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:
Pt = R * ( vn-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
Pt = | (Fr - C*(i)) * (vn-t+1) |
Discount Bonds
First Order Modified Approx.
P(in)
=
P(io)*[1 - (in - io)(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) * ( vn )
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 + sn)n
=
(1 + sn-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
It (Interest Due)
=
(i) * Bt-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
Bt
=
L(1+i)t - R(s angle t)
Write-Down for bond
same as principal repaid for loans
(Coupon Payment - Interest Earned)
Pt = | (Fr - C*(i)) * (vn-t+1) |
Premium bonds
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)
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(vn) [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( vn ) ] / 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
It = R * (1−vn−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
Bt
=
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
Pt (Principal Repaid)
=
Rt (loan repayment) - It (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:
Pt+k
/
(1 + i)k
Principal Repaid when R is Level
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
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 / ( i2 ) )
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( vn ) ]
/
i
Annuity-Due PV to AV relation
(A double-dot angle n) =
(S double-dot angle n) * ( vn )
Σ t * vt+1 * CFt
/
Σ vt * CFt
Modified Duration Formula
a'(t) / a(t)
Force Of Interest
Pt ( principal repaid for bonds at time t )
Pt ( principal repaid for bonds at time t )
=
| ( Fr - C(i) )*( vn-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:
Pt = R * ( vn-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
Pt = | (Fr - C*(i)) * (vn-t+1) |
Discount Bonds
Write-Up for bond
P(in)
=
P(io)*[1 - (in - io)(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) * ( vn )
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 + sn)n
=
(1 + sn-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
It (Interest Due)
=
(i) * Bt-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
Bt
=
L(1+i)t - R(s angle t)
Retrospective Formula
(Coupon Payment - Interest Earned)
Pt = | (Fr - C*(i)) * (vn-t+1) |
Premium bonds
Write-Down for bond
same as principal repaid for loans
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
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(vn) [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( vn ) ] / 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
It = R * (1−vn−t+1)
P(Price of bond)
=
C(Redemption value)
+
(Fr - C * i)(a angle n)
Bonds: Premium/Discount Formula
Bt
=
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
Pt (Principal Repaid)
=
Rt (loan repayment) - It (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:
Pt+k
/
(1 + i)k
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)
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 / ( i2 ) )
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( vn ) ]
/
i
Unit Increasing Annuity
[ (Ia) angle n]
(S double-dot angle n) * ( vn )
Annuity-Due PV to AV relation
(A double-dot angle n) =
Modified Duration Formula
Σ t * vt+1 * CFt
/
Σ vt * CFt
Force Of Interest
a'(t) / a(t)
Pt ( principal repaid for bonds at time t )
=
| ( Fr - C(i) )*( vn-t+1 ) |
Pt ( 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