text: # Theory of Interest - Formula Sheet III 1. Bonds Notation: P
  
  ```Theory of Interest - Formula Sheet III
  1. Bonds
  Notation:
  P - price.
  F - par value or face amount.
  C - redemption value, that is the amount paid at a redemption date.
  r - coupon rate (per coupon payment).
  F r - amount of the coupon.
  g - modified coupon rate, defined by F r = Cg.
  i - yield rate (to maturity) of a bond.
  n - number of coupon payments.
  K - present value (computed at the yield rate) of the redemption value.
  G - base amount of a bond, defined by Gi = F r.
  The formulas:
  • basic formula: P = F ran| + Cv n .
  • premium/discount formula: P = C + (F r − Ci)an| .
  • base formula: P = G + (C − G)v n .
  • Makeham formula: P = K + gi (C − K).
  • An approximate yield formula:
  i∼
  g − nk
  ,
  1 + n+1
  2n k
  where k = P −C
  C . A further approximation called the bond salesman’s method is obtained
  by replacing n+1
  2n with 1/2 in the above formula.
  2. Approximate Methods
  Notation:
  L - amount borrowed.
  K - finance charge.
  R - periodic payment =
  L+K
  n .
  n - total number of payments.
  m - annual payment frequency.
  i(m) - APR implied by the finance charge.
  Approximations to i(m) :
  mK
  i(m) ∼ Pn−1
  ,
  t=0 Bt/m
  where Bt/m is the unpaid balance at time t/m.
  The following four methods use different approximations to the denominator in the above
  formula.
  • Maximum Yield Method:
  • Minimum Yield Method:
  • Constant Ratio Method:
  Pn−1
  n+1
  n−1
  t=0 Bt/m = 2 L − 2 K.
  Pn−1
  n+1
  n−1
  t=0 Bt/m = 2 L + 2 K.
  Pn−1
  n+1
  t=0 Bt/m = 2 L.
  • Direct Ratio Method (Rule of 78):
  Pn−1
  t=0
  Bt/m =
  n+1
  2 L
  +
  n−1
  6 K.
  3. Depreciation methods
  Notation:
  A - asset value at the beginning of n periods.
  S - salvage value at the end of n periods.
  Bt - book value at the end of the tth period (B0 = A, Bn = S).
  Dt - depreciation charge during tth period: Dt = Bt−1 − Bt .
  Four common depreciation methods:
  • Sinking Fund Method: Bt = A −
  • Straight Line Method: Bt = A −
  A−S
  sn|j st|j .
  A−S
  n
  t.
  • Constant Percentage Method (AKA Compound Discount, Declining Balance):
  Bt = (1 − d)t A, where d = 1 − (S/A)1/n .
  • Sum of the Digits Method: Bt = A −
  k(k + 1)/2.
  A−S
  Sn
  (Sn − Sn−t ), where Sk = 1 + &middot; &middot; &middot; + k =
  ```
explanation: The document is a formula sheet for the Theory of Interest, covering topics such as bonds, approximate methods for calculating yield, and depreciation methods. It provides various notations and formulas for each topic but does not contain any specific questions or answers to questions.
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distilabel_metadata.content: # Theory of Interest - Formula Sheet III 1. Bonds Notation: P
  
  ```Theory of Interest - Formula Sheet III
  1. Bonds
  Notation:
  P - price.
  F - par value or face amount.
  C - redemption value, that is the amount paid at a redemption date.
  r - coupon rate (per coupon payment).
  F r - amount of the coupon.
  g - modified coupon rate, defined by F r = Cg.
  i - yield rate (to maturity) of a bond.
  n - number of coupon payments.
  K - present value (computed at the yield rate) of the redemption value.
  G - base amount of a bond, defined by Gi = F r.
  The formulas:
  • basic formula: P = F ran| + Cv n .
  • premium/discount formula: P = C + (F r − Ci)an| .
  • base formula: P = G + (C − G)v n .
  • Makeham formula: P = K + gi (C − K).
  • An approximate yield formula:
  i∼
  g − nk
  ,
  1 + n+1
  2n k
  where k = P −C
  C . A further approximation called the bond salesman’s method is obtained
  by replacing n+1
  2n with 1/2 in the above formula.
  2. Approximate Methods
  Notation:
  L - amount borrowed.
  K - finance charge.
  R - periodic payment =
  L+K
  n .
  n - total number of payments.
  m - annual payment frequency.
  i(m) - APR implied by the finance charge.
  Approximations to i(m) :
  mK
  i(m) ∼ Pn−1
  ,
  t=0 Bt/m
  where Bt/m is the unpaid balance at time t/m.
  The following four methods use different approximations to the denominator in the above
  formula.
  • Maximum Yield Method:
  • Minimum Yield Method:
  • Constant Ratio Method:
  Pn−1
  n+1
  n−1
  t=0 Bt/m = 2 L − 2 K.
  Pn−1
  n+1
  n−1
  t=0 Bt/m = 2 L + 2 K.
  Pn−1
  n+1
  t=0 Bt/m = 2 L.
  • Direct Ratio Method (Rule of 78):
  Pn−1
  t=0
  Bt/m =
  n+1
  2 L
  +
  n−1
  6 K.
  3. Depreciation methods
  Notation:
  A - asset value at the beginning of n periods.
  S - salvage value at the end of n periods.
  Bt - book value at the end of the tth period (B0 = A, Bn = S).
  Dt - depreciation charge during tth period: Dt = Bt−1 − Bt .
  Four common depreciation methods:
  • Sinking Fund Method: Bt = A −
  • Straight Line Method: Bt = A −
  A−S
  sn|j st|j .
  A−S
  n
  t.
  • Constant Percentage Method (AKA Compound Discount, Declining Balance):
  Bt = (1 − d)t A, where d = 1 − (S/A)1/n .
  • Sum of the Digits Method: Bt = A −
  k(k + 1)/2.
  A−S
  Sn
  (Sn − Sn−t ), where Sk = 1 + &middot; &middot; &middot; + k =
  ```
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a formula sheet for the Theory of Interest, covering topics such as bonds, approximate methods for calculating yield, and depreciation methods. It provides various notations and formulas for each topic but does not contain any specific questions or answers to questions.
  
  Label: no-question