rate of return, r
- r = |risk free
+ liquidity +. . .
{z rate} + |inflation + default
{z
}
time delay
risk factors
- Gross Return, R ; Net Return, r and % Return.
Ex: 2.05 percent→ r = 0.0205, R = (1 + r ) = 1.0205 and %
return =r ∗ 100 = 2.05%:
- Nominal & Real rates: Rnominal = Rreal ∗ Rinflation
Ex: Deposit which pays a 2% interest, and inflation is 2%.
Rreal = 1.02
1.02 = 1. There is no compensation for delaying
consumption as r = R − 1 = 0%.
Capitalization and Discounting
Capitalization
FV = PV (1 + tr ) simple
FV = PV (1 + r )t compound
PV =
FV
(1+r )t
PV =
compound
FV
(1+tr )
simple
Discounting
simple and compound interest
Simple Interest → FV = PV (1 + r (n))
Compound Interest → FV = PV (1 + r )n or FV = PV (R)n
FV= Future Value, PV= Present Value, r=interest Rate, n=years
Example: PV=1000, r=8%, n=4, FV?
a) Simple interest: interest on only the original amount of principal
FV = 1000(1 + (0.08)(4)) = 1320.00
b) Compound interest: interest on the initial principal as well as
upon the interest that has already accrued (reinvesting interest
earned):
FV = 1000(1 + 0.08)4 = 1360.48
Within Period Compounding or reinvesting
earned interest
a) Annual compounding (standard as rates are expressed always in
annual basis):
n
X
CFt
PV =
(1 + r )t
t=0
b) m compounding, m=1 (annual), m=2 (semiannual), m=4
(quarterly), m=12 (monthly):
PV =
n
X
i=1
CFi
(1 + mr )mti
c) Continuous compounding:
PV =
n
X
i=1
CFi e −rti
Continuous Compounding
1 n
= 2.7182 . . .
e == lim 1 +
n→∞
n
FV
(1 + mr )mt
1
PV = FV
r
r
(1 + mr )mt r
PV =
r
PV = FV
1
(1 +
1
e tr
PV = FV e −tr
PV = FV
1 tr mr
m
r
)
With Intermediate Cash Flows
cT
c2
c1
c0
cT −1
Capitalization
PV =
Pn
ct
t=1 (1+r )t
Discounting
FV =
Pn
1 ct (1
+ r )t
Annuities & Perpetuities
Annuities and Perpetuities are geometric sequences. The initial
1
CF
and ratio 1+r
. Perpetuities are annuities where
term is 1+r
n → ∞.
PV =
CF
CF
CF
CF
+
+
+ ··· +
1
2
3
(1 + r )
(1 + r )
(1 + r )
(1 + r )n
If CF has a constant growth g every period; the initial term is
CF0 (1+g )
and the ratio becomes 1+g
1+r
1+r :
PV =
CF0 (1 + g ) CF0 (1 + g )2 CF0 (1 + g )3
CF0 (1 + g )n
+
+
+
·
·
·
+
(1 + r )1
(1 + r )2
(1 + r )3
(1 + r )n
Sum of terms of a Geometric Sequence
A geometric sequence such as {1, 21 , 41 , 18 } can be expressed as
gn = a, ak, ak 2 , ak 3 , . . . , ak m
and the sum of the terms can be calculated as follows:
n
X
gn = a + ak + ak 2 + ak 3 + · · · + ak m
1
k
n
X
1
gn − k
n
X
1
n
X
1
n
X
n=1
1
gn = ak + ak 2 + ak 3 + · · · + ak m+1
gn = a − ak m+1
∞
gn = a
X
1
1 − kn
; when n → ∞
gn = a
1−k
1−k
n=3, a=1, k=1/2 and m=n-1
n=1
1
Geometric Series Convergency Solutions
annuity
without growth
PV =
CF
r
1−
perpetuity
1
(1+r )n
PV =
CF
r
n
with growth
PV =
CF −CF ( 1+g
1+r )
r −g
PV =
CF (1+g )
r −g
Time Value of Money Example
FV =1000 e , r=10% per year-nominal rate
Simple interest (not reinvesting interests) PV =
t = 1 year → PV =
t = 2 years → PV =
1000
(1+(1∗0.1))
1000
(1+(2∗0.1))
= 909.09
= 833.33
Compounded (reinvesting interests) PV =
t=1 year, m=2 → PV =
FV
(1+tr )
1000
)2∗1
(1+ 0.1
2
FV
(1+ mr )mt
= 907.02
t=1 year, continously (m → ∞) → PV =
t=2 years, quarterly (m=4)→ PV =
1000
e 0.1∗1
1000
(1+ 0.1
)4∗2
4
= 904.83
= 820.75
Effective Annual Rate, EAR
if m is the number of compounding periods in a year:
r m
= (1 + EAR)
1+
m
Effective Annual Rate, EAR = (1 +
r m
m)
− 1.
Is the rate expressed in an annual basis and taking into account
compounding interest. Are useful to compare rates.
Example r=10% nominal rate compounded:
annually results in a EAR of 10%
semiannually results in a EAR of (1 +
quarterly results in a EAR of (1 +
0.1 2
2 )
0.1 4
4 )
− 1 = 10.25%
− 1 = 10.38%
continuously results in a EAR of e 0.10 − 1 = 10.52%
SPANISH TIN and TAE
In Spain the EAR is known as TAE or tasa anual equivalente, and
is a common way to express interest rates for bank consumer loans
and includes bank commissions if any. The nominal rate (known as
TIN or tasa de interés nominal) does not include commissions.
Ex.1: TIN 10% no commisions and quarterly compounding: TAE
4
= (1 + 0.1
4 ) − 1 = 10.38%.
Ex.2: An Spanish retail bank offers a loan to buy a car which costs
(PVP) 10280 e paying 84 monthly payments of 145 e and a
commission of study of 257 e . Which is the TIN and TAE?
10280 =
84
X
t=1
145
(1 +
10280 = 257 +
t
TIN 12 12
12 )
→ TIN = 4.93%
84
X
145
t=1
(1 + TAE ) 12
t
→ TAE = 5.85%