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Prospective Method

Liquidity Preferance Theory / Opportunity Cost Theory 

weighted average of individual asset's duration 

Annuity-Immediate Present Value

Accumulation function for Constant Force of Interest

Duration of Portfolio

Payer

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

Terminology Bond Amortization: 

Write-down (Premium Bonds)

Write-Up (Discount Bonds) 

Expectation theory 

Settlement dates 

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

Loan Amoritization:

P_t = R * ( v^n-t+1 )

A(t)

=

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

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

Loan Amortization 

Coupons

Loan consist of what two components

Write-Up for bond

First Order Modified Approx. 

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

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

Accumulation Function For Variable Force of Interest

To find R (swap rate) 

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

Annuity-Immediate PV to AV relation

(A angle n) =

Annuity-Due AV to PV relation

(S double-dot angle n) =

delta = force of interest constant

Discount Bond Formula

Loan: Outstanding Balance at time t

is equal to 

Notional amount 

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

unit decreasing annuity 

Market Segmentation Theory 

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

Discounting Factor with (d)

( 1 - d )

Yield to Maturity

Amount Function Nominal Discount Rate

Annuity-Due Present Value

Internal Rate of Return (IRR)

floating rate

[ 1 - ( (1 + k) / ( 1 +i ) )ⁿ ]

/

( i - k)

i = interest

k = number of payments in progression/percentage

n = number of total payments

Implied Forward Rate 

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

Terminology Bonds Amortization: Book value

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

Effective Discount Rate defined as

What is opportunity cost of capital?

Given a price 

swap rate 

Calculating Interest Due 

Effective Interest Rate defined as

Annuity-Immediate AV to PV relation

(S angle n)=

A bond's BOOK VALUE  at time t 

is equal to 

Interest Rate Swap

Accreting Swap 

Solving for bonds

Spot Rates

Annuity-Immediate geometric progression

Given a yield rate 

Terminology Bond Amortization: Coupon payment

Retrospective Formula 

Write-Down for bond

same as principal repaid for loans

Calculating Outstanding Balance end of period

Redemption Value

Bonds: Basic Formula 

Callable Bonds

Settlement period 

Block Payment Annuity

Swap Term / Swap Tenor 

Bond's Yield Rate

P and Q formula for annuity increasing Arithmetic Progression

Interest earned on a Bond calculation

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

Finding additional payment

(Annuity Immediate AV)

Face amount / par value

Amount Function Effective Discount Rate

Bank Reserve Requirement

e^{(delta) * (t)}

 =

( 1 + i )^t

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

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

Premium Bond Formula 

Finding additional payment

(Annuity Immediate PV )

Default Risk 

Amount Function Simple Interest

A(t)

=

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

Loan Amortization

I_t = R * (1−v^n−t+1)

Bonds: Premium/Discount Formula

Prospective Method Formula

Bank's Reserve

Accumulation Function Simple Interest

Annuity-Immediate Accumulated Value

Force Of Interest

Amount Function Effective Interest Rate

Receiver 

Amortizing Swap 

Amount Function Nominal Interest Rate

Final Amortization Value for Bond

Calculating Principal Repaid 

Continous Varying Annuities PV

Annuity-Due Accumulated Value

Loan Amoritization:

P_t+k 

/

(1 + i)^k

Net Present Value (NPV)

Preferred Habit Theory 

Retrospective Method

Discounting Factor with (i)

Call Price

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

Perpetuity-Immediate increasing

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

Relationship between Constant Force Of Interest and Constant effective interest rate

Unit Increasing Annuity
[ (Ia) angle n]

Annuity-Due PV to AV relation

(A double-dot angle n) =

Σ t * v^t+1 * CF_t 

/

Σ v^t * CF_t

a'(t) / a(t)

P_t ( principal repaid for bonds at time t )

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

Default Risk Premium 

forward-looking based on future cash flow

time-t outstanding loan balance

=

PV(remaining loan payments with i)

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

Duration of Portfolio

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

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

weighted average of individual asset's duration 

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

Accumulation function for Constant Force of Interest

Loan Amortization: Principal Repaid

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.

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

Amount Function Nominal Discount Rate

Principal Repaid when R is level 

Amount Function Effective Interest Rate

Annuity-Immediate Accumulated Value

repaying a loan with payments at regular intervals

Periodic interest payments which form an annuity 

1.) Interest Due

2.) Principal Repaid

Absolute value of write-down 

P_t = | (Fr - C*(i)) * (v^n-t+1)  |

Discount Bonds

P(i_n)

=

P(i_o)*[1 - (i_n - i_o)(ModD)]

Annuity-Due Accumulated Value

Amount Function Simple Interest

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

PV(Variable) = PV(Fixed) 

Accumulation Function For Variable Force of Interest

(S angle n) * ( vⁿ )

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

ln (1 + i)

(Redemption - Price)

or

( C*(i) - Fr ) * (a angle n) 

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

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

Effective Interest Rate

(Da) angle n 

= 

[ n - (a angle n) ] 

/ 

i

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

Annuity-Immediate Present Value

( 1 - d )

Discounting Factor with (d)

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

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

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

also called yield rate

it's the rate that 

produces a NPV of 0

also known as breakeven 

also known as the variable rate

Annuity-Immediate geometric progression

(1 + s_n)ⁿ 

= 

(1 + s_n-1)^n-1 * (1 + f_{[n-1, n]})

Effective Discount Rate

Loan Amortization: Outstanding balance

(1 - d)

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

Rate of return

on an equally-risky asset

an investor could have earned

if he or she had NOT invested in the project. 

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

known as the fixed rate 

I_t (Interest Due)

= 

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

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

(A angle n) * ( ( 1 + i )ⁿ )

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

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

if notional amount increases over time

Step 1: Identify Cash Flows

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

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

[ 1 - ( (1 + k) / ( 1 +i ) )ⁿ ]

/

( i - k)

i = interest

k = number of payments in progression/percentage

n = number of total payments

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

Loan Amortization: Loan Payment 

B_t

 =

L(1+i)^t - R(s angle t)

(Coupon Payment - Interest Earned)

P_t = | (Fr - C*(i)) * (v^n-t+1)  |

Premium bonds

B_t (Outstanding loan balance time t) 

=

 B_t-1 (Outstanding loan balance time t-1)  - P_t (Principal repaid time t)

One large payment at the end of the bond term

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 ]

Can be terminated before the redemption dates labeled as call dates

is the time between settlement dates 

Method 1: Start with payments furthest from the comparison date

2: Make adjustments when moving towards the comparison date 

specified period of the swap 

Rate of Return from investor's point of view

constant rate discounting future cash flow 

PV = P*(a angle n)

+

[ (Q)*((a angle n) - n( vⁿ ) ] / i 

P = first term

Q = constant amount of increase for each payment

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

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

Accumulation Function Simple Interest

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

+

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

=

 0

not a cash flow

used to determine coupon amount

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

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

Relationship between Constant Force Of Interest and Constant effective interest rate

(1 + i)

interest rate used  = 

cost of capital

or

opportunity cost of capital

(Price - Redemption)

or

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

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

+

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

=

0

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

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

Amount Function Nominal Interest Rate

Interest Due when R is Level

P(Price of bond) 

= 

C(Redemption value) 

+ 

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

B_t 

=

R(a angle (n-t))

The actual amount of deposit with the central bank

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

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

a'(t) / a(t)

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

counter party who receives the fix rate 

and

pays the variable rate 

if the notional amount decreases over time 

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

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

P_t (Principal Repaid)

=

R_t (loan repayment)  - I_t (interest due)

PV = [ integration from 0 to n]

of

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

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

Principal Repaid when R is Level

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 

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. 

backward-looking; based on previous cash flows; 

time t outstanding loan balance

=

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

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

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

Discounting Factor with (i)

(Ia) angle infinity

=

 ( ( P / i ) + ( Q / ( i² ) )

Amount Function Effective Discount Rate

e^(delta) = ( 1 + i )

P = Q

[ (Ia) angle n ]

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

/

i

(S double-dot angle n) * ( vⁿ )

Modified Duration Formula

Force Of Interest

P_t ( principal repaid for bonds at time t )

=

| ( Fr - C(i) )*( v^n-t+1 ) |

Annuity-Due Present Value

compensates investors for the possibility that a borrower will default