FNAN 303 Formulas and Notes (p. 1 of 6)
Value in t periods with simple interest:
C0 × [1 + (simple interest rate per period × t)] = C0 + (C0 × simple interest rate per period × t)
FVt = C0 × (1+r)t
Financial calculator: In either BEGIN or END mode, FV is the future value in N periods from the reference point
(time 0) of a cash flow equal to -PV at the reference point with an interest rate, return, etc. of I% per period
FVt = Ck × (1+r)(t-k)
PV0 = PV = Ct / (1 + r)t
Financial calculator: In either BEGIN or END mode, PV is the opposite of the present value as of the reference
point (time 0) of a cash flow equal to FV that takes place in N periods from the reference point, with a discount rate
of I% per period
PV0 = PV = C0 + [C1/(1+r)1] + [C2/(1+r)2] + … + [Ct-1/(1+r)t-1] + [Ct/(1+r)t]
PV for a fixed perpetuity = [C/(1+r)] + [C/(1+r)2] + [C/(1+r)3] + … = C / r
Rate of return for a fixed perpetuity = r = C /PV
Cash flow for a fixed perpetuity = C = PV × r
PV for a growing perpetuity = C1/(1+r) + [C1(1+g)]/(1+r)2 + [C1(1+g)2]/(1+r)3 + … = C1 / (r – g)
Rate of return for a growing perpetuity = r = [C1 / PV] + g
First cash flow for a growing perpetuity = C1 = PV × (r – g)
Growth rate for a growing perpetuity = g = r – [C1 / PV]
Ck = C1 × (1 + g)k – 1 which is the same as Ct = C1 × (1 + g)(t – 1)
Also, Cb = Ca × (1 + g)(b-a) so g = [(Cb / Ca)[1/(b-a)]] – 1
PV for an annuity = [C/(1+r)] + [C/(1+r)2] + … + [C/(1+r)t]
= C × [{1 – 1/(1+r)t } / r ] = C × [(1/r) – 1/{r(1+r)t}] = (C/r) × [1 – (1/{(1+r)t})]
Financial calculator: In END mode, PV is the opposite of the present value as of the reference point (time 0) of a
series of N regular cash flows equal to PMT per period where the first regular cash flow takes place 1 period from
the reference point, the last cash flow takes place N periods from the reference point, and the discount rate is I% per
period
PV for an annuity due = (1+r) × PV for an annuity = C + [C/(1+r)] + [C/(1+r)2] + [C/(1+r)3] + … + [C/(1+r)t-1]
= (1+r) × C × [{1 – 1/(1+r)t } / r ] = (1+r) × C × [(1/r) – 1/{r(1+r)t}] = C + (C/r) × [1 – (1/{(1+r)t-1})])
Financial calculator: In BEGIN mode, PV is the opposite of the present value as of the reference point (time 0) of a
series of N regular cash flows equal to PMT per period where the first regular cash flow takes place at the reference
point, the last cash flow takes place N-1 periods from the reference point, and the discount rate is I% per period
PV0 = PV = PVk / (1+r)k
FVt = [C0 × (1+r)t] + [C1 × (1+r)t-1] + [C2 × (1+r)t-2] + … + [Ck × (1+r)t-k] + … + [Ct-1 × (1 + r)1] + [Ct]
FV for an annuity = [C1 × (1+r)t-1] + [C2 × (1+r)t-2] + … + Ct
= (1+r)t × C × [{1 – 1/(1+r)t} / r] = C × [{(1+ r)t – 1} / r] = (1+r)t × C × [(1/r) – 1/{r(1+r)t}]
Financial calculator: In END mode, FV is the future value in N periods from the reference point (time 0) of a series
of N regular cash flows equal to –PMT per period where the first regular cash flow takes place 1 period from the
reference point, the last cash flow takes place N periods from the reference point, and the interest rate, return, etc. is
I% per period
FV for an annuity due = (1+r) × FV for an annuity = [C0 × (1+r)t] + [C1 × (1+r)t-1] + … + [Ct-1 × (1+r)1]
= (1+r)t+1 × C × [{1 – 1/(1+r)t} / r] = (1+r) × C × [{(1+r)t – 1} / r] = (1+r)t+1 × C × [(1/r) – 1/{r(1+r)t}]
Financial calculator: In BEGIN mode, FV is the future value in N periods from the reference point (time 0) of a
series of N regular cash flows equal to -PMT per period where the first regular cash flow takes place at the
reference point, the last cash flow takes place N-1 periods from the reference point, and the interest rate, return, etc.
is I% per period
FNAN 303 Formulas and Notes (p. 2 of 6)
APR = annual percentage rate = # periods in a year × periodic rate = # periods in a year × [(1 + EAR)1/# periods in a year – 1]
EAR = Effective annual rate = [(1 + periodic rate)# of periods in a year] – 1 = [1 + (APR/# periods per year)]# periods per year – 1
Periodic rate = [APR / # periods per year] = [(1 + EAR)(1 / # of periods in a year) – 1]
EAR with continuous compounding = (eAPR) – 1
Bond value = [cpn/(1+r)1] + [cpn/(1+r)2] +…+ [cpn/(1+r)t] + [face/(1+r)t]
= {cpn × [{1 – 1/(1+r)t} / r]} + {face/(1+r)t} = {cpn × [(1/r) – 1/{r(1+r)t}]} + {face/(1+r)t}
Financial calculator: Bond value equals –PV, where PV is the opposite of the present value as of the reference point
(time 0) of N coupon payments equal to PMT per period where each coupon equals the coupon rate multiplied by the
face value divided by the number of coupons per year, the first coupon is paid 1 period from the reference point (END
mode), N is the number of coupons paid before maturity and equals number of coupons per year multiplied by the
number of years to maturity, the discount rate is I% per period, where I% equals the bond’s yield–to-maturity divided
by the number of coupons per year, and FV is the face (or par) value of the bond
r for a bond = discount rate per period, where a period equals 1 year divided by the number of coupons per year
Coupon payment = (coupon rate × face value) / number of coupons per year
= total aggregate dollar amount of coupons per year / number of coupons per year
Total aggregate dollar amount of coupons per year = coupon rate × face value = coupon rate × par value
= coupon payment × number of coupons per year
Current yield = total aggregate dollar amount of coupons per year / bond value
= (coupon rate × face value) / bond value
Capital gain = ending value – initial value
