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Fm formula sheet 2022

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Financial Mathematics (Math 375)

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University of Wisconsin-River Falls

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coachingactuaries Copyright © 2022 Coaching Actuaries. All Rights Reserved. 1

Exam FM

updated 03/23/

coachingactuaries

INTEREST MEASUREMENT

Effective Rate of Interest

𝑖𝑖! = 𝐴𝐴(𝑡𝑡) − 𝐴𝐴(𝑡𝑡 − 1)𝐴𝐴(𝑡𝑡 − 1)

Effective Rate of Discount 𝑀𝑀! = 𝐴𝐴(𝑡𝑡) − 𝐴𝐴(𝑡𝑡 − 1)𝐴𝐴(𝑡𝑡)

Accumulation Function and Amount Function 𝐴𝐴(𝑡𝑡) = 𝐴𝐴(0) ∙ 𝑀𝑀(𝑡𝑡)

All-in-One Relationship Formula

(1 + 𝑖𝑖)! = .1 + 𝑖𝑖

(#) 𝑚𝑚 0

#! = (1 − 𝑀𝑀)%!

= .1 − 𝑀𝑀

(#) 𝑚𝑚 0

%#! = 𝑒𝑒&!

Simple Interest 𝑀𝑀(𝑡𝑡) = 1 + 𝑖𝑖𝑡𝑡

Variable Force of Interest 𝛿𝛿! = 𝑀𝑀

'(𝑡𝑡)

𝑀𝑀(𝑡𝑡)

Accumulate 1 from time 𝑡𝑡( 𝑡𝑡𝑀𝑀 𝑡𝑡𝑖𝑖𝑚𝑚𝑒𝑒 𝑡𝑡):

𝐴𝐴𝑉𝑉 = exp .9 𝛿𝛿* 𝑀𝑀𝑑𝑑

!! !"

0

Discount Factor 𝑣𝑣 =

1 1 + 𝑖𝑖 = 1 − 𝑀𝑀

𝑀𝑀 =

𝑖𝑖

1 + 𝑖𝑖 = 𝑖𝑖𝑣𝑣

ANNUITIES

Annuity-Immediate 𝑃𝑃𝑉𝑉 = 𝑀𝑀+| = 𝑣𝑣 + 𝑣𝑣) + ⋯ + 𝑣𝑣+

= 1 − 𝑣𝑣

𝑖𝑖 𝐴𝐴𝑉𝑉 = 𝑠𝑠+| = 1 + (1 + 𝑖𝑖) + ⋯ + (1 + 𝑖𝑖)+%( = (1 + 𝑖𝑖)

+ − 1

𝑖𝑖

Annuity-Due 𝑃𝑃𝑉𝑉 = 𝑀𝑀̈+| = 1 + 𝑣𝑣 + 𝑣𝑣) + ⋯ + 𝑣𝑣+%( = 1 − 𝑣𝑣

𝑀𝑀 𝐴𝐴𝑉𝑉 = 𝑠𝑠̈+| = (1 + 𝑖𝑖) + (1 + 𝑖𝑖)) + ⋯ + (1 + 𝑖𝑖)+

= (1 + 𝑖𝑖)

+ − 1

𝑀𝑀

Immediate vs. Due 𝑀𝑀̈+| = 𝑀𝑀+|(1 + 𝑖𝑖) = 1 + 𝑀𝑀+%(| 𝑠𝑠̈+| = 𝑠𝑠+|(1 + 𝑖𝑖) = 𝑠𝑠+-(| − 1

Deferred Annuity m-year deferred n-year annuity-immediate: 𝑃𝑃𝑉𝑉 = #|𝑀𝑀 +| = 𝑣𝑣# ⋅ 𝑀𝑀+| = 𝑀𝑀#-+| − 𝑀𝑀#|

Perpetuity ï Perpetuity-immediate: 𝑃𝑃𝑉𝑉 = 𝑀𝑀/| = 𝑣𝑣 + 𝑣𝑣) + ⋯ = 1𝑖𝑖 ï Perpetuity-due: 𝑃𝑃𝑉𝑉 = 𝑀𝑀̈/| = 1 + 𝑣𝑣 + 𝑣𝑣) + ⋯ = 1𝑀𝑀 𝑀𝑀̈/| = 1 + 𝑀𝑀/|

MORE GENERAL ANNUITIES

j-effective method is used when payments are more or less frequent than the interest period.

“j-effective” Method Convert the given interest rate to the equivalent effective interest rate for the period between each payment.

Example: To find the present value of 𝑛𝑛 monthly payments given annual effective rate of 𝑖𝑖, define 𝑗𝑗 as the monthly effective rate where 𝑗𝑗 = (1 + 𝑖𝑖)( ()⁄ − 1. Then apply 𝑃𝑃𝑉𝑉 = 𝑀𝑀+| using 𝑗𝑗.

Payments in Arithmetic Progression ï PV of n-year annuity-immediate with payments of 𝑃𝑃, 𝑃𝑃 + 𝑄𝑄, 𝑃𝑃 + 2𝑄𝑄, ... , 𝑃𝑃 + (𝑛𝑛 − 1)𝑄𝑄

𝑃𝑃𝑉𝑉 = 𝑃𝑃𝑀𝑀+| + 𝑄𝑄 𝑀𝑀+|

111 − 𝑛𝑛𝑣𝑣+

𝑖𝑖

Calculator-friendly version: 𝑃𝑃𝑉𝑉 = G𝑃𝑃 + 𝑄𝑄𝑖𝑖 H 𝑀𝑀+| 111 + G− 𝑄𝑄𝑛𝑛𝑖𝑖 H 𝑣𝑣+ 𝑁𝑁 = 𝑛𝑛, 𝐼𝐼 𝑌𝑌⁄ = 𝑖𝑖 (in %), 𝑃𝑃𝑀𝑀𝑃𝑃 = 𝑃𝑃 + 𝑄𝑄𝑖𝑖 , 𝐶𝐶𝑉𝑉 = − 𝑄𝑄𝑛𝑛𝑖𝑖

ï PV of n-year annuity-immediate with payments of 1, 2, 3, ... , 𝑛𝑛

Unit increasing: (𝐼𝐼𝑀𝑀)+| = 𝑀𝑀̈+|

− 𝑛𝑛𝑣𝑣+

𝑖𝑖

P&Q version: 𝑃𝑃 = 1, 𝑄𝑄 = 1, 𝑁𝑁 = 𝑛𝑛

ï PV of n-year annuity-immediate with payments of 𝑛𝑛, 𝑛𝑛 − 1, 𝑛𝑛 − 2, ... , 1 Unit decreasing: (𝑀𝑀𝑀𝑀)+| = 𝑛𝑛 − 𝑀𝑀 𝑖𝑖+| P&Q version: 𝑃𝑃 = 𝑛𝑛, 𝑄𝑄 = −1, 𝑁𝑁 = 𝑛𝑛

ï PV of perpetuity-immediate and perpetuity-due with payments of 1, 2, 3, ... (𝐼𝐼𝑀𝑀)/| = 1𝑖𝑖𝑀𝑀 = 1𝑖𝑖 + 1𝑖𝑖)

(𝐼𝐼𝑀𝑀̈)/| = 1𝑀𝑀)

INTEREST MEASUREMENT ANNUITIES MORE GENERAL ANNUITIES

an sn

$

1 1 1 1

... n–1 n

...

2

a !!

!s!

n 1 1

1 1

... n–1 n

...

2 $

coachingactuaries Copyright © 2022 Coaching Actuaries. All Rights Reserved. 2

Payments in Geometric Progression PV of an n-year annuity-immediate with payments of 1, (1 + 𝑘𝑘), (1 + 𝑘𝑘)), ... , (1 + 𝑘𝑘)+%(

𝑃𝑃𝑉𝑉 = 1 − _1 + 𝑘𝑘1 + 𝑖𝑖

`

𝑖𝑖 − 𝑘𝑘 , 𝑖𝑖 ≠ 𝑘𝑘

Level and Increasing Continuous Annuity

𝑀𝑀b+| = 9 𝑣𝑣!

𝑀𝑀𝑡𝑡 = 1 − 𝑣𝑣

𝛿𝛿 = 𝑖𝑖𝛿𝛿 𝑀𝑀+|

(𝐼𝐼̅𝑀𝑀b)+| = 9 𝑡𝑡𝑣𝑣!

𝑀𝑀𝑡𝑡 = 𝑀𝑀

b+| − 𝑛𝑛𝑣𝑣+ 𝛿𝛿

YIELD RATES

Two methods for comparing investments: ï Net Present Value (NPV): Sum the present value of cash inflows and cash outflows. Choose investment with greatest positive NPV. ï Internal Rate of Return (IRR): The rate such that the present value of cash inflows is equal to the present value of cash outflows. Choose investment with greatest IRR.

LOANS

Outstanding Balance Calculation ï Prospective: 𝐵𝐵! = 𝑅𝑅𝑀𝑀+%!|, Present value of future level payments of 𝑅𝑅. ï Retrospective: 𝐵𝐵! = 𝐿𝐿(1 + 𝑖𝑖)! − 𝑅𝑅𝑠𝑠!| Accumulated value of original loan amount L minus accumulated value of all past payments.

Loan Amortization For a loan of 𝑀𝑀+| repaid with n payments of 1: Period 𝑡𝑡 Interest (𝐼𝐼!) 1 − 𝑣𝑣+%!-( Principal repaid (𝑃𝑃!) 𝑣𝑣+%!-( Total 1

General Formulas for Amortized Loan with Level/Non-Level Payments 𝐼𝐼! = 𝑖𝑖 ⋅ 𝐵𝐵!%( 𝐵𝐵! = 𝐵𝐵!%((1 + 𝑖𝑖) − 𝑅𝑅! = 𝐵𝐵!%( − 𝑃𝑃! 𝑃𝑃! = 𝑅𝑅! − 𝐼𝐼! 𝑃𝑃!-3 = 𝑃𝑃! (1 + 𝑖𝑖) 3 (only for Level Payments)

BONDS

Bond Pricing Formulas 𝑃𝑃 Price of bond 𝐶𝐶 Par value (face amount) of bond (not a cash flow) 𝐹𝐹 Coupon rate per payment period 𝐶𝐶𝐹𝐹 Amount of each coupon payment 𝐶𝐶 Redemption value of bond (𝐶𝐶 = 𝐶𝐶 unless otherwise stated) 𝑖𝑖 Interest rate per payment period 𝑛𝑛 Number of coupon payments Basic Formula 𝑃𝑃 = 𝐶𝐶𝐹𝐹𝑀𝑀+| 4 + 𝐶𝐶𝑣𝑣+ Premium/Discount Formula: 𝑃𝑃 = 𝐶𝐶 + (𝐶𝐶𝐹𝐹 − 𝐶𝐶𝑖𝑖)𝑀𝑀+| 4

Premium vs. Discount Premium Discount

Condition

𝑃𝑃 > 𝐶𝐶

or 𝐶𝐶𝐹𝐹 > 𝐶𝐶𝑖𝑖

𝑃𝑃 < 𝐶𝐶

or 𝐶𝐶𝐹𝐹 < 𝐶𝐶𝑖𝑖

Amortization Process

Write- Down

Write-Up

Amount

|(𝐶𝐶𝐹𝐹 − 𝐶𝐶𝑖𝑖) ⋅ 𝑣𝑣+%!-(|

= |𝐵𝐵!%( − 𝐵𝐵!| = |𝐶𝐶𝐹𝐹 − 𝐼𝐼!|

General Formulas for Bond Amortization ï Book value: 𝐵𝐵! = 𝐶𝐶𝐹𝐹𝑀𝑀+%!| 4 + 𝐶𝐶𝑣𝑣+%! = 𝐶𝐶 + (𝐶𝐶𝐹𝐹 − 𝐶𝐶𝑖𝑖)𝑀𝑀+%!| 4 ï Interest earned = 𝑖𝑖𝐵𝐵!%(

Callable Bonds Calculate the lowest price for all possible redemption dates at a certain yield rate. This is the highest price that guarantees this yield rate. ï Premium bond – call the bond on the FIRST possible date. ï Discount bond – call the bond on the LAST possible date.

SPOT RATES AND

FORWARD RATES

𝑠𝑠! is the 𝑡𝑡-year spot rate. 𝑓𝑓[!",!!] is the forward rate from time 𝑡𝑡( to time 𝑡𝑡), expressed annually.

(1 + 𝑠𝑠+)+ ⋅ m1 + 𝑓𝑓[+,+-#] n# = (1 + 𝑠𝑠+-#)+-#

(1 + 𝑠𝑠+)+ = m1 + 𝑓𝑓[2,(] n ⋅ m1 + 𝑓𝑓[(,)] n ⋯ m1 + 𝑓𝑓[+%(,+] n

YIELD RATES

LOANS

BONDS

Prospective Discounting Future Payments

Retrospective Accumulating Past Payments

0 t n L

SPOT RATES AND

FORWARD RATES

0 n n+m

(1+sn+m)n+m

(1+sn)n (1+f[n,n+m])m

1 2 ... n ...

n–

(1+sn)n

(1+f[0,1]) (1+f[1,2]) (1+f[n–1,n])

0

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Fm formula sheet 2022

Course: Financial Mathematics (Math 375)

University: University of Wisconsin-River Falls

Was this document helpful?

Exam FM

updated 03/23/22

www.coachingactuaries.com

INTERESTMEASUREMENT



EffectiveRateofInterest

𝑖𝑖!=𝐴𝐴(𝑡𝑡)−𝐴𝐴(𝑡𝑡−1)

𝐴𝐴(𝑡𝑡−1)



EffectiveRateofDiscount

𝑀𝑀!=𝐴𝐴(𝑡𝑡)−𝐴𝐴(𝑡𝑡−1)

𝐴𝐴(𝑡𝑡)



AccumulationFunctionand

AmountFunction

𝐴𝐴(𝑡𝑡)=𝐴𝐴(0)∙𝑀𝑀(𝑡𝑡)



All-in-OneRelationshipFormula

(1+𝑖𝑖)!=.1+𝑖𝑖(#)

𝑚𝑚0#! =(1−𝑀𝑀)%!

=.1−𝑀𝑀(#)

𝑚𝑚0%#! =𝑒𝑒&!



SimpleInterest

𝑀𝑀(𝑡𝑡)=1+𝑖𝑖𝑡𝑡



VariableForceofInterest

𝛿𝛿!=𝑀𝑀'(𝑡𝑡)

𝑀𝑀(𝑡𝑡)

Accumulate1fromtime𝑡𝑡(𝑡𝑡𝑀𝑀𝑡𝑡𝑖𝑖𝑚𝑚𝑒𝑒𝑡𝑡):

𝐴𝐴𝑉𝑉=exp.9𝛿𝛿*𝑀𝑀𝑑𝑑

!"0



DiscountFactor

𝑣𝑣= 1

1+𝑖𝑖 =1−𝑀𝑀

𝑀𝑀= 𝑖𝑖

1+𝑖𝑖 =𝑖𝑖𝑣𝑣

ANNUITIES



Annuity-Immediate

𝑃𝑃𝑉𝑉=𝑀𝑀+|

=𝑣𝑣+𝑣𝑣)+⋯+𝑣𝑣+

=1−𝑣𝑣+

𝑖𝑖

𝐴𝐴𝑉𝑉=𝑠𝑠+|

=1+(1+𝑖𝑖)+⋯+(1+𝑖𝑖)+%(

=(1+𝑖𝑖)+−1

𝑖𝑖



Annuity-Due

𝑃𝑃𝑉𝑉=𝑀𝑀+|

=1+𝑣𝑣+𝑣𝑣)+⋯+𝑣𝑣+%(

=1−𝑣𝑣+

𝑀𝑀

𝐴𝐴𝑉𝑉=𝑠𝑠+|

=(1+𝑖𝑖)+(1+𝑖𝑖))+⋯+(1+𝑖𝑖)+

=(1+𝑖𝑖)+−1

𝑀𝑀



Immediatevs.Due

𝑀𝑀+| =𝑀𝑀+|(1+𝑖𝑖)=1+𝑀𝑀+%(|

𝑠𝑠+| =𝑠𝑠+|(1+𝑖𝑖)=𝑠𝑠+-(| −1



DeferredAnnuity

m-yeardeferredn-yearannuity-immediate:

𝑃𝑃𝑉𝑉= 𝑀𝑀



+| =𝑣𝑣#⋅𝑀𝑀+| =𝑀𝑀#-+| −𝑀𝑀#|



Perpetuity

• Perpetuity-immediate:

𝑃𝑃𝑉𝑉=𝑀𝑀/| =𝑣𝑣+𝑣𝑣)+⋯=1

𝑖𝑖

• Perpetuity-due:

𝑃𝑃𝑉𝑉=𝑀𝑀/| =1+𝑣𝑣+𝑣𝑣)+⋯=1

𝑀𝑀

𝑀𝑀/| =1+𝑀𝑀/|

MOREGENERALANNUITIES



j-effectivemethodisusedwhenpayments

aremoreorlessfrequentthanthe

interestperiod.



“j-effective”Method

Convertthegiveninterestratetothe

equivalenteffectiveinterestrateforthe

periodbetweeneachpayment.



Example:Tofindthepresentvalueof𝑛𝑛

monthlypaymentsgivenannualeffective

rateof𝑖𝑖,define𝑗𝑗asthemonthlyeffective

ratewhere𝑗𝑗=(1+𝑖𝑖)(()

⁄−1.

Thenapply𝑃𝑃𝑉𝑉=𝑀𝑀+|using𝑗𝑗.



PaymentsinArithmeticProgression

• PVofn-yearannuity-immediatewith

paymentsof

𝑃𝑃,𝑃𝑃+𝑄𝑄,𝑃𝑃+2𝑄𝑄,…,𝑃𝑃+(𝑛𝑛−1)𝑄𝑄

𝑃𝑃𝑉𝑉=𝑃𝑃𝑀𝑀+| +𝑄𝑄𝑀𝑀+|

−𝑛𝑛𝑣𝑣+

𝑖𝑖

Calculator-friendlyversion:

𝑃𝑃𝑉𝑉=G𝑃𝑃+𝑄𝑄

𝑖𝑖H𝑀𝑀+|

+G−𝑄𝑄𝑛𝑛

𝑖𝑖H𝑣𝑣+

𝑁𝑁=𝑛𝑛,𝐼𝐼 𝑌𝑌

⁄=𝑖𝑖(in



%),

𝑃𝑃𝑀𝑀𝑃𝑃=𝑃𝑃+𝑄𝑄

𝑖𝑖,𝐶𝐶𝑉𝑉=−𝑄𝑄𝑛𝑛

𝑖𝑖



• PVofn-yearannuity-immediatewith

paymentsof1,2,3,…,𝑛𝑛

Unitincreasing:(𝐼𝐼𝑀𝑀)+| =𝑀𝑀+| −𝑛𝑛𝑣𝑣+

𝑖𝑖

P&Qversion:𝑃𝑃=1,𝑄𝑄=1,𝑁𝑁=𝑛𝑛



• PVofn-yearannuity-immediatewith

paymentsof𝑛𝑛,𝑛𝑛−1,𝑛𝑛−2,…,1

Unitdecreasing:(𝑀𝑀𝑀𝑀)+| =𝑛𝑛−𝑀𝑀+|

𝑖𝑖

P&Qversion:𝑃𝑃=𝑛𝑛,𝑄𝑄=−1,𝑁𝑁=𝑛𝑛



• PVofperpetuity-immediateand

perpetuity-duewithpaymentsof1,2,3,…

(𝐼𝐼𝑀𝑀)/| =1

𝑖𝑖𝑀𝑀=1

𝑖𝑖+1

𝑖𝑖)

(𝐼𝐼𝑀𝑀)/| =1

𝑀𝑀)



INTEREST MEASUREMENT ANNUITIES MORE GENERAL ANNUITIES

1 1 1

nn–1

…

1 1

nn–1

…

Fm formula sheet 2022

Financial Mathematics (Math 375)

University of Wisconsin-River Falls

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Exam FM

updated 03/23/

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INTEREST MEASUREMENT

= .1 − 𝑀𝑀

'(𝑡𝑡)

𝑀𝑀(𝑡𝑡)

0

1

1 + 𝑖𝑖 = 1 − 𝑀𝑀

𝑀𝑀 =

𝑖𝑖

1 + 𝑖𝑖 = 𝑖𝑖𝑣𝑣

ANNUITIES

+ − 1

𝑖𝑖

+ − 1

𝑀𝑀

MORE GENERAL ANNUITIES

111 − 𝑛𝑛𝑣𝑣+

𝑖𝑖

− 𝑛𝑛𝑣𝑣+

𝑖𝑖

INTEREST MEASUREMENT ANNUITIES MORE GENERAL ANNUITIES

an sn

$

1

1 1 1

...

2

a !!

!s!

1 1

...

2

$

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𝑃𝑃𝑉𝑉 = 1 − _1 + 𝑘𝑘1 + 𝑖𝑖

`

𝑀𝑀𝑡𝑡 = 1 − 𝑣𝑣

𝑀𝑀𝑡𝑡 = 𝑀𝑀

YIELD RATES

LOANS

BONDS

𝑃𝑃 > 𝐶𝐶

𝑃𝑃 < 𝐶𝐶

|(𝐶𝐶𝐹𝐹 − 𝐶𝐶𝑖𝑖) ⋅ 𝑣𝑣+%!-(|

= |𝐵𝐵!%( − 𝐵𝐵!| = |𝐶𝐶𝐹𝐹 − 𝐼𝐼!|

SPOT RATES AND

FORWARD RATES

YIELD RATES

LOANS

BONDS

SPOT RATES AND

FORWARD RATES

(1+sn+m)n+m

(1+sn)n (1+f[n,n+m])m

(1+sn)n

(1+f[0,1]) (1+f[1,2]) (1+f[n–1,n])

0

Fm formula sheet 2022

Course: Financial Mathematics (Math 375)

University: University of Wisconsin-River Falls

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