ST226 Ultimate Exam Formula Sheet - All Topics

ST226 EXAM FORMULA SHEET
Complete Reference - All High-Yield Topics
Ch 1: Time Value of Money Perpetuities Ch 3: Loan Schedules

Interest Rate Relations Outstanding Balance
1 1 1
a∞ = i
, ä∞ = d
, ā∞ = δ
1 ˆ Retrospective: L(t) = L0 (1 + i)t − R · st
ˆ v= (discount factor)
1+i ˆ Prospective: L(t) = R · an−t

i
ˆ d= = 1 − v (discount rate)
1+i Equal Payments
ˆ δ = ln(1 + i) (force of interest) ˆ Level payment: R = L0
an
ˆ 1+i= eδ , v= e−δ ˆ Interest in period k: Ik = L(k − 1) · i
Relationships ˆ Principal in period k: Pk = R − Ik
Single Payment
ˆ PV: C · v t = C
än = (1 + i)an , ān = i
a
Ch 5: Bonds (10-15%)
(1+i)t δ n
ˆ AV: C · (1 + i)t
− 0t δ(u)du
R Price Formulas
ˆ Variable: C · e
ˆ No tax: P = Cg · an + C · v n
Continuous Payment Stream ˆ Income tax: P = Cg(1 − t1 )an + Cv n
ˆ With CGT:
Z T Rs P = Cg(1 − t1 )an + Cv n − t2 (C − P )v n
PV = ρ(s) · e− 0 δ(u)du
ds
0 NO CGT if: (1 − t1 )g = i
Increasing (payments 1,2,3,...,n)
Constant ρ and δ:
T Premium/Discount
Z
PV = ρ e−δs ds = ρ · āT än −nv n
0 ˆ (Ia)n = i ˆ g > i: Premium (P > C)
ˆ g < i: Discount (P < C)
Ch 2: Annuities Certain (15-20%) ˆ (Iä)n = (1 + i)(Ia)n ˆ g = i: Par (P = C)

Three Main Types ān −nv n
ˆ (Iā)n = δ
Yields
1 − vn Cg
ˆ Imm: an = (end of period) Coupon yield = P
i
1 − vn
ˆ Due: än = (start of period) Ch 7: Life Contingencies (25-30%)
d
1 − vn
ˆ Cont: ān = (continuous) Core Notation
δ
lx+t
Deferred (starts after m years) ˆ t px = survive t years: lx
Accumulated Values ˆ t qx = die within t years: 1 − t px
n n
ˆ u| qx = survive u, die in year u + 1:
(1+i) −1 (1+i) −1
sn = i
, s̈n = d m| an = v m an = am+n − am u px · qx+u (when n = 1 implied)