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 + · · · + k =