋅ v + (1 + q ) ⋅ v + (1 + q ) ⋅ v +  + (1 + q ) 2

3

1 − (1 + q ) v n ⋅v = i−q n

n

Useful Identities an+ k | = an | + v n ak |

v n − v m = i am | − an |  n a 2 |

1 = v + i an | n

=

n a |

a2 n | an |

1 − v 2n = = 1 + vn n 1− v

( Da )n | + ( Ia )n | = (n + 1) an |  s2 n |  sn |

=

 s2 n |  sn |

(1 + i )2n − 1 n = = (1 + i ) + 1 n (1 + i ) − 1

If the interest rate varies: an | =

1 1 1 + + + a (1) a (2 ) a (n )

 sn | =

a(n ) a(1)

+

a(n ) 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

 sn(m| ) =

( m ) an |

i

i

( m ) sn |

n(m| ) = a

d 

 sn(m| ) =

 a d (m ) n |

d  d (m ) 1m

(2) Use an equivalent interest rate effective over the payment period:  j = (1 + i ) an(m| i) = an |  j

If the payments are If the payments are

 sn(m| i) = sn |  j

1 2

n

n(m| i) = a n |  j a

m

2

 ,

2 m

2

 ,… ,

(m )

n m

2

−1

 sn(m| i ) = sn | j

 ,  ,… , , then the present value is ( Ia )n | i = m m m

1

sn |

n | i − nv n a

i (m ) n(m| i) − nv n a (m )

, then the present value is ( I (m )a )n | i =

i (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

 P   A

≡

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.

 P k 

≡

principle paid in payment  P (k )

 I k 

≡

interest paid in payment  P (k )

Note that  B0 = L

Useful Equations for Level Payments  L

 L =  P   A ⋅ an | i

 P   A =

Prospective Method 

 Bk  = P   A ⋅ an −k | i

Retrospective Method 

 Bk  = L(1 + i ) − P   A ⋅ s k | i

an | i

k 

 Bk +t  =  Bk (1 + i ) and   P k +t  = P k  (1 + i ) t 

 P   A =  P  k  + I k 

t 

(

n − k +1  I k  = i ⋅ Bk −1 = P   A 1 − v

)

n −k +1  P k  = P   A − I k  =  P   A ⋅ v

Useful Equations for Non-Level Payments  L = P (1)v + P (2 )v 2 +  + P (n )v n

 Bk  = P ( k +1)v + P (k +2 )v 2 +  + P (n )v n−k  = Bk −1 (1 + i ) − P (k )

 I k  = i ⋅ Bk −1

 P k  = P ( k ) − I k  = Bk −1 − Bk 

 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

 P S 

≡

periodic outlay by the borrower = interest payment to lender + SFD

S k 

≡

sinking fund balance after  k -th deposit

 Lk 

≡

net loan balance at time k 

Useful Equations  L

 L = DS  ⋅ sn | j

 DS  =

 sk | j S k  = DS  ⋅ sk |  j = L  s n |  j

 Lk  = L − DS  ⋅ sk |  j

 sn | j

 P S  =  Li + DS  =  Li +

 L  sn |  j

k −1

 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

 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 

 BV k 

≡

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

≡

Present value of the redemption value

 Fr 

≡

modified coupon rate

 g  =

C 

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 ) a n | i n − k 

 P − C = P k  v k −1 + v k −2 +  + v + 1 + (1 + i ) +  + (1 + i )

) = P  a k 

k −1 | i

+ sn−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 ) a n | 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 ) a n | i

if   F  = C , then  P = F  1 + (r − i ) an | i

Bond Amortized  k −m

 BV k  =  Fran −k | i + Cvin−k 

(

 BV k  =  BV m (1 + i )

)

 I k  = i ⋅ BV k −1 =  Fr 1 − v n−k +1 + iC v n−k +1

If  F  = C , then

 P k +t   P k 

− Fr ⋅ sk −m | i

 Fr = I k  + P k 

 P k  = Fr v n−k +1 − iC v n −k +1

= (1 + i )

t 

Write-Up during the first k years (Discount) ≡ BV k  − P  Write-Down during the first k years (Premium) ≡ P − BV k  Write-Up/Write -Down in general during time m to time k , (k  > m ) ≡ BV k  − BV m WDk  = ( Fr − iC ) v n −k +1

WU k  = ( iC − Fr ) v n − k +1

Makeham’s Formula  P = K  +

 g  i

(C − K )

if   F  = C , then  P = K  +

r  i

( F  − K )

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  ≡  P 0 (1 + i )

t 

Accrued Interest ≡ t ( Fr )

 P  ≡ Price Plus Accrued − Accrued Interest =  P 0 (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 C 0 , C 1 , C 2 , … , C n , the IRR is a solution for  i of the equation C 0 +

C 1

C 2

+

(1 + i ) (1 + i )

2

+…+

C n

(1 + i )

n

=0

or

C 0 + C 1v + C 2 v 2 + … + C n v n = 0

Time Weighted Rates of Interest (TWR) C k ′

≡

Contribution at time t k 

 Bk ′

≡

Fund value at time t k  before the contribution C k ′ is made

 jk 

≡

Effective rate over  [t k −1 , t k ]

1 +  jk  =

 Bk ′

TWR 

 Bk ′ −1 + C k ′ −1

→

1 + i = (1 +  j1 )(1 +  j2 ) ⋅  (1 +  jm )

Dollar Weighted Rates of Interest (DWR)  A

≡

Initial fund balance

 B

≡

Final fund balance

 I 

≡

Interest earned 

C t 

≡

Contribution or withdrawal at time t  (cash flows)

C  Net 

≡

Net contribution

C  Net  =

∑ C 

DWR 

→

 B =  A + C  Net  + I 

t 

i=

 I 

 A +

∑ C  (1 − t ) t 

⇒  I  = B − A − C  Net 

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

i =1

(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 ) ⇒ Constant Yield on the Bond = IRR  2. → IRR → CPT 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  → CPT ⇒ Constant Yield on the Bond = I/Y 2. → 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 n

(1 + sn ) 1 +  f n = n −1 (1 + sn−1 )

⇒

n n (1 + sn ) = (1 + f n )(1 + sn−1 )

−1

 Duration  Duration — a measure of sensitivity of a financial asset to changes in interest rates

Investment Cash Flows

C 1 , C 2 , … , C n

Investment Price

 P (i) = vC 1 + v 2C 2 +  + v n C n =

wt  =

Weights for Macaulay Duration

v t C t 

=

 P (i)

v t C t  vC 1 + v 2 C 2 +  + v n C n

∑ v C  t 

t 

t >0

v t C t 

=

∑ v C  ∑ tv C  = ∑ v C  t 

t 

t >0

t 

t 

 D M  = 1 ⋅ w1 + 2 ⋅ w2 +  + n ⋅ wn

Macaulay Duration

t >0

t 

t 

t >0

Modified Duration

 D = −

Duration of a Level Payment Investment

 D =

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  n  Fr ( Ia )n | i + nCv  D M  =  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  = W 1 D1 + W 2 D2 +  + W n Dn

where

W k  =

 X k   X 1 +  X 2 +  +  X n

Convexity and Approximations in Change of Price Convexity and Estimate Change in Price

C =

1 d 2 P (i )  P (i)

di

2

≈

 P (i − Δi ) − 2 P (i ) + P (i + Δi ) 2

 P (i ) ⋅ (Δi )

 P ′(i)  P (i )Δi = − D ⋅ P (i )Δi  P (i)  P (i) 2  P ′′(i) 2 Δ P ≈  P ′(i)Δi + Δi = − D ⋅ P (i)Δi +C  ⋅ Δi

Δ P = P (i + Δi ) − P (i ) ≈ P ′(i )Δi = 2

2

 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 (2) Duration Matching (3) Greater Convexity for Assets

 A(io ) = L(io ) d  di

=

 A(i )

d 2 di 2

i0

d   L(i ) di

>

 A(i ) i0

d 2 di 2

i0

 L(i ) i0

Immunization in terms of the asset and liability amounts at time t… (1) PV Matching

∑ A v = ∑ L v t >0

(2) Duration Matching

t  t  i0

∑ tA v = ∑ tL v t  t  i0

t >0

(3) Greater Convexity for Assets

t >0

t  t  i0

t >0

t  t  i0

∑ t  A v = ∑ t  L v 2

t >0

t  t  i0

2

t >0

t  t  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 C 0 . 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 )

n

= sn y = k + i ( Is )k j





 AV  of initial investment

AV  of reinvestment

• Suppose you make an initial investment of  C 0 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 n

C0 (1 + y ) = C0 + k ( i X C0 ) + iY iX C0 ( Is )k −1 i

 Z  

Sum of principal and interest after k periods

⇒

n

(1 + y ) = 1 + ki X + 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.

⎛   y (m )  ⎞ ⎟⎟  P (1 +  y ) =  P ⎜⎜1 + m ⎝   ⎠

m⋅n

n

=  Frsk | i + C 

⇒

(1 + y )n =

 Frs k | i + C   Fra k | 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  F 1 , r 1 and C 1 denote the par value, coupon rate and redemption value, respectively, for the bond with the longest duration. Denote F 2 , r 2 and C 2 for the bond with the next longest duration, and so on. C 1

Step1- Purchase

 F 1r 1 + C 1

Step2- This gives  F 1r 1 ⋅

of the bond. This is a percentage. C 1

 F 1r 1 + C 1

, a fractional amount of the coupon the period before.

C 1 Step 3- Determine the amount left we need to match. C 2 − F 1r 1 ⋅  F 1r 1 + C 1

Step 4- Purchase

C 1 C 2 − F 1r 1 ⋅  F 1r 1 + C 1  F 2 r 2 + C 2

of the bond.

Price of the bond to match liabilities is:

C 1 C2 − F1r 1 ⋅  F1r1 + C1  F2 r2 + C2

 P 2 +

C 1 F1 r1 + C 1

 P  1 . This matches

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

i =1