Prospective Method

**forward-looking based on future cash flow**

**time-t outstanding loan balance**

**=**

**PV(remaining loan payments with i)**

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.

weighted average of individual asset's duration

Duration of Portfolio

Annuity-Immediate Present Value

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

Accumulation function for Constant Force of Interest

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

Duration of Portfolio

weighted average of individual asset's duration

Payer

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

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

Accumulation function for Constant Force of Interest

**Terminology Bond Amortization: **

Write-down (*Premium Bonds*)

Write-Up (*Discount Bonds*)

**Loan Amortization: **Principal Repaid

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.

Settlement dates

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

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

Amount Function Nominal Discount Rate

Loan Amoritization:

P_{t} = R * ( v^{n-t+1} )

Principal Repaid when R is level

**A(t) **

**=**

**A(0)*(1 + i) ^{t}**

Amount Function Effective Interest Rate

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

Annuity-Immediate Accumulated Value

Loan Amortization

repaying a loan with payments at regular intervals

Coupons

Periodic interest payments which form an annuity

Loan consist of what two components

**1.) Interest Due**

**2.) Principal Repaid**

Write-Up for bond

Absolute value of write-down

P_{t }= **|** (Fr - C*(i)) * (v^{n-t+1}) **|**

**Discount Bonds**

First Order Modified Approx.

**P(i _{n}) **

**=**

**P(i _{o})*[1 - (i_{n} - i_{o})(ModD)]**

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

Annuity-Due Accumulated Value

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

Amount Function Simple Interest

Accumulation Function For Variable Force of Interest

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

To find R (*swap rate*)

PV(Variable) = PV(Fixed)

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

Accumulation Function For Variable Force of Interest

Annuity-Immediate PV to AV relation

(A angle n) =

(S angle n) * ( v** ^{n }**)

Annuity-Due AV to PV relation

(S double-dot angle n) =

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

delta = force of interest constant

ln (1 + i)

Discount Bond Formula

(Redemption - Price)

or

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

**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 **

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**

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

Effective Interest Rate

unit decreasing annuity

**(Da) angle n **

**= **

**[ n - (a angle n) ] **

**/ **

**i**

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

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

Annuity-Immediate Present Value

Discounting Factor with (d)

( 1 - d )

( 1 - d )

Discounting Factor with (d)

Yield to Maturity

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

Amount Function Nominal Discount Rate

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

Annuity-Due Present Value

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

Internal Rate of Return (IRR)

**also called yield rate**

**it's the rate that **

**produces a NPV of 0**

**also known as breakeven **

floating rate

also known as the variable rate

**[ 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

Implied Forward Rate

**(1 + s _{n})^{n} **

**= **

**(1 + s _{n-1})^{n-1} * (1 + f_{[n-1, n]})**

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

Effective Discount Rate

**Terminology Bonds Amortization: **Book value

**Loan Amortization: **Outstanding balance

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

(1 - d)

Effective Discount Rate defined as

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

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. **

Given a price

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

swap rate

known as the fixed rate

Calculating Interest Due

**I _{t} (Interest Due)**

**= **

**(i) * B _{t-1} (outstanding balance of previous period)**

Effective Interest Rate defined as

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

Annuity-Immediate AV to PV relation

(S angle n)=

(A angle n) * ( ( 1 + i )^{n })

**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 **

Interest Rate Swap

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

Accreting Swap

if notional amount increases over time

Solving for bonds

Step 1: Identify Cash Flows

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

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

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

Given a yield rate

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

**Terminology Bond Amortization: **Coupon payment

**Loan Amortization: **Loan Payment

Retrospective Formula

**B _{t}**

** =**

**L(1+i) ^{t} - R(s angle t)**

Write-Down for bond

same as principal repaid for loans

(Coupon Payment - Interest Earned)

**P _{t} = | (Fr - C*(i)) * (v^{n-t+1}) |**

Premium bonds

Calculating Outstanding Balance end of period

**B _{t} (Outstanding loan balance time t) **

**=**

** B _{t-1} (Outstanding loan balance time t-1) - P_{t} (Principal repaid time t)**

Redemption Value

One large payment at the end of the bond term

**Bonds: **Basic Formula

**P( Price of bond) **

**=**

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

*+*

**C(v ^{n}) [Pv at 0 of the redemption value ]**

Callable Bonds

Can be terminated before the redemption dates labeled as call dates

Settlement period

is the time between settlement dates

Block Payment Annuity

Method 1: Start with payments furthest from the comparison date

2: Make adjustments when moving towards the comparison date

Swap Term / Swap Tenor

specified period of the swap

Bond's Yield Rate

Rate of Return from investor's point of view

constant rate discounting future cash flow

P and Q formula for annuity increasing Arithmetic Progression

**PV = P*(a angle n)**

**+**

**[ (Q)*((a angle n) - n( v ^{n }) ] / i **

P = first term

Q = constant amount of increase for each payment

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

Interest earned on a Bond calculation

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

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

Accumulation Function Simple Interest

Finding additional payment

(Annuity Immediate AV)

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

+

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

=

0

Face amount / par value

not a cash flow

used to determine coupon amount

Amount Function Effective Discount Rate

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

Bank Reserve Requirement

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

**e ^{(}^{delta) * (t)}**

** =**

**( 1 + i ) ^{t}**

Relationship between Constant Force Of Interest and Constant effective interest rate

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

(1 + i)

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

**interest rate used = **

**cost of capital **

**or **

**opportunity cost of capital**

Premium Bond Formula

(Price - Redemption)

or

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

Finding additional payment

(Annuity Immediate PV )

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

+

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

=

0

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.

Amount Function Simple Interest

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

**A(t) **

**=**

**A(0) * ( 1 + (i/m) ) ^{(mt)}**

Amount Function Nominal Interest Rate

Loan Amortization

I_{t} = R * (1−v^{n−t+1})

Interest Due when R is Level

**Bonds: **Premium/Discount Formula

P(*Price of bond*)

=

C(*Redemption value) *

+

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

Prospective Method Formula

**B _{t} **

**=**

**R(a angle (n-t))**

Bank's Reserve

The actual amount of deposit with the central bank

Accumulation Function Simple Interest

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

Annuity-Immediate Accumulated Value

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

Force Of Interest

a'(t) / a(t)

Amount Function Effective Interest Rate

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

Receiver

counter party who receives the fix rate

and

pays the variable rate

Amortizing Swap

if the notional amount decreases over time

Amount Function Nominal Interest Rate

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

**Final Amortization Value for Bond**

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

Calculating Principal Repaid

**P _{t} (Principal Repaid)**

**=**

**R _{t} (loan repayment) - I_{t }(interest due)**

Continous Varying Annuities PV

**PV = [ integration from 0 to n] **

**of**

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

Annuity-Due Accumulated Value

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

Loan Amoritization:

P_{t+k}

/

(1 + i)^{k}

Principal Repaid when R is Level

Net Present Value (NPV)

**NPV = Σ CF _{t} * (v^{t}) **

**summation from (t = 0) to n **

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

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.

Retrospective Method

**backward-looking; based on previous cash flows; **

**time t outstanding loan balance **

**=**

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

Discounting Factor with (i)

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

Call Price

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

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

Discounting Factor with (i)

Perpetuity-Immediate increasing

**(Ia) angle infinity **

**=**

** ( ( P / i ) + ( Q / ( i ^{2 }) )**

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

Amount Function Effective Discount Rate

Relationship between Constant Force Of Interest and Constant effective interest rate

e^(delta) = ( 1 + i )

Unit Increasing Annuity

[ (Ia) angle n]

P = Q

[ (Ia) angle n ]

=

**(P) * [ ( a double dot angle n ) - n( v ^{n }) ]**

**/**

**i**

Annuity-Due PV to AV relation

(A double-dot angle n) =

(S double-dot angle n) * ( v** ^{n }**)

**Σ t * v ^{t+1} * CF_{t} **

**/**

**Σ v ^{t} * CF_{t}**

Modified Duration Formula

a'(t) / a(t)

Force Of Interest

P_{t} ( *principal repaid for bonds at time t )*

P_{t} ( *principal repaid for bonds at time t )*

*=*

**| **( Fr - C(i) )*( v^{n-t+1} ) **|**

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

Annuity-Due Present Value

Default Risk Premium

compensates investors for the possibility that a borrower will default

**forward-looking based on future cash flow**

**time-t outstanding loan balance**

**=**

**PV(remaining loan payments with i)**

Prospective Method

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

Duration of Portfolio

weighted average of individual asset's duration

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

Annuity-Immediate Present Value

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

Accumulation function for Constant Force of Interest

weighted average of individual asset's duration

Duration of Portfolio

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

Payer

Accumulation function for Constant Force of Interest

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

**Loan Amortization: **Principal Repaid

**Terminology Bond Amortization: **

Write-down (*Premium Bonds*)

Write-Up (*Discount Bonds*)

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

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

Settlement dates

Amount Function Nominal Discount Rate

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

Principal Repaid when R is level

Loan Amoritization:

P_{t} = R * ( v^{n-t+1} )

Amount Function Effective Interest Rate

**A(t) **

**=**

**A(0)*(1 + i) ^{t}**

Annuity-Immediate Accumulated Value

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

repaying a loan with payments at regular intervals

Loan Amortization

Periodic interest payments which form an annuity

Coupons

**1.) Interest Due**

**2.) Principal Repaid**

Loan consist of what two components

Absolute value of write-down

P_{t }= **|** (Fr - C*(i)) * (v^{n-t+1}) **|**

**Discount Bonds**

Write-Up for bond

**P(i _{n}) **

**=**

**P(i _{o})*[1 - (i_{n} - i_{o})(ModD)]**

First Order Modified Approx.

Annuity-Due Accumulated Value

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

Amount Function Simple Interest

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

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

Accumulation Function For Variable Force of Interest

PV(Variable) = PV(Fixed)

To find R (*swap rate*)

Accumulation Function For Variable Force of Interest

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

(S angle n) * ( v** ^{n }**)

Annuity-Immediate PV to AV relation

(A angle n) =

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

Annuity-Due AV to PV relation

(S double-dot angle n) =

ln (1 + i)

delta = force of interest constant

(Redemption - Price)

or

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

Discount Bond Formula

**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 *

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

Effective Interest Rate

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

**(Da) angle n **

**= **

**[ n - (a angle n) ] **

**/ **

**i**

unit decreasing annuity

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

Annuity-Immediate Present Value

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

( 1 - d )

Discounting Factor with (d)

Discounting Factor with (d)

( 1 - d )

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

Yield to Maturity

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

Amount Function Nominal Discount Rate

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

Annuity-Due Present Value

**also called yield rate**

**it's the rate that **

**produces a NPV of 0**

**also known as breakeven **

Internal Rate of Return (IRR)

also known as the variable rate

floating rate

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

**(1 + s _{n})^{n} **

**= **

**(1 + s _{n-1})^{n-1} * (1 + f_{[n-1, n]})**

Implied Forward Rate

Effective Discount Rate

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

**Loan Amortization: **Outstanding balance

**Terminology Bonds Amortization: **Book value

(1 - d)

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

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

Effective Discount Rate defined as

**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?

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

Given a price

known as the fixed rate

swap rate

**I _{t} (Interest Due)**

**= **

**(i) * B _{t-1} (outstanding balance of previous period)**

Calculating Interest Due

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

Effective Interest Rate defined as

(A angle n) * ( ( 1 + i )^{n })

Annuity-Immediate AV to PV relation

(S angle n)=

**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 **

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

Interest Rate Swap

if notional amount increases over time

Accreting Swap

Step 1: Identify Cash Flows

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

Solving for bonds

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

**[ 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

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

Given a yield rate

**Loan Amortization: **Loan Payment

**Terminology Bond Amortization: **Coupon payment

**B _{t}**

** =**

**L(1+i) ^{t} - R(s angle t)**

Retrospective Formula

(Coupon Payment - Interest Earned)

**P _{t} = | (Fr - C*(i)) * (v^{n-t+1}) |**

Premium bonds

Write-Down for bond

same as principal repaid for loans

**B _{t} (Outstanding loan balance time t) **

**=**

** B _{t-1} (Outstanding loan balance time t-1) - P_{t} (Principal repaid time t)**

Calculating Outstanding Balance end of period

One large payment at the end of the bond term

Redemption Value

**P( Price of bond) **

**=**

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

*+*

**C(v ^{n}) [Pv at 0 of the redemption value ]**

**Bonds: **Basic Formula

Can be terminated before the redemption dates labeled as call dates

Callable Bonds

is the time between settlement dates

Settlement period

Method 1: Start with payments furthest from the comparison date

2: Make adjustments when moving towards the comparison date

Block Payment Annuity

specified period of the swap

Swap Term / Swap Tenor

Rate of Return from investor's point of view

constant rate discounting future cash flow

Bond's Yield Rate

**PV = P*(a angle n)**

**+**

**[ (Q)*((a angle n) - n( v ^{n }) ] / 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

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

Interest earned on a Bond calculation

Accumulation Function Simple Interest

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

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

+

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

=

0

Finding additional payment

(Annuity Immediate AV)

not a cash flow

used to determine coupon amount

Face amount / par value

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

Amount Function Effective Discount Rate

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

Bank Reserve Requirement

Relationship between Constant Force Of Interest and Constant effective interest rate

**e ^{(}^{delta) * (t)}**

** =**

**( 1 + i ) ^{t}**

(1 + i)

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

**interest rate used = **

**cost of capital **

**or **

**opportunity cost of capital**

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

(Price - Redemption)

or

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

Premium Bond Formula

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

+

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

=

0

Finding additional payment

(Annuity Immediate PV )

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

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

Amount Function Simple Interest

Amount Function Nominal Interest Rate

**A(t) **

**=**

**A(0) * ( 1 + (i/m) ) ^{(mt)}**

Interest Due when R is Level

Loan Amortization

I_{t} = R * (1−v^{n−t+1})

P(*Price of bond*)

=

C(*Redemption value) *

+

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

**Bonds: **Premium/Discount Formula

**B _{t} **

**=**

**R(a angle (n-t))**

Prospective Method Formula

The actual amount of deposit with the central bank

Bank's Reserve

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

Accumulation Function Simple Interest

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

Annuity-Immediate Accumulated Value

a'(t) / a(t)

Force Of Interest

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

Amount Function Effective Interest Rate

counter party who receives the fix rate

and

pays the variable rate

Receiver

if the notional amount decreases over time

Amortizing Swap

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

Amount Function Nominal Interest Rate

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

**Final Amortization Value for Bond**

**P _{t} (Principal Repaid)**

**=**

**R _{t} (loan repayment) - I_{t }(interest due)**

Calculating Principal Repaid

**PV = [ integration from 0 to n] **

**of**

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

Continous Varying Annuities PV

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

Annuity-Due Accumulated Value

Principal Repaid when R is Level

Loan Amoritization:

P_{t+k}

/

(1 + i)^{k}

**NPV = Σ CF _{t} * (v^{t}) **

**summation from (t = 0) to n **

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

Net Present Value (NPV)

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

**backward-looking; based on previous cash flows; **

**time t outstanding loan balance **

**=**

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

Retrospective Method

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

Discounting Factor with (i)

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

Call Price

Discounting Factor with (i)

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

**(Ia) angle infinity **

**=**

** ( ( P / i ) + ( Q / ( i ^{2 }) )**

Perpetuity-Immediate increasing

Amount Function Effective Discount Rate

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

e^(delta) = ( 1 + i )

Relationship between Constant Force Of Interest and Constant effective interest rate

P = Q

[ (Ia) angle n ]

=

**(P) * [ ( a double dot angle n ) - n( v ^{n }) ]**

**/**

**i**

Unit Increasing Annuity

[ (Ia) angle n]

(S double-dot angle n) * ( v** ^{n }**)

Annuity-Due PV to AV relation

(A double-dot angle n) =

Modified Duration Formula

**Σ t * v ^{t+1} * CF_{t} **

**/**

**Σ v ^{t} * CF_{t}**

Force Of Interest

a'(t) / a(t)

P_{t} ( *principal repaid for bonds at time t )*

*=*

**| **( Fr - C(i) )*( v^{n-t+1} ) **|**

P_{t} ( *principal repaid for bonds at time t )*

Annuity-Due Present Value

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

compensates investors for the possibility that a borrower will default

Default Risk Premium