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Fm formula sheet - N/A
SUNY Broome Community College
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INTEREST MEASUREMENT
Effective Rate of Interest 끫殬"=
끫歨(끫毂)− 끫歨(끫毂 − 1)
끫歨(끫毂 − 1)
Effective Rate of Discount
끫殀"=
끫歨(끫毂)− 끫歨(끫毂 − 1)
끫歨(끫毂)
Accumulation Function and Amount Function 끫歨(끫毂)= 끫歨( 0 )∙ 끫殜(끫毂)
All -in-One Relationship Formula
(1 + 끫殬)"= /1 +
끫殬( 0 )
끫殴 2
0" =(1 − 끫殀)3"
= /1 −
끫殀( 0 )
끫殴 2
30" = 끫殤6"
Simple Interest 끫殜(끫毂)= 1 +끫殬끫毂
Variable Force of Interest 끫毾"=끫殜
8 (끫毂)
끫殜(끫毂)
_Accumulate 1 from time _ 끫毂 9 끫毂끫殀 끫毂끫殬끫殴끫殤 끫毂; _: _
끫歨끫殒 = exp /@ 끫毾A 끫殀끫殢
"C "D
2
Discount Factor 끫毆 =
1
1 + 끫殬= 1 − 끫殀
끫殀 =1 + 끫殬끫殬 = 끫殬끫毆
ANNUITIES
Annuity-Immediate 끫殆끫殒 = 끫殜G| = 끫毆 + 끫毆;+ ⋯ + 끫毆G =
1 − 끫毆G
끫殬
끫歨끫殒 = 끫毀G|
= 1 +(1 + 끫殬)+ ⋯ +(1 + 끫殬)G
=
(1 + 끫殬)G− 1
끫殬
Annuity-Due 끫殆끫殒 = 끫殜̈G| = 1 + 끫毆 + 끫毆;+ ⋯ + 끫毆G =
1 − 끫毆G
끫殀
끫歨끫殒 = 끫毀̈G|
=(1 + 끫殬)+(1 + 끫殬);+ ⋯ +(1 + 끫殬)G
=
(1 + 끫殬)G− 1
끫殀
Immediate vs. Due 끫殜̈G|= 끫殜G|(1 + 끫殬)= 1 + 끫殜G39| 끫毀̈G|= 끫毀G|(1 + 끫殬)= 끫毀GL9|− 1
Deferred Annuity _m- year deferred n-year annuity-immediate: _ 끫殆끫殒 = 끫殜0| G|= 끫毆 0 ⋅ 끫殜G|= 끫殜0LG|− 끫殜0|
Perpetuity ï _Perpetuity-immediate: _ 끫殆끫殒 = 끫殜N|= 끫毆 + 끫毆;+ ⋯ =
1
끫殬
ï _Perpetuity-due: _ 끫殆끫殒 = 끫殜̈N|= 1 + 끫毆 + 끫毆;+ ⋯ =
1
끫殀
끫殜̈N|= 1 + 끫殜N|
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 끫殶 monthly payments given annual effective rate of 끫殬, define 끫殮 as the monthly effective rate where 끫殮 =(1 + 끫殬)9 9;⁄ − 1. Then apply 끫殆끫殒 = 끫殜G| using 끫殮.
Payments in Arithmetic Progression ï _PV of n-year annuity-immediate with _ _payments of _ 끫殆, 끫殆 + 끫殈, 끫殆 + 2끫殈, ... , 끫殆 +(끫殶 − 1)끫殈
끫殆끫殒 = 끫殆끫殜G|+ 끫殈
끫殜G|VVV− 끫殶끫毆G
끫殬
Calculator-friendly version: 끫殆끫殒 = W끫殆 +
끫殈
끫殬X 끫殜G|VVV+ W−
끫殈끫殶
끫殬X 끫毆
G
끫殂 = 끫殶, 끫歸 끫殘⁄ = 끫殬 (in %),
끫殆끫殀끫殆= 끫殆 +
끫殈
끫殬, 끫歲끫殒 = −
끫殈끫殶
끫殬
ï _PV of n-year annuity-immediate with _ _payments of _ 1, 2, 3, ... , 끫殶
Unit increasing: (끫歸끫殜)G|=
끫殜̈G|− 끫殶끫毆G
끫殬
P&Q version: 끫殆 = 1, 끫殈 = 1, 끫殂 = 끫殶
ï _PV of n-year annuity-immediate with _ _payments of _ 끫殶, 끫殶 − 1, 끫殶 − 2, ... , 1 Unit decreasing:(끫殀끫殜)G|=
끫殶 − 끫殜G|
끫殬
P&Q version: 끫殆 = 끫殶, 끫殈 = −1, 끫殂 = 끫殶
ï _PV of perpetuity-immediate and _ _perpetuity-due with payments of _ 1, 2, 3, ... (끫歸끫殜)N|= 1 끫殬끫殀=
1
끫殬+
1
끫殬;
(끫歸끫殜̈)N|=끫殀 1 ;
INTEREST MEASUREMENT ANNUITIES MORE GENERAL ANNUITIES
an sn
$
1
1 1 1
... –1 nn
...
2
a !!
n
! s!
n 1 1
1 1
... –1 nn
...
2
$
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Payments in Geometric Progression _PV of an n-year annuity-immediate with payments of _ 1,(1 + 끫殰),(1 + 끫殰);, ... ,(1 + 끫殰)G
끫殆끫殒 =
1 − n1 + 끫殰1 + 끫殬o
G
끫殬 − 끫殰 , 끫殬 ≠ 끫殰
Level and Increasing Continuous Annuity 끫殜VG|=@끫毆"
G q
끫殀끫毂 =
1 − 끫毆G
끫毾 =
끫殬
끫毾끫殜G|
(끫歸끫殜V̅)G|=@끫毂끫毆"
G q
끫殀끫毂 =
끫殜VG|− 끫殶끫毆G
끫毾
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.
LOAN AMORTIZATION
Outstanding Balance Calculation ï Prospective: 끫歪"= 끫殊끫殜G3"|, Present value of future level payments of 끫殊. ï Retrospective: 끫歪"= 끫歾(1 + 끫殬)"− 끫殊끫毀"| Accumulated value of original loan amount _L _ minus accumulated value of all past payments.
Loan Amortization For a loan of 끫殜G| repaid with n payments of 1: Period 끫毂 Interest (끫歸") 1 − 끫毆G3"L Principal repaid (끫殆") 끫毆G3"L Total 1
General Formulas for Amortized Loan with Level/Non -Level Payments 끫歸"= 끫殬 ⋅ 끫歪" 끫歪"= 끫歪"39(1 + 끫殬)− 끫殊"= 끫歪"39− 끫殆" 끫殆"= 끫殊"− 끫歸" 끫殆"Lv= 끫殆"(1 + 끫殬)v (only for Level Payments)
BONDS
**Bond Pricing Formulas ** 끫殆 Price of bond 끫歲 Par value (face amount) of bond (not a cash flow) 끫歲 Coupon rate per payment period 끫歲끫歲 Amount of each coupon payment 끫歬 Redemption value of bond (끫歲 = 끫歬 unless otherwise stated) 끫殬 Interest rate per payment period 끫殶 Number of coupon payments Basic Formula 끫殆 = 끫歲끫歲끫殜G|y+ 끫歬끫毆G Premium/Discount Formula : 끫殆 = 끫歬 +(끫歲끫歲 − 끫歬끫殬)끫殜G|y
Premium vs. Discount
Premium Discount
Condition
끫殆 > 끫歬
or 끫歲끫歲 > 끫歬끫殬
끫殆 < 끫歬
or 끫歲끫歲 < 끫歬끫殬
Amortization Process
Write- Down
Write-Up
Amount
|(끫歲끫歲 − 끫歬끫殬)⋅ 끫毆G3"L9|
=|끫歪"39− 끫歪"|=|끫歲끫歲 − 끫歸"|
General Formulas for Bond Amortization ï Book value: 끫歪"= 끫歲끫歲끫殜G3"|y+ 끫歬끫毆G3" = 끫歬 +(끫歲끫歲 − 끫歬끫殬)끫殜G3"|y ï Interest earned = 끫殬끫歪"
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.
YIELD RATES
LOAN AMORTIZATION BONDS
Bt
Prospective Discounting Future Payments
Retrospective Accumulating Past Payments
0 t n L
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INTEREST MEASUREMENT OF A
FUND
Dollar-weighted Interest Rate The yield rate computation depends on the amount invested. Method: ï Calculate interest: 끫歸 = 끫歪 − 끫歨 − 끫歬 끫歨: Amount at the beginning of period 끫歪: Amount at the end of period 끫歬: Deposit/withdrawal ï Calculate dollar-weighted interest rate: 끫殬ôö=
끫歸
끫歨 +∑끫歬"(1 − 끫毂)
Time -weighted Interest Rate The yield rate computation depends on successive sub-intervals of the year each time a deposit or withdrawal is made. Method: 1 + 끫殬úö= W끫歨끫歪; 9
X ⋅ W끫歨끫歪â ;
X ⋅ W끫歨끫歪ù â
X ⋅ ... ⋅ W끫歪끫歨G
G
X
Date 1 Date
Account Before CF
끫歨 9 끫歨;
Cash Flow (CF) 끫歬 9 끫歬;
Account After CF
끫歪 9 = 끫歨 9 + 끫歬 9 끫歪;= 끫歨;+ 끫歬;
DURATION AND CONVEXITY
Duration
끫殀끫殜끫殀끫殀= −
끫殆 8 (끫毾)
끫殆(끫毾)=
∑G"ûq끫毂 ⋅ 끫毆"⋅ 끫歬끫歲" ∑G"ûq끫毆"⋅ 끫歬끫歲"
끫殀끫殀끫殀끫殀= −끫殆
8 (끫殬)
끫殆(끫殬)=
∑G"ûq끫毂 ⋅ 끫毆"L9⋅ 끫歬끫歲" ∑G"ûq끫毆"⋅ 끫歬끫歲" 끫殀끫殀끫殀끫殀= 끫殀끫殜끫殀끫殀⋅ 끫毆
끫殀끫殜끫殀끫殀 끫殶-year zero-coupon bond
끫殶
Geometrically increasing perpetuity
1 + 끫殬
끫殬 − 끫殰
끫殶-year par bond 끫殜̈G|
First -order Modified Approximation 끫殆(끫殬G)≈ 끫殆(끫殬†)⋅ [1 −(끫殬G− 끫殬†)(끫殀끫殀끫殀끫殀)]
First -order Macaulay Approximation
끫殆(끫殬G)≈ 끫殆(끫殬†)⋅ W
1 + 끫殬†
1 + 끫殬GX
°é¢ô
Passage of Time Given that the future cash flows are the same at time 끫毂 9 and time 끫毂;: 끫殀끫殜끫殀 끫殀"C= 끫殀끫殜끫殀 끫殀"D−(끫毂;− 끫毂 9 ) 끫殀끫殀끫殀끫殀"C= 끫殀끫殀끫殀끫殀"D− 끫毆(끫毂;− 끫毂 9 )
Duration of a portfolio For a portfolio of m securities where invested amount 끫殆 = 끫殆 9 + 끫殆;+ ⋯ + 끫殆 0 at time 0: 끫殀끫殜끫殀끫殀£=
끫殆 9
끫殆 끫殀끫殜끫殀끫殀 9 + ⋯ +
끫殆 0
끫殆 끫殀끫殜끫殀끫殀 0
Convexity
끫殀끫殀끫殀끫歬=
끫殆 88 (끫殬)
끫殆(끫殬)=
∑G"ûq끫毂 ⋅(끫毂 + 1)⋅ 끫毆"L;⋅ 끫歬끫歲" ∑G"ûq끫毆"⋅ 끫歬끫歲"
끫殀끫殜끫殀끫歬=
끫殆 88 (끫毾)
끫殆(끫毾)=
∑G"ûq끫毂;⋅ 끫毆"⋅ 끫歬끫歲" ∑G"ûq끫毆"⋅ 끫歬끫歲" 끫殀끫殀끫殀끫歬= 끫毆;(끫殀끫殜끫殀끫歬+ 끫殀끫殜끫殀끫殀)
끫殀끫殜끫殀끫歬(끫殶-year zero-coupon bond)= 끫殶;
IMMUNIZATION
Redington and Full Immunization
Redington Full 끫殆끫殒§••å"•= 끫殆끫殒¶yéßy®y"yå• 끫殀끫殜끫殀끫殀§= 끫殀끫殜끫殀끫殀¶ or 끫殆§ 8 = 끫殆¶ 8
끫歬§> 끫歬¶
or 끫殆§ 88 > 끫殆¶ 88
There has to be asset cash flows before and after each liability cash flow.
Immunizes against small changes in 끫殬
Immunizes against any changes in 끫殬
Immunization Shortcut ( works for immunization questions that have asset cash flows before and after the liability cash flow )
- Identify the asset allocation at the time the liability occurs by equating face amounts (prices) and durations. 끫毈 =
끫毂;− 끫毂¶
끫毂;− 끫毂 9
끫毂 9 Shorter bond duration 끫毂; Longer bond duration 끫毂¶ Liability duration 끫毈 Shorter bond's weight 1 − 끫毈 Longer bond's weight
- Adjust for interest to the asset maturity date.
INTEREST MEASUREMENT
OF A FUND
DURATION AND CONVEXITY IMMUNIZATION
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BA -II PLUS CALCULATOR GUIDELINE
Basic Operations
ENTER (SET) : Send value to a
variable (option)
↑ ↓ : Navigate through variables
2ND : Access secondary functions (yellow)
STO + 09 : Send on-screen value
into memory
RCL + 09 : Recall value from a memory
Time Value of Money (TVM) Good for handling annuities, loans and bonds. Note: Be careful with signs of cash flows. N : Number of periods I/Y : Effective interest rate per period (in %) PV : Present value PMT : Amount of each payment of an annuity FV : Future value CPT + (one of above): Solve for unknown 2ND + BGN , 2ND + SET , 2ND + QUIT : Switch between annuity immediate and annuity due 2ND + P/Y : _Please keep P/Y and C/Y as _ 2ND + CLR TVM : Clear TVM worksheet 2ND + AMORT : Amortization (See Below)
_For bonds _ n끫殆 =끫歲끫歲끫殜G|y+ 끫歬끫毆G o: N = 끫殶; I/Y = 끫殬; PV = −끫殆; PMT = 끫歲끫歲; FV = 끫歬.
Cash Flow Worksheet ( CF , NPV , IRR ) Good for non-level series of payments. Input ( CF ) CF 0 : Cash flow at time Cn: nth cash flow Fn: Frequency of the cash flow
Output ( NPV , IRR ) I: Effective interest rate (in %) NPV + CPT : Solve for net present value IRR + CPT : Solve for internal rate of return
Amortization Schedule ( 2ND + AMORT ) Good for finding outstanding balance of the loan and interest/principal portion of certain payments. Note: BA-II Plus requires computing the unknown TVM variable before entering into AMORT function. P1: Starting period P2: Ending period BAL: Remaining balance of the loan after P PRN: Sum of the principal repaid from P1 to P INT: Sum of the interest paid from P1 to P
BA-II PLUS CALCULATOR GUIDE
Fm formula sheet - N/A
University: SUNY Broome Community College
