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1

Prospective Method

forward-looking based on future cash flow

time-t outstanding loan balance

=

PV(remaining loan payments with i)

2

Liquidity Preferance Theory / Opportunity Cost Theory 

To persuade lenders to lend for a longer time, borrowers will have to pay higher interest rate as an incentive.

3

weighted average of individual asset's duration 

Duration of Portfolio

4

Annuity-Immediate Present Value

A angle n, with i. ( 1 - v^(n) ) / i

5

Accumulation function for Constant Force of Interest

a(t) = e ^(delta * time)

6

Duration of Portfolio

weighted average of individual asset's duration 

7

Payer

party who agrees to pay the fixed rate and receive the variable rates 

8

a(t) = e ^(delta * time)

Accumulation function for Constant Force of Interest

9

Terminology Bond Amortization: 

Write-down (Premium Bonds)

Write-Up (Discount Bonds

Loan Amortization: Principal Repaid

10

Expectation theory 

Interest rate for a long term investment provides future expectation for interest on short term investments 

for example; consider 2 year loan with higher interest rate then a 1 year loan. Then one year from now the interest rate on the 1 year loan is expected to be higher than the current interest of the 1 year loan.

11

Settlement dates 

specified dates during the swap tenor when the interest payments are exchanged 

12

A(t) = A(0) * ( 1 - (d/m) )^(-mt)

Amount Function Nominal Discount Rate

13

Loan Amoritization:

Pt = R * ( vn-t+1 )

Principal Repaid when R is level 

14

15

A(t)

=

A(0)*(1 + i)t

Amount Function Effective Interest Rate

16

S angle n, with i. ( ( 1 + i )^(n) - 1 ) / i

Annuity-Immediate Accumulated Value

17

Loan Amortization 

repaying a loan with payments at regular intervals

18

Coupons

Periodic interest payments which form an annuity 

19

Loan consist of what two components

1.) Interest Due

2.) Principal Repaid

20

Write-Up for bond

Absolute value of write-down 

P= | (Fr - C*(i)) * (vn-t+1)  |

Discount Bonds

21

First Order Modified Approx. 

P(in)

=

P(io)*[1 - (in - io)(ModD)]

22

S double dot angle n, with i. ( ( 1 + i )^(n) - 1 ) / d

Annuity-Due Accumulated Value

23

A(t) = A(0)*(1 + i(t))

Amount Function Simple Interest

24

Accumulation Function For Variable Force of Interest

a(t) = e^( integration from 0 to t) of delta

25

To find R (swap rate

PV(Variable) = PV(Fixed) 

26

a(t) = e^( integration from 0 to t) of delta

Accumulation Function For Variable Force of Interest

27

Annuity-Immediate PV to AV relation

(A angle n) =

(S angle n) * ( v)

28

Annuity-Due AV to PV relation

(S double-dot angle n) =

(A double-dot angle n) * ( ( 1 + i ))

29

delta = force of interest constant

ln (1 + i)

30

Discount Bond Formula

(Redemption - Price)

or

( C*(i) - Fr ) * (a angle n) 

31

Loan: Outstanding Balance at time t

is equal to 

Present value at time t of it's remaining cash flow using the loan's interest rate 

32

Notional amount 

is the amount used to determine interest payments and swap payments;

to obtain interest payment or swap payment multiply the notional amount by the respective interest rate 

USUALLY THE LOAN AMOUNT

33

(A(t) - A(t-1)) / A(t-1)

Effective Interest Rate

34

unit decreasing annuity 

(Da) angle n 

[ n - (a angle n) ] 

i

35

Market Segmentation Theory 

Borrowers and Lenders have different preferances on how long they want to borrow or lend for, thus causing different interest rates for different terms 

36

A angle n, with i. ( 1 - v^(n) ) / i

Annuity-Immediate Present Value

37

Discounting Factor with (d)

( 1 - d )

38

( 1 - d )

Discounting Factor with (d)

39

Yield to Maturity

Level annual effective rate of interest which equates cash inflows to cash outflows

40

Amount Function Nominal Discount Rate

A(t) = A(0) * ( 1 - (d/m) )^(-mt)

41

Annuity-Due Present Value

A double dot angle n, with i. ( 1 - v^(n) ) / d

42

Internal Rate of Return (IRR)

also called yield rate

it's the rate that 

produces a NPV of 0

also known as breakeven 

43

floating rate

also known as the variable rate

44

[ 1 - ( (1 + k) / ( 1 +i ) )n ]

/

( i - k)

i = interest

k = number of payments in progression/percentage

n = number of total payments

Annuity-Immediate geometric progression

45

Implied Forward Rate 

(1 + sn)n 

(1 + sn-1)n-1 * (1 + f[n-1, n])

46

(A(t) - A(t-1)) / A(t)

Effective Discount Rate

47

Terminology Bonds Amortization: Book value

Loan Amortization: Outstanding balance

48

(1 + i)^(-1) discount factor

(1 - d)

49

Effective Discount Rate defined as

(A(t) - A(t-1)) / A(t)

50

What is opportunity cost of capital?

Rate of return

on an equally-risky asset

an investor could have earned

if he or she had NOT invested in the project. 

51

Given a price 

lowest yield rate calculated is the minimum yield that an investor would earn

52

swap rate 

known as the fixed rate 

53

Calculating Interest Due 

It (Interest Due)

(i) * Bt-1 (outstanding balance of previous period)

54

Effective Interest Rate defined as

(A(t) - A(t-1)) / A(t-1)

55

Annuity-Immediate AV to PV relation

(S angle n)=

(A angle n) * ( ( 1 + i ))

56
Bonds

A bond's BOOK VALUE  at time t 

is equal to 

Present value at time t of its remaining cash flows using the bond's initial yield rate 

57

Interest Rate Swap

agreement between two parties in which both parties agree to exchange a series of cash flow based on interest rates

58

Accreting Swap 

if notional amount increases over time

59

Solving for bonds

Step 1: Identify Cash Flows

Step 2: Calculate the bond price as the PV of future cash flow at time 0 

60

Spot Rates

Annual Effective Yield rates on a zero-coupon bond

Measures the yield from the beginning of the investment to the end of the single cash flow

 

 

61

Annuity-Immediate geometric progression

[ 1 - ( (1 + k) / ( 1 +i ) )n ]

/

( i - k)

i = interest

k = number of payments in progression/percentage

n = number of total payments

62

Given a yield rate 

the lowest price calculated is the maximum price that an investor would pay 

63

Terminology Bond Amortization: Coupon payment

Loan Amortization: Loan Payment 

64

Retrospective Formula 

Bt

 =

L(1+i)t - R(s angle t)

65

Write-Down for bond

same as principal repaid for loans

(Coupon Payment - Interest Earned)

Pt = | (Fr - C*(i)) * (vn-t+1)  |

Premium bonds

66

Calculating Outstanding Balance end of period

Bt (Outstanding loan balance time t) 

=

 Bt-1 (Outstanding loan balance time t-1)  - Pt (Principal repaid time t)

67

Redemption Value

One large payment at the end of the bond term

68

Bonds: Basic Formula 

P(Price of bond

=

Fr(a angle n) [Pv at time 0 of coupon payments over n periods] 

+

C(vn) [Pv at 0 of the redemption value ]

69

Callable Bonds

Can be terminated before the redemption dates labeled as call dates

70

Settlement period 

is the time between settlement dates 

71

Block Payment Annuity

Method 1: Start with payments furthest from the comparison date

2: Make adjustments when moving towards the comparison date 

72

Swap Term / Swap Tenor 

specified period of the swap 

73

Bond's Yield Rate

Rate of Return from investor's point of view

constant rate discounting future cash flow 

74

P and Q formula for annuity increasing Arithmetic Progression

PV = P*(a angle n)

+

[ (Q)*((a angle n) - n( v) ] / i 

P = first term

Q = constant amount of increase for each payment

n = number of payments one period before the first payment date

75

Interest earned on a Bond calculation

(BOOK VALUE of PREVIOUS PERIOD) * (YIELD RATE) 

76

a(t) = (1 + i(t))

Accumulation Function Simple Interest

77

Finding additional payment

(Annuity Immediate AV)

[ (pmt) * (annuity-immediate(AV)) ]

+

[ (fv) - (PV)(AV factor) ]

=

 0

78

Face amount / par value

not a cash flow

used to determine coupon amount

79

Amount Function Effective Discount Rate

A(t) = A(0) * ( 1 - d )-t

80

Bank Reserve Requirement

A central bank requires every bank to deposit and maintain a specific amount of money with it

81

e(delta) * (t)

 =

( 1 + i )t

Relationship between Constant Force Of Interest and Constant effective interest rate

82

(1 - d)^(-1) accumulation factor

(1 + i)

83

What interest rate should be used to discount the cash flow for NPV? 

interest rate used  = 

cost of capital

or

opportunity cost of capital

84

Premium Bond Formula 

(Price - Redemption)

or

(Fr - C*(i)) * (a angle n)

85

Finding additional payment

(Annuity Immediate PV )

[ (pmt) * ( annuity - immediate( PV ) ) ] 

+

[ (fv)(discounted factor) - ( PV ) ]

=

0

86

Default Risk 

Is the risk of loan defaulting, the borrower is unable to make a promised payment at the contracted time and for the contracted amount. 

87

Amount Function Simple Interest

A(t) = A(0)*(1 + i(t))

88

A(t)

=

A(0) * ( 1 + (i/m) )(mt)

Amount Function Nominal Interest Rate

89

 

Loan Amortization

It = R * (1−vn−t+1)

Interest Due when R is Level

90

Bonds: Premium/Discount Formula

P(Price of bond

C(Redemption value) 

(Fr - C * i)(a angle n)

91

Prospective Method Formula

Bt 

=

R(a angle (n-t))

92

Bank's Reserve

The actual amount of deposit with the central bank

93

Accumulation Function Simple Interest

a(t) = (1 + i(t))

94

Annuity-Immediate Accumulated Value

S angle n, with i. ( ( 1 + i )^(n) - 1 ) / i

95

Force Of Interest

a'(t) / a(t)

96

Amount Function Effective Interest Rate

A(t) = A(0)*(1 + i)^t

97

Receiver 

counter party who receives the fix rate 

and

pays the variable rate 

98

Amortizing Swap 

if the notional amount decreases over time 

99

Amount Function Nominal Interest Rate

A(t) = A(0) * ( 1 + (i/m) )^(mt)

100

Final Amortization Value for Bond

The book value at the end of the bond term which is the redemption value C

101

Calculating Principal Repaid 

Pt (Principal Repaid)

=

Rt (loan repayment)  - I(interest due)

102

Continous Varying Annuities PV

PV = [ integration from 0 to n]

of

 [ f(t) * e - (integration from o to t) of ( delta ) ]

103

Annuity-Due Accumulated Value

S double dot angle n, with i. ( ( 1 + i )^(n) - 1 ) / d

104

Loan Amoritization:

Pt+k 

/

(1 + i)k

Principal Repaid when R is Level

105

Net Present Value (NPV)

NPV = Σ CFt * (vt)  

summation from (t = 0) to n 

NPV = sum of the present value of all cash flow over ( n ) years 

106

Preferred Habit Theory 

Borrowers and Lenders if given enough incentive will be willing to go out of their boundaries. 

For example, a lender who only wants to lend for a certain time may be willing to lend for a longer time if given enough incentives. 

107

Retrospective Method

backward-looking; based on previous cash flows; 

time t outstanding loan balance

=

AV(Orginal loan with i) - AV(loan payments)

108

Discounting Factor with (i)

v = ( 1 / ( 1 + i )^(n))

109

Call Price

additional cash flow the investor receives if the bond terminates on a call date

110

v = ( 1 / ( 1 + i )^(n))

Discounting Factor with (i)

111

Perpetuity-Immediate increasing

(Ia) angle infinity

=

 ( ( P / i ) + ( Q / ( i) )

112

A(t) = A(0)*(1-d)^(-t)

Amount Function Effective Discount Rate

113

114

Relationship between Constant Force Of Interest and Constant effective interest rate

e^(delta) = ( 1 + i )

115

Unit Increasing Annuity
[ (Ia) angle n]

P = Q

[ (Ia) angle n ]

=
(P) * [ ( a double dot angle n ) - n( v) ]

/

i

116

Annuity-Due PV to AV relation

(A double-dot angle n) =

(S double-dot angle n) * ( v)

117

Σ t * vt+1 * CFt 

/

Σ vt * CFt

Modified Duration Formula

118

a'(t) / a(t)

Force Of Interest

119

Pt ( principal repaid for bonds at time t )

Pt ( principal repaid for bonds at time t )

=

| ( Fr - C(i) )*( vn-t+1 ) |

120

A double dot angle n, with i. ( 1 - v^(n) ) / d

Annuity-Due Present Value

121

Default Risk Premium 

compensates investors for the possibility that a borrower will default

122
Reversed Cards

forward-looking based on future cash flow

time-t outstanding loan balance

=

PV(remaining loan payments with i)

Prospective Method

123
Reversed Cards

To persuade lenders to lend for a longer time, borrowers will have to pay higher interest rate as an incentive.

Liquidity Preferance Theory / Opportunity Cost Theory 

124
Reversed Cards

Duration of Portfolio

weighted average of individual asset's duration 

125
Reversed Cards

A angle n, with i. ( 1 - v^(n) ) / i

Annuity-Immediate Present Value

126
Reversed Cards

a(t) = e ^(delta * time)

Accumulation function for Constant Force of Interest

127
Reversed Cards

weighted average of individual asset's duration 

Duration of Portfolio

128
Reversed Cards

party who agrees to pay the fixed rate and receive the variable rates 

Payer

129
Reversed Cards

Accumulation function for Constant Force of Interest

a(t) = e ^(delta * time)

130
Reversed Cards

Loan Amortization: Principal Repaid

Terminology Bond Amortization: 

Write-down (Premium Bonds)

Write-Up (Discount Bonds

131
Reversed Cards

Interest rate for a long term investment provides future expectation for interest on short term investments 

for example; consider 2 year loan with higher interest rate then a 1 year loan. Then one year from now the interest rate on the 1 year loan is expected to be higher than the current interest of the 1 year loan.

Expectation theory 

132
Reversed Cards

specified dates during the swap tenor when the interest payments are exchanged 

Settlement dates 

133
Reversed Cards

Amount Function Nominal Discount Rate

A(t) = A(0) * ( 1 - (d/m) )^(-mt)

134
Reversed Cards

Principal Repaid when R is level 

Loan Amoritization:

Pt = R * ( vn-t+1 )

135
Reversed Cards

136
Reversed Cards

Amount Function Effective Interest Rate

A(t)

=

A(0)*(1 + i)t

137
Reversed Cards

Annuity-Immediate Accumulated Value

S angle n, with i. ( ( 1 + i )^(n) - 1 ) / i

138
Reversed Cards

repaying a loan with payments at regular intervals

Loan Amortization 

139
Reversed Cards

Periodic interest payments which form an annuity 

Coupons

140
Reversed Cards

1.) Interest Due

2.) Principal Repaid

Loan consist of what two components

141
Reversed Cards

Absolute value of write-down 

P= | (Fr - C*(i)) * (vn-t+1)  |

Discount Bonds

Write-Up for bond

142
Reversed Cards

P(in)

=

P(io)*[1 - (in - io)(ModD)]

First Order Modified Approx. 

143
Reversed Cards

Annuity-Due Accumulated Value

S double dot angle n, with i. ( ( 1 + i )^(n) - 1 ) / d

144
Reversed Cards

Amount Function Simple Interest

A(t) = A(0)*(1 + i(t))

145
Reversed Cards

a(t) = e^( integration from 0 to t) of delta

Accumulation Function For Variable Force of Interest

146
Reversed Cards

PV(Variable) = PV(Fixed) 

To find R (swap rate

147
Reversed Cards

Accumulation Function For Variable Force of Interest

a(t) = e^( integration from 0 to t) of delta

148
Reversed Cards

(S angle n) * ( v)

Annuity-Immediate PV to AV relation

(A angle n) =

149
Reversed Cards

(A double-dot angle n) * ( ( 1 + i ))

Annuity-Due AV to PV relation

(S double-dot angle n) =

150
Reversed Cards

ln (1 + i)

delta = force of interest constant

151
Reversed Cards

(Redemption - Price)

or

( C*(i) - Fr ) * (a angle n) 

Discount Bond Formula

152
Reversed Cards

Present value at time t of it's remaining cash flow using the loan's interest rate 

Loan: Outstanding Balance at time t

is equal to 

153
Reversed Cards

is the amount used to determine interest payments and swap payments;

to obtain interest payment or swap payment multiply the notional amount by the respective interest rate 

USUALLY THE LOAN AMOUNT

Notional amount 

154
Reversed Cards

Effective Interest Rate

(A(t) - A(t-1)) / A(t-1)

155
Reversed Cards

(Da) angle n 

[ n - (a angle n) ] 

i

unit decreasing annuity 

156
Reversed Cards

Borrowers and Lenders have different preferances on how long they want to borrow or lend for, thus causing different interest rates for different terms 

Market Segmentation Theory 

157
Reversed Cards

Annuity-Immediate Present Value

A angle n, with i. ( 1 - v^(n) ) / i

158
Reversed Cards

( 1 - d )

Discounting Factor with (d)

159
Reversed Cards

Discounting Factor with (d)

( 1 - d )

160
Reversed Cards

Level annual effective rate of interest which equates cash inflows to cash outflows

Yield to Maturity

161
Reversed Cards

A(t) = A(0) * ( 1 - (d/m) )^(-mt)

Amount Function Nominal Discount Rate

162
Reversed Cards

A double dot angle n, with i. ( 1 - v^(n) ) / d

Annuity-Due Present Value

163
Reversed Cards

also called yield rate

it's the rate that 

produces a NPV of 0

also known as breakeven 

Internal Rate of Return (IRR)

164
Reversed Cards

also known as the variable rate

floating rate

165
Reversed Cards

Annuity-Immediate geometric progression

[ 1 - ( (1 + k) / ( 1 +i ) )n ]

/

( i - k)

i = interest

k = number of payments in progression/percentage

n = number of total payments

166
Reversed Cards

(1 + sn)n 

(1 + sn-1)n-1 * (1 + f[n-1, n])

Implied Forward Rate 

167
Reversed Cards

Effective Discount Rate

(A(t) - A(t-1)) / A(t)

168
Reversed Cards

Loan Amortization: Outstanding balance

Terminology Bonds Amortization: Book value

169
Reversed Cards

(1 - d)

(1 + i)^(-1) discount factor

170
Reversed Cards

(A(t) - A(t-1)) / A(t)

Effective Discount Rate defined as

171
Reversed Cards

Rate of return

on an equally-risky asset

an investor could have earned

if he or she had NOT invested in the project. 

What is opportunity cost of capital?

172
Reversed Cards

lowest yield rate calculated is the minimum yield that an investor would earn

Given a price 

173
Reversed Cards

known as the fixed rate 

swap rate 

174
Reversed Cards

It (Interest Due)

(i) * Bt-1 (outstanding balance of previous period)

Calculating Interest Due 

175
Reversed Cards

(A(t) - A(t-1)) / A(t-1)

Effective Interest Rate defined as

176
Reversed Cards

(A angle n) * ( ( 1 + i ))

Annuity-Immediate AV to PV relation

(S angle n)=

177
Reversed Cards

Present value at time t of its remaining cash flows using the bond's initial yield rate 

A bond's BOOK VALUE  at time t 

is equal to 

178
Reversed Cards

agreement between two parties in which both parties agree to exchange a series of cash flow based on interest rates

Interest Rate Swap

179
Reversed Cards

if notional amount increases over time

Accreting Swap 

180
Reversed Cards

Step 1: Identify Cash Flows

Step 2: Calculate the bond price as the PV of future cash flow at time 0 

Solving for bonds

181
Reversed Cards

Annual Effective Yield rates on a zero-coupon bond

Measures the yield from the beginning of the investment to the end of the single cash flow

 

 

Spot Rates

182
Reversed Cards

[ 1 - ( (1 + k) / ( 1 +i ) )n ]

/

( i - k)

i = interest

k = number of payments in progression/percentage

n = number of total payments

Annuity-Immediate geometric progression

183
Reversed Cards

the lowest price calculated is the maximum price that an investor would pay 

Given a yield rate 

184
Reversed Cards

Loan Amortization: Loan Payment 

Terminology Bond Amortization: Coupon payment

185
Reversed Cards

Bt

 =

L(1+i)t - R(s angle t)

Retrospective Formula 

186
Reversed Cards

(Coupon Payment - Interest Earned)

Pt = | (Fr - C*(i)) * (vn-t+1)  |

Premium bonds

Write-Down for bond

same as principal repaid for loans

187
Reversed Cards

Bt (Outstanding loan balance time t) 

=

 Bt-1 (Outstanding loan balance time t-1)  - Pt (Principal repaid time t)

Calculating Outstanding Balance end of period

188
Reversed Cards

One large payment at the end of the bond term

Redemption Value

189
Reversed Cards

P(Price of bond

=

Fr(a angle n) [Pv at time 0 of coupon payments over n periods] 

+

C(vn) [Pv at 0 of the redemption value ]

Bonds: Basic Formula 

190
Reversed Cards

Can be terminated before the redemption dates labeled as call dates

Callable Bonds

191
Reversed Cards

is the time between settlement dates 

Settlement period 

192
Reversed Cards

Method 1: Start with payments furthest from the comparison date

2: Make adjustments when moving towards the comparison date 

Block Payment Annuity

193
Reversed Cards

specified period of the swap 

Swap Term / Swap Tenor 

194
Reversed Cards

Rate of Return from investor's point of view

constant rate discounting future cash flow 

Bond's Yield Rate

195
Reversed Cards

PV = P*(a angle n)

+

[ (Q)*((a angle n) - n( v) ] / i 

P = first term

Q = constant amount of increase for each payment

n = number of payments one period before the first payment date

P and Q formula for annuity increasing Arithmetic Progression

196
Reversed Cards

(BOOK VALUE of PREVIOUS PERIOD) * (YIELD RATE) 

Interest earned on a Bond calculation

197
Reversed Cards

Accumulation Function Simple Interest

a(t) = (1 + i(t))

198
Reversed Cards

[ (pmt) * (annuity-immediate(AV)) ]

+

[ (fv) - (PV)(AV factor) ]

=

 0

Finding additional payment

(Annuity Immediate AV)

199
Reversed Cards

not a cash flow

used to determine coupon amount

Face amount / par value

200
Reversed Cards

A(t) = A(0) * ( 1 - d )-t

Amount Function Effective Discount Rate

201
Reversed Cards

A central bank requires every bank to deposit and maintain a specific amount of money with it

Bank Reserve Requirement

202
Reversed Cards

Relationship between Constant Force Of Interest and Constant effective interest rate

e(delta) * (t)

 =

( 1 + i )t

203
Reversed Cards

(1 + i)

(1 - d)^(-1) accumulation factor

204
Reversed Cards

interest rate used  = 

cost of capital

or

opportunity cost of capital

What interest rate should be used to discount the cash flow for NPV? 

205
Reversed Cards

(Price - Redemption)

or

(Fr - C*(i)) * (a angle n)

Premium Bond Formula 

206
Reversed Cards

[ (pmt) * ( annuity - immediate( PV ) ) ] 

+

[ (fv)(discounted factor) - ( PV ) ]

=

0

Finding additional payment

(Annuity Immediate PV )

207
Reversed Cards

Is the risk of loan defaulting, the borrower is unable to make a promised payment at the contracted time and for the contracted amount. 

Default Risk 

208
Reversed Cards

A(t) = A(0)*(1 + i(t))

Amount Function Simple Interest

209
Reversed Cards

Amount Function Nominal Interest Rate

A(t)

=

A(0) * ( 1 + (i/m) )(mt)

210
Reversed Cards

Interest Due when R is Level

 

Loan Amortization

It = R * (1−vn−t+1)

211
Reversed Cards

P(Price of bond

C(Redemption value) 

(Fr - C * i)(a angle n)

Bonds: Premium/Discount Formula

212
Reversed Cards

Bt 

=

R(a angle (n-t))

Prospective Method Formula

213
Reversed Cards

The actual amount of deposit with the central bank

Bank's Reserve

214
Reversed Cards

a(t) = (1 + i(t))

Accumulation Function Simple Interest

215
Reversed Cards

S angle n, with i. ( ( 1 + i )^(n) - 1 ) / i

Annuity-Immediate Accumulated Value

216
Reversed Cards

a'(t) / a(t)

Force Of Interest

217
Reversed Cards

A(t) = A(0)*(1 + i)^t

Amount Function Effective Interest Rate

218
Reversed Cards

counter party who receives the fix rate 

and

pays the variable rate 

Receiver 

219
Reversed Cards

if the notional amount decreases over time 

Amortizing Swap 

220
Reversed Cards

A(t) = A(0) * ( 1 + (i/m) )^(mt)

Amount Function Nominal Interest Rate

221
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The book value at the end of the bond term which is the redemption value C

Final Amortization Value for Bond

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Pt (Principal Repaid)

=

Rt (loan repayment)  - I(interest due)

Calculating Principal Repaid 

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PV = [ integration from 0 to n]

of

 [ f(t) * e - (integration from o to t) of ( delta ) ]

Continous Varying Annuities PV

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S double dot angle n, with i. ( ( 1 + i )^(n) - 1 ) / d

Annuity-Due Accumulated Value

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Principal Repaid when R is Level

Loan Amoritization:

Pt+k 

/

(1 + i)k

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NPV = Σ CFt * (vt)  

summation from (t = 0) to n 

NPV = sum of the present value of all cash flow over ( n ) years 

Net Present Value (NPV)

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Borrowers and Lenders if given enough incentive will be willing to go out of their boundaries. 

For example, a lender who only wants to lend for a certain time may be willing to lend for a longer time if given enough incentives. 

Preferred Habit Theory 

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backward-looking; based on previous cash flows; 

time t outstanding loan balance

=

AV(Orginal loan with i) - AV(loan payments)

Retrospective Method

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v = ( 1 / ( 1 + i )^(n))

Discounting Factor with (i)

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additional cash flow the investor receives if the bond terminates on a call date

Call Price

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Discounting Factor with (i)

v = ( 1 / ( 1 + i )^(n))

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(Ia) angle infinity

=

 ( ( P / i ) + ( Q / ( i) )

Perpetuity-Immediate increasing

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Amount Function Effective Discount Rate

A(t) = A(0)*(1-d)^(-t)

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e^(delta) = ( 1 + i )

Relationship between Constant Force Of Interest and Constant effective interest rate

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P = Q

[ (Ia) angle n ]

=
(P) * [ ( a double dot angle n ) - n( v) ]

/

i

Unit Increasing Annuity
[ (Ia) angle n]

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(S double-dot angle n) * ( v)

Annuity-Due PV to AV relation

(A double-dot angle n) =

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Modified Duration Formula

Σ t * vt+1 * CFt 

/

Σ vt * CFt

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Force Of Interest

a'(t) / a(t)

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Pt ( principal repaid for bonds at time t )

=

| ( Fr - C(i) )*( vn-t+1 ) |

Pt ( principal repaid for bonds at time t )

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Annuity-Due Present Value

A double dot angle n, with i. ( 1 - v^(n) ) / d

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compensates investors for the possibility that a borrower will default

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