Appendix — Reference Information
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This appendix includes supplemental information to help you use your
BA II PLUSé and BA II PLUSé PROFESSIONAL calculator:
• Formulas
• Error conditions
• Accuracy information
• IRR (internal-rate-of-return) calculations
• Algebraic operating system (AOS™)
• Battery information
• In case of difficulty
• TI product service and warranty information
Formulas
This section lists formulas used internally by the calculator.
Time Value of Money
where: PMT Ā0
y =C/Y P P/Y
x =(.01 Q I/Y) P C/Y
C/Y =compounding periods per year
P/Y =payment periods per year
I/Y =interest rate per year
where: PMT =0
The iteration used to compute i:
i ey lnx + 1= –1
i –FV PV1 N= – 1
0 PV PMT Gi
1 1 + i–N –
i
= + ----------------------------- + FV 1 + i–N
I/Y =
where: x = i
y =P/Y P C/Y
Gi = 1 + i Q k
where: k =0 for end-of-period payments
k =1 for beginning-of-period payments
where: i ƒ0
N = L(PV + FV) P PMT
where: i =0
where: i ƒ0
PMT = L(PV + FV) P N
where: i =0
where: i ƒ0
PV = L(FV + PMT Q N)
where: i =0
100 C Y ey lnx + 1⁄– 1
N
PMT Gi – FV i
PMT Gi + PV i
----------------------------------------------
ln
ln1 + i
= ---------------------------------------------------------
PMT
–i
Gi
----- PV PV + FV
1 + iN – 1
= + ---------------------------
PV
PMT Gi
i
------------------------ – FV 1
1 + iN ------------------
PMT Gi
i
= – ------------------------
where: i ƒ0
FV = L(PV + PMT Q N)
where: i =0
Amortization
If computing bal(), pmt2 = npmt
Let bal(0) = RND(PV)
Iterate from m = 1 to pmt2
then: bal( ) =bal(pmt2)
GPrn( ) =bal(pmt2) N bal(pmt1)
GInt( ) =(pmt2 N pmt1 +1) Q RND(PMT) N GPrn( )
where: RND =round the display to the number of decimal
places selected
RND12 =round to 12 decimal places
Balance, principal, and interest are dependent on the values of PMT, PV,
I/Y, and pmt1 and pmt2.
Cash Flow
where:
FV
PMT Gi
i
------------------------ 1 + iN – PV
PMT Gi
i
+ ------------------------
=
Im = RNDRND12–i balm – 1
balm= balm – 1– Im + RNDPMT
NPV CF0 CFj1 + i
-Sj – 1 1 1 + i
-nj –
i
----------------------------------
j = 1
N
= +
Sj
ni
i = 1
jj 1
≥
0 j 0 =
=
Net present value depends on the values of the initial cash flow (CF0),
subsequent cash flows (CFj), frequency of each cash flow (nj), and the
specified interest rate (i).
where: i is the periodic interest rate used in the calculation of NPV.
where: is the frequency of the kth cash flow.
IRR = 100 i, where i satisfies npv() = 0
Internal rate of return depends on the values of the initial cash flow
(CF0) and the subsequent cash flows (CFj).
i = I/Y 100
The calculator uses this formula to compute the modified internal rate of
return:
where: positive = positive values in the cash flows
negative = negative values in the cash flows
N = number of cash flows
rrate = reinvestment rate
frate = finance rate
NPV (values, rate) = Net present value of the values in the rate
described
NFV = 1 + ip NPV
p nk
k = 1
N
=
nk
MOD – NPV (positive, rrate
NPV (negative, frate)
-----------------------------------------------------
1 ⁄N
= 1 + rrate– 1
Bonds1
Price (given yield) with one coupon period or less to redemption:
where: PRI =dollar price per $100 par value
RV =redemption value of the security per $100 par value (RV =
100 except in those cases where call or put features must be
considered)
R =annual interest rate (as a decimal; CPN _ 100)
M =number of coupon periods per year standard for the
particular security involved (set to 1 or 2 in Bond worksheet)
DSR =number of days from settlement date to redemption date
(maturity date, call date, put date, etc.)
E =number of days in coupon period in which the settlement
date falls
Y =annual yield (as a decimal) on investment with security held
to redemption (YLD P 100)
A =number of days from beginning of coupon period to
settlement date (accrued days)
Note: The first term computes present value of the redemption amount,
including interest, based on the yield for the invested period. The second
term computes the accrued interest agreed to be paid to the seller.
Yield (given price) with one coupon period or less to redemption:
1. Source for bond formulas (except duration): Lynch, John J., Jr., and Jan H. Mayle.
Standard Securities Calculation Methods. New York: Securities Industry Association,
1986.
PRI
RV 100 R
M
+ ------------------
1 DSR
E
----------- YM ----
+
-------------------------------------- AE
--- 100 R
M
= – ------------------
Y
RV
100
-------- RM
+ ----
PRI
100
--------- AE
--- RM----
+
–
PRI
100
--------- AE
--- RM ----
+
--------------------------------------------------------------------------- M E
DSR
= --------------
Price (given yield) with more than one coupon period to redemption:
where: N =number of coupons payable between settlement date and
redemption date (maturity date, call date, put date, etc.). (If this
number contains a fraction, raise it to the next whole number;
for example, 2.4 = 3)
DSC =number of days from settlement date to next coupon date
K =summation counter
Note: The first term computes present value of the redemption amount,
not including interest. The second term computes the present values for
all future coupon payments. The third term computes the accrued
interest agreed to be paid to the seller.
Yield (given price) with more than one coupon period to redemption:
Yield is found through an iterative search process using the “Price with
more than one coupon period to redemption” formula.
Accrued interest for securities with standard coupons or interest at
maturity:
where: AI =accrued interest
PAR =par value (principal amount to be paid at maturity)
Modified duration:2
2. Source for duration: Strong, Robert A., Portfolio Construction, Management, and
Protection, South-Western College Publishing, Cincinnati, Ohio, 2000.
PRI
RV
1 YM - + ---
N – 1 DSC
E
+ -----------
------------------------------------------
100 RM
---- AE
– ---
100 RM
----
1 YM
+ ----
K – 1 DSC
E
+ -----------
------------------------------------------
K = 1
N
= +
AI PAR RM
---- AE
= ---
Modified Duration Duration
1 YM
+ ----
= ------------------------
where Duration is calculated using one of the following formulas used to
calculate Macaulay duration:
• For a bond price with one coupon period or less to redemption:
• For a bond price with more than one coupon period to redemption:
Depreciation
RDV = CST N SAL N accumulated depreciation
Values for DEP, RDV, CST, and SAL are rounded to the number of
decimals you choose to be displayed.
In the following formulas, FSTYR = (13 N MO1) P 12.
Straight-line depreciation
First year:
Last year or more: DEP = RDV
Dur 1 YM - + ---
Dsr
Rv 100 R
M
+ ------------------
1 Dsr Y
E M
-------------------
+
2 -----------------------------------------
E M Pri
= ⋅---------------------------------------------------------------
CST – SAL
LIF
---------------------------
CST – SAL
LIF
--------------------------- FSTYR
Sum-of-the-years’-digits depreciation
First year:
Last year or more: DEP = RDV
Declining-balance depreciation
where: RBV is for YR - 1
First year:
Unless ; then use RDV Q FSTYR
If DEP > RDV, use DEP = RDV
If computing last year, DEP = RDV
Statistics
Note: Formulas apply to both x and y.
Standard deviation with n weighting (sx):
LIF + 2 – YR – FSTYRCST – SAL
LIF LIF + 1 2
------------------------------------------------------------------------------------------------------
LIF CST – SAL
LIF LIF + 1 2
------------------------------------------------------------ FSTYR
RBV DB%
LIF 100
-------------------------------
CST DB%
LIF 100
------------------------------ FSTYR
CST DB%
LIF 100
------------------------------ RDV
1 ⁄2
x2
x
2
n
– -------------------
n
----------------------------------------
Standard deviation with n-1 weighting (sx):
Mean:
Regressions
Formulas apply to all regression models using transformed data.
Interest Rate Conversions
where: x =.01 Q NOM P CˆY
where: x =.01 Q EFF
Percent Change
1 ⁄2
x2
x
2
n
– -------------------
n – 1
----------------------------------------
x
x
n
= ---------------
b
nxy – yx
nx2x2 –
= ---------------------------------------------------------
a
y – bx
n
= ---------------------------------
r
bx
y
= --------
EFF = 100 eC ⁄Y Inx 1– 1
NOM = 100 C ⁄Y e1 C ⁄Y In x + 1– 1
NEW OLD 1 %CH
100
+ -------------
#PD
=
where: OLD =old value
NEW =new value
%CH =percent change
#PD =number of periods
Profit Margin
Breakeven
PFT = P Q N (FC + VC Q)
where: PFT =profit
P =price
FC =fixed cost
VC =variable cost
Q =quantity
Days between Dates
With the Date worksheet, you can enter or compute a date within the
range January 1, 1950, through December 31, 2049.
Actual/actual day-count method
Note: The method assumes the actual number of days per month and
per year.
DBD (days between dates) = number of days II - number of days I
Number of Days I= (Y1 - YB) Q 365
+ (number of days MB to M1)
+ DT1
+
Number of Days II=(Y2 - YB) Q 365
+ (number of days MB to M2)
+ DT2
+
Gross Profit Margin Selling Price – Cost
Selling Price
= ----------------------------------------------- 100
Y1 – YB
4
------------------------
Y2 – YB
4
------------------------
where: M1 =month of first date
DT1 =day of first date
Y1 =year of first date
M2 =month of second date
DT2 =day of second date
Y2 =year of second date
MB =base month (January)
DB =base day (1)
YB =base year (first year after leap year)
30/360 day-count method3
Note: The method assumes 30 days per month and 360 days per year.
where: M1 =month of first date
DT1 =day of first date
Y1 =year of first date
M2 =month of second date
DT2 =day of second date
Y2 =year of second date
Note: If DT1 is 31, change DT1 to 30. If DT2 is 31 and DT1 is 30 or 31,
change DT2 to 30; otherwise, leave it at 31.
3. Source for 30/360 day-count method formula: Lynch, John J., Jr., and Jan H. Mayle.
Standard Securities Calculation Methods. New York: Securities Industry Association,
1986
DBD = Y2 – Y1 360 + M2 + M1 30 + DT2 – DT1
Error Messages
Note: To clear an error message, press P.
This appendix includes supplemental information to help you use your
BA II PLUSé and BA II PLUSé PROFESSIONAL calculator:
• Formulas
• Error conditions
• Accuracy information
• IRR (internal-rate-of-return) calculations
• Algebraic operating system (AOS™)
• Battery information
• In case of difficulty
• TI product service and warranty information
Formulas
This section lists formulas used internally by the calculator.
Time Value of Money
where: PMT Ā0
y =C/Y P P/Y
x =(.01 Q I/Y) P C/Y
C/Y =compounding periods per year
P/Y =payment periods per year
I/Y =interest rate per year
where: PMT =0
The iteration used to compute i:
i ey lnx + 1= –1
i –FV PV1 N= – 1
0 PV PMT Gi
1 1 + i–N –
i
= + ----------------------------- + FV 1 + i–N
I/Y =
where: x = i
y =P/Y P C/Y
Gi = 1 + i Q k
where: k =0 for end-of-period payments
k =1 for beginning-of-period payments
where: i ƒ0
N = L(PV + FV) P PMT
where: i =0
where: i ƒ0
PMT = L(PV + FV) P N
where: i =0
where: i ƒ0
PV = L(FV + PMT Q N)
where: i =0
100 C Y ey lnx + 1⁄– 1
N
PMT Gi – FV i
PMT Gi + PV i
----------------------------------------------
ln
ln1 + i
= ---------------------------------------------------------
PMT
–i
Gi
----- PV PV + FV
1 + iN – 1
= + ---------------------------
PV
PMT Gi
i
------------------------ – FV 1
1 + iN ------------------
PMT Gi
i
= – ------------------------
where: i ƒ0
FV = L(PV + PMT Q N)
where: i =0
Amortization
If computing bal(), pmt2 = npmt
Let bal(0) = RND(PV)
Iterate from m = 1 to pmt2
then: bal( ) =bal(pmt2)
GPrn( ) =bal(pmt2) N bal(pmt1)
GInt( ) =(pmt2 N pmt1 +1) Q RND(PMT) N GPrn( )
where: RND =round the display to the number of decimal
places selected
RND12 =round to 12 decimal places
Balance, principal, and interest are dependent on the values of PMT, PV,
I/Y, and pmt1 and pmt2.
Cash Flow
where:
FV
PMT Gi
i
------------------------ 1 + iN – PV
PMT Gi
i
+ ------------------------
=
Im = RNDRND12–i balm – 1
balm= balm – 1– Im + RNDPMT
NPV CF0 CFj1 + i
-Sj – 1 1 1 + i
-nj –
i
----------------------------------
j = 1
N
= +
Sj
ni
i = 1
jj 1
≥
0 j 0 =
=
Net present value depends on the values of the initial cash flow (CF0),
subsequent cash flows (CFj), frequency of each cash flow (nj), and the
specified interest rate (i).
where: i is the periodic interest rate used in the calculation of NPV.
where: is the frequency of the kth cash flow.
IRR = 100 i, where i satisfies npv() = 0
Internal rate of return depends on the values of the initial cash flow
(CF0) and the subsequent cash flows (CFj).
i = I/Y 100
The calculator uses this formula to compute the modified internal rate of
return:
where: positive = positive values in the cash flows
negative = negative values in the cash flows
N = number of cash flows
rrate = reinvestment rate
frate = finance rate
NPV (values, rate) = Net present value of the values in the rate
described
NFV = 1 + ip NPV
p nk
k = 1
N
=
nk
MOD – NPV (positive, rrate
NPV (negative, frate)
-----------------------------------------------------
1 ⁄N
= 1 + rrate– 1
Bonds1
Price (given yield) with one coupon period or less to redemption:
where: PRI =dollar price per $100 par value
RV =redemption value of the security per $100 par value (RV =
100 except in those cases where call or put features must be
considered)
R =annual interest rate (as a decimal; CPN _ 100)
M =number of coupon periods per year standard for the
particular security involved (set to 1 or 2 in Bond worksheet)
DSR =number of days from settlement date to redemption date
(maturity date, call date, put date, etc.)
E =number of days in coupon period in which the settlement
date falls
Y =annual yield (as a decimal) on investment with security held
to redemption (YLD P 100)
A =number of days from beginning of coupon period to
settlement date (accrued days)
Note: The first term computes present value of the redemption amount,
including interest, based on the yield for the invested period. The second
term computes the accrued interest agreed to be paid to the seller.
Yield (given price) with one coupon period or less to redemption:
1. Source for bond formulas (except duration): Lynch, John J., Jr., and Jan H. Mayle.
Standard Securities Calculation Methods. New York: Securities Industry Association,
1986.
PRI
RV 100 R
M
+ ------------------
1 DSR
E
----------- YM ----
+
-------------------------------------- AE
--- 100 R
M
= – ------------------
Y
RV
100
-------- RM
+ ----
PRI
100
--------- AE
--- RM----
+
–
PRI
100
--------- AE
--- RM ----
+
--------------------------------------------------------------------------- M E
DSR
= --------------
Price (given yield) with more than one coupon period to redemption:
where: N =number of coupons payable between settlement date and
redemption date (maturity date, call date, put date, etc.). (If this
number contains a fraction, raise it to the next whole number;
for example, 2.4 = 3)
DSC =number of days from settlement date to next coupon date
K =summation counter
Note: The first term computes present value of the redemption amount,
not including interest. The second term computes the present values for
all future coupon payments. The third term computes the accrued
interest agreed to be paid to the seller.
Yield (given price) with more than one coupon period to redemption:
Yield is found through an iterative search process using the “Price with
more than one coupon period to redemption” formula.
Accrued interest for securities with standard coupons or interest at
maturity:
where: AI =accrued interest
PAR =par value (principal amount to be paid at maturity)
Modified duration:2
2. Source for duration: Strong, Robert A., Portfolio Construction, Management, and
Protection, South-Western College Publishing, Cincinnati, Ohio, 2000.
PRI
RV
1 YM - + ---
N – 1 DSC
E
+ -----------
------------------------------------------
100 RM
---- AE
– ---
100 RM
----
1 YM
+ ----
K – 1 DSC
E
+ -----------
------------------------------------------
K = 1
N
= +
AI PAR RM
---- AE
= ---
Modified Duration Duration
1 YM
+ ----
= ------------------------
where Duration is calculated using one of the following formulas used to
calculate Macaulay duration:
• For a bond price with one coupon period or less to redemption:
• For a bond price with more than one coupon period to redemption:
Depreciation
RDV = CST N SAL N accumulated depreciation
Values for DEP, RDV, CST, and SAL are rounded to the number of
decimals you choose to be displayed.
In the following formulas, FSTYR = (13 N MO1) P 12.
Straight-line depreciation
First year:
Last year or more: DEP = RDV
Dur 1 YM - + ---
Dsr
Rv 100 R
M
+ ------------------
1 Dsr Y
E M
-------------------
+
2 -----------------------------------------
E M Pri
= ⋅---------------------------------------------------------------
CST – SAL
LIF
---------------------------
CST – SAL
LIF
--------------------------- FSTYR
Sum-of-the-years’-digits depreciation
First year:
Last year or more: DEP = RDV
Declining-balance depreciation
where: RBV is for YR - 1
First year:
Unless ; then use RDV Q FSTYR
If DEP > RDV, use DEP = RDV
If computing last year, DEP = RDV
Statistics
Note: Formulas apply to both x and y.
Standard deviation with n weighting (sx):
LIF + 2 – YR – FSTYRCST – SAL
LIF LIF + 1 2
------------------------------------------------------------------------------------------------------
LIF CST – SAL
LIF LIF + 1 2
------------------------------------------------------------ FSTYR
RBV DB%
LIF 100
-------------------------------
CST DB%
LIF 100
------------------------------ FSTYR
CST DB%
LIF 100
------------------------------ RDV
1 ⁄2
x2
x
2
n
– -------------------
n
----------------------------------------
Standard deviation with n-1 weighting (sx):
Mean:
Regressions
Formulas apply to all regression models using transformed data.
Interest Rate Conversions
where: x =.01 Q NOM P CˆY
where: x =.01 Q EFF
Percent Change
1 ⁄2
x2
x
2
n
– -------------------
n – 1
----------------------------------------
x
x
n
= ---------------
b
nxy – yx
nx2x2 –
= ---------------------------------------------------------
a
y – bx
n
= ---------------------------------
r
bx
y
= --------
EFF = 100 eC ⁄Y Inx 1– 1
NOM = 100 C ⁄Y e1 C ⁄Y In x + 1– 1
NEW OLD 1 %CH
100
+ -------------
#PD
=
where: OLD =old value
NEW =new value
%CH =percent change
#PD =number of periods
Profit Margin
Breakeven
PFT = P Q N (FC + VC Q)
where: PFT =profit
P =price
FC =fixed cost
VC =variable cost
Q =quantity
Days between Dates
With the Date worksheet, you can enter or compute a date within the
range January 1, 1950, through December 31, 2049.
Actual/actual day-count method
Note: The method assumes the actual number of days per month and
per year.
DBD (days between dates) = number of days II - number of days I
Number of Days I= (Y1 - YB) Q 365
+ (number of days MB to M1)
+ DT1
+
Number of Days II=(Y2 - YB) Q 365
+ (number of days MB to M2)
+ DT2
+
Gross Profit Margin Selling Price – Cost
Selling Price
= ----------------------------------------------- 100
Y1 – YB
4
------------------------
Y2 – YB
4
------------------------
where: M1 =month of first date
DT1 =day of first date
Y1 =year of first date
M2 =month of second date
DT2 =day of second date
Y2 =year of second date
MB =base month (January)
DB =base day (1)
YB =base year (first year after leap year)
30/360 day-count method3
Note: The method assumes 30 days per month and 360 days per year.
where: M1 =month of first date
DT1 =day of first date
Y1 =year of first date
M2 =month of second date
DT2 =day of second date
Y2 =year of second date
Note: If DT1 is 31, change DT1 to 30. If DT2 is 31 and DT1 is 30 or 31,
change DT2 to 30; otherwise, leave it at 31.
3. Source for 30/360 day-count method formula: Lynch, John J., Jr., and Jan H. Mayle.
Standard Securities Calculation Methods. New York: Securities Industry Association,
1986
DBD = Y2 – Y1 360 + M2 + M1 30 + DT2 – DT1
Error Messages
Note: To clear an error message, press P.