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 =
  ```