Exam FM Summary Sheet

I.

Interest Rate Measurement: 

Accumulation Function, a(t): Accumulated value @ time (t) of an original investment of $1



Amount Function, A(t): Accumulated value @ time (t) of an original investment of $K







 () =  ∙ ()

 =       Effective Rate of Discount,  =    1 −  =  =   1 +  =  For simple interest,  ( ) = 1 +  ∙  For compound interest,  ( ) = (1 +  ) Effective Rate of Interest,









(

( ()

)

)

(

)

()

(



()

)

( )

Nominal Rates: o

  =  ()  =       =  ()  =     ( )

(

o o

)

( )

( )

o

o

To convert between nominal rates, use the equivalence of o f rates: 







(1 +

() ) = ( 1 + () )  

( ) ( )    (1 − ) = (1 − )   ( )   1 +   = () 

Force of Interest: o

Measure of interest @ individual moments of time

o

Interest payable continuously

o

 ( )   ( )   = () = () 

For Simple Interest:



For Compound interest:

 = 1 + ∙ 

 = ln(1+ )



Real rate of interest: o

o

II.

The real rate of interest takes into account the rate of inflation = r

  = 

Valuation of Annuities: 

Annuity Immediate:



Annuity Due:



Relationship between immediate & due:

  o ,  =   o   ,  =  (1 +  )

  o , ̈ =  ()      ̈ , = (1 + ) = o   ̈     o

Note that an annuity due is equivalent to an annuity immediate evaluated one period later 









Deferred Annuities: (n payments deferred for m periods) o



(1 + ) 

  =  − 

Perpetuity Immediate: o



| =   =  − 

Accumulated Value of (n) payments (m) periods after last payment: o



 (1 + ) = ̈  (1 + ) =  ̈ ̈ =  + 1  =  ̈ + 1

 = 

Perpetuity Due: o

̈ = 





Annuities Payable m

 each 

Each period has (m) payments of 

o



o

This can be also attempted as an ordinary annuity with (

( )

  = ( ) 

Arithmetically Increasing Annuities:

 

o o o

    = ̈ 

Using BAII Plus: 

Set mode to BGN



N=n



PMT=1



FV=-n



CPT, PV



Divide by ( i )

 =  ̈      ̈  ̈ =   ̈ =  ̈

Increasing Perpetuity: o o



:

o

o



thly

 =  ̈ = ()

Decreasing Annuities: o o o o

 =   ̈ = (1 + ) =       =   ̈ = ( 1 + ) (



)

Using BAII Plus: 

Set mode to END



N=n



PMT=-1



PV=-n



CPT, FV



Divide by ( i )

 × ) payments



Geometrically Varying Annuities: o o

o

o



Geometric Perpetuity Immediate: o



Present value is only defined for

 < ⇒ 

Continuous Annuities: o o



( )    =  =  ∙     If  = ⇒  =   If  = ⇒ =  (1 +  )  

     = ∫   =  =    ̅  = () =  

Continuous Varying Annuities: o o o

o

  =      = ̅ 

  

If the payment @ time (t) = ( ) and the force of interest is

 = ∫ () ∫ 

:

III.

Project Appraisals & Loans: 

Net Present Value: o



o

o

IRR is the interest rate at which NPV = 0

Measures the success of the investor Internal balances do not matter

Measures the success if the broker Lengths of internal intervals do not matter

  = 

Amortization: (Level payment s  ) o o o o

IV.

)

Portfolio Rate of Return: o



(

Time-weighted rate of return: o



)

Dollar-weighted rate of return: o



(

Internal Rate of Return: o



  = ∑   

 = (1 −  )  =   =  (1 + )  = 

Bond Valuation: 

Notation: o o o o o

 ≡   ≡ ,   ≡   ≡   ≡   =   ≡ ,  − −  ≡ , ℎ ,ℎ  =   

o o





Yields: o

Nominal Yield: Annualized Coupon Rate 

o

Current Yield: Ratio of annualized coupon to price 

o



 =  

Yield-to-Maturity: Actual annualized yield rate.

Price of a bond: o



 =  

 =  + 

(Frank)

Amortizing a Bond: o

 > ( > ) → ℎ,−ℎℎ .   =  = − = ( − )  =  −   < ( < ) → ℎ,−ℎℎ.   =  =  −  =  +  



o

 

o

IM PORTANT NOTE ABOUT AM ORTIZING A BOND: The book value at any poi nt in t im e is equal  to a hypot hetical redemption value had the bond been redeemed at t hat ti me. It also equals a  hypothetical price at whi ch the bond is bought at that point i n tim e.



Callable Bonds: o

For a fixed redemption value:

 → ℎℎ .  → ℎ . For different calling prices: ℎ .  

o



V.

Cashflow Duration: 

Price as a function of yield: o



Modified Duration: o



 ( )   = − () = (∑ ∑)(  )

Macaulay Duration: o o o



 = ∑  (1 +  )

 () ∑    = − () = 

ℎ,  = ̈   = 

Relationship between Mod D & Mac D: o o

 =lim →   =   (



Redington Immunization: o o o



 =   =   =  =    >  > 

Full Immunization: o o o



)

 =   =   ℎ &ℎℎ

Stock Valuation: o o

 =  For geometrically increasing dividends: 

  =  