Time Value of Money
(Text reference: Chapter 4)
Topics
Background
One period case - single cash flow
Multi-period case - single cash flow
Multi-period case - compounding periods
Multi-period case - multiple cash flows
Perpetuities
Annuities
Mortgages
Amortization schedules
AFM 271 - Time Value of Money
Slide 1
Background
the economic value of a cash flow depends on when it
occurs (i.e. the timing of the cash flow)
⇒ $1 today > $1 tomorrow > $1 a year from now > . . .
present value calculations allow us to determine the
value today of a stream of cash flows to be rec’d/paid in
the future, by taking into account the time value of
money
this is an important concept which is widely used both
in corporate and in personal financial decision-making
AFM 271 - Time Value of Money
Slide 2
Cont’d
notation:
PV = present value = value today of a stream of
cash flows
FV = future value = value at some future time of a
stream of cash flows
C = cash flow
r = interest rate (a.k.a. discount rate)
T = number of years
m = number of periods in a year
n = total number of periods (n = m × T )
we will often apply subscripts to indicate time, e.g. Ct is
a cash flow occurring at time t
AFM 271 - Time Value of Money
Slide 3
One Period Case - Single Cash Flow
cash flow assumptions: timing and size are given in all
our examples, and for now there is no risk
as long as we measure investment alternatives at the
same point in time, we can rank them consistently
example: you have $100,000. An investment costs
C0 = $100,000 today, and returns C1 = $104,000 in a
year. A bank offers a 5% annual interest rate on
deposits. Should you make the investment (option A) or
put your money in the bank (option B)?
FV analysis:
PV analysis: PV = C1 /(1 + r)
AFM 271 - Time Value of Money
Slide 4
Cont’d
net present value: NPV = (PV of future cash flows) - (cost of
investment today)
NPV < 0 ⇒ don’t invest (since the cost of the investment
today exceeds the value today of all its future cash flows)
calculate NPV for options A and B:
observations:
PV analysis and FV analysis both yield the same
conclusion; the difference is only the time at which the cash
flows are compared
NPV also gives the same conclusion (A and B cost the
same, so we are really just comparing their PVs)
AFM 271 - Time Value of Money
Slide 5
Cont’d
another example: a piece of land costs $20,000 and will
be worth $21,500 a year from now. The bank pays a 4%
annual interest rate. Should you invest?
PV analysis:
FV analysis:
NPV analysis:
AFM 271 - Time Value of Money
Slide 6
Multi-Period Case - Single Cash Flow
two types of interest:
simple interest: interest earned only on original
principal
FV after 1 yr = C0 +C0 × r
FV after 2 yrs = C0 +C0 × r +C0 × r
..
.
n terms
z
}|
{
FV after n yrs = C0 + C0 × r +C0 × r + · · · +C0 × r
= C0 × (1 + n × r)
AFM 271 - Time Value of Money
Slide 7
Cont’d
compound interest: interest earned on original
principal and on previously earned interest
FV after 1 yr = C0 × (1 + r)
FV after 2 yrs = C0 × (1 + r) × (1 + r)
..
.
n terms
z
}|
{
FV after n yrs = C0 × (1 + r) × (1 + r) × · · · × (1 + r)
= C0 × (1 + r)n
AFM 271 - Time Value of Money
Slide 8
Cont’d
the power of compounding ($100 deposited at 6% and
10%)
Time (years)
6% simple
6% compound
10% simple
10% compound
1
106
106
110
110
2
112
112.36
120
121
4
124
126.25
140
146.41
10
160
179.08
200
259.37
50
400
1842.02
600
11739.09
100
700
33930.21
1100
1378061.23
other examples:
text p. 85: Julius Caesar lent one penny to someone;
assuming 6% annual interest, what would be owed
on this loan 2,000 years later?
the island of Manhattan was purchased in 1626 for
the equivalent of $24; what would this amount be
worth in 2005, assuming 5% annual interest?
AFM 271 - Time Value of Money
Slide 9
Cont’d
discounting moves a cash flow that is expected to occur
in the future back to today (i.e. finding PV)
compounding moves a cash flow from present to future
value amounts (i.e. finding FV)
cash flows at different points in time cannot be
compared or aggregated, unless first brought to the
same point in time (via discounting and/or
compounding)
example: C0 = 120, C5 = 130. Adding these up to 250 is
meaningless (like adding £120 + $130), but we can find
the value of the combined CFs at any point in time by
discounting and/or compounding (like using exchange
rates for currency conversion).
AFM 271 - Time Value of Money
Slide 10
Cont’d
some examples:
how much must be invested today in order to receive $1,000
in 5 years if interest is compounded at 7% per annum?
what is the annual compound interest rate equivalent to a
simple interest rate of 6% for a five year investment?
suppose a bank’s annual compound interest rate is 4%.
What is the PV of $100 invested for 5 years at 4% simple
interest?
AFM 271 - Time Value of Money
Slide 11
Multi-Period Case - Compounding Periods
stated annual rate: (SAR)
not the whole story unless the compound frequency is given
effective annual rate: (EAR)
if compounding occurs m times per year, then
EAR = (1 + SAR/m)m − 1
example: SAR = 10%, C0 = $1
annual compounding
FV1 = $1 × (1 + 0.10) = $1.10
EAR = 10%
semi-annual compounding
FV1 = $1 × (1 + 0.10/2)2 = $1.1025
EAR = 10.25%
quarterly compounding
FV1 = $1 × (1 + 0.10/4)4 = $1.1038
EAR = 10.38%
AFM 271 - Time Value of Money
Slide 12
Cont’d
we can also have monthly, weekly, daily compounding, etc.
example: Mastercard statement says that the annual interest
rate is 18.4%, the daily interest rate is .05041%, and interest is
compounded daily. What is EAR?
m×T
SAR
in the limit: limm→∞ 1 + m
= eSAR×T
this is called continuous compounding:
FV1 = $1 × e0.10 = $1.1052
EAR = 10.52%
why does EAR increase as compounding period decreases
(i.e. as compounding frequency increases)?
AFM 271 - Time Value of Money
Slide 13
Cont’d
converting between compounding frequencies: e.g. 10%
compounded semi-annually is equivalent to what rate
compounded weekly?
compounding over several years at non-annual compounding
frequencies: FVT = C0 × (1 + SAR/m)m×T
FV of $100 invested for 6 years at 5% compounded
monthly:
continuous compounding over many years: FVT = C0 × eSAR×T
C7 = $1,000, what is PV today if interest rate is 5.5%
compounded continuously?
AFM 271 - Time Value of Money
Slide 14
Multi-Period Case - Multiple Cash Flows
time line:
t =0
t=1
t=2
t=3
C0
C1
C2
C3
...
...
...
t=k
t=n
...
Ck
Cn
t = k implies end of year k, beginning of year k + 1
C3
Cn
C2
C1
+ (1+r)
PV = C0 + 1+r
2 + (1+r)3 + · · · + (1+r)n
e.g. if r = 6%, calculate PV of the following cash flows:
t =0
t=1
-$100 $150
t=2
t=3
-$80
$300
AFM 271 - Time Value of Money
Slide 15
Cont’d
Summarizing to here:
simple interest: FV = C0 × (1 + n × r)
discrete compounding/discounting:
SAR mT
FVT = C0 × 1 +
m
= C0 × (1 + r)n
PV0 =
CT
1 + SAR
m
CT
=
(1 + r)n
mT
continuous compounding/discounting: FVT = C0 × erT ,
PV0 = CT × e−rT
with multiple CFs, treat each one separately and add (as in the
example on slide 15). There are no convenient formulas if cash
flows vary in general, but there are for “nice” cash flows.
AFM 271 - Time Value of Money
Slide 16
Perpetuities
a perpetuity is a stream of equal cash flows that occur
every period forever, 1st payment one period from now:
t=0
t=1
t=2
t=3
t=4
t =5
C0 = 0
C1 = C
C2 = C
C3 = C
C4 = C
C5 = C
PV of a perpetuity is
C
C
C
+
+···
+
(1 + r) (1 + r)2 (1 + r)3
C
=
r
PV =
e.g. if r = 3%, find PV of a perpetuity paying $200/year:
AFM 271 - Time Value of Money
Slide 17
Cont’d
a growing perpetuity is a stream of cash flows that
occurs every period forever, has 1st payment one
period from now, and grows at a rate of g per period:
0
1
2
3
4
0
C
C(1 + g)
C(1 + g)2
C(1 + g)3
PV of a growing perpetuity is
C(1 + g) C(1 + g)2 C(1 + g)3
C
+
+
+
+···
PV =
(1 + r) (1 + r)2
(1 + r)3
(1 + r)4
C
=
(r − g)
AFM 271 - Time Value of Money
Slide 18
Cont’d
the formula above is valid only if g < r
e.g. you wish to purchase a preferred share of ABC Co.
The share is expected to pay a dividend of $2.50 next
year, thereafter growing at a rate of 3%. Assume an
interest rate of 8%. How much should you be prepared
to pay for the share? (Assume that you will never
receive your initial capital back.)
note that we could have g < 0 (a decreasing perpetuity)
- reconsider the example above but assume dividends
will decline at a rate of 2% per year forever:
AFM 271 - Time Value of Money
Slide 19
Annuities
ordinary annuity (a.k.a. annuity in arrears): a stream of
equal cash flows that occur every period, for a specified
number of periods, at the end of each period. Examples
include car leases, mortgages, pensions, etc. Our PV
formula below (slide 22) assumes that the 1st cash flow
is one period from now.
e.g. find PV of an annuity paying $1,000 per year for 4
years, r = 7%:
0
1
2
3
4
5
0
1,000
1,000
1,000
1,000
0
AFM 271 - Time Value of Money
Slide 20
Cont’d
e.g. find FV at the end of year 4 of an annuity paying
$1,000 per year for 4 years, r = 7%:
0
1
2
3
4
5
0
1,000
1,000
1,000
1,000
0
AFM 271 - Time Value of Money
Slide 21
Cont’d
a simplifying formula for the PV of an annuity - find a
formula for the PV of an ordinary annuity paying C for a
total of n periods, assuming an interest rate of r per
period:
AFM 271 - Time Value of Money
1 − (1 + r)−n
PV of annuity = C ×
r
Slide 22
Cont’d
a simplifying formula for the FV of an annuity - find a
formula for the FV (at the end of n periods) of an
ordinary annuity paying C for a total of n periods,
assuming an interest rate of r per period:
(1 + r)n − 1
FV of annuity after n periods = C ×
r
exercise: use the formulas on slides 22 and 23 to verify
the calculations on slides 20 and 21
AFM 271 - Time Value of Money
Slide 23
Cont’d
note that our PV formulas for annuities and perpetuities
all assume that the 1st payment is one period from now
and that the rate of interest per period is r
ordinary annuity examples:
you have an outstanding balance on your Visa
account of $2,400. If you can only afford to repay at
the rate of $150 per month, how long will it take you
to repay the entire amount if the interest rate on the
outstanding balance is 24% compounded monthly?
AFM 271 - Time Value of Money
Slide 24
Cont’d
redo the previous example, but assume that the
interest rate is 24% compounded semi-annually:
you are 25 years old and wish to have savings of
$200,000 by the time you retire on your 55th
birthday. You are willing to put money aside for next
20 years. Assuming an interest rate of 6% until your
retirement, how much must you set each year?
AFM 271 - Time Value of Money
Slide 25
Cont’d
annuity due (a.k.a. annuity in advance): payments are
at the start of each period
PV annuity due = (1 + r) × PV ordinary annuity
PV(FV) of annuity due > PV(FV) of ordinary annuity
e.g. Sue has won a lottery, which pays $25,000 per
year for 8 years, starting today. Calculate PV today
and FV (in 8 years) of the lottery payout (assume
r = 5%).
AFM 271 - Time Value of Money
Slide 26
Cont’d
delayed annuity: payments start after a delay by a
certain number of periods
this involves a two step calculation; e.g. Bill will
graduate in 4 years, and will thereafter earn an
annual income of $75,000 for 35 years (assume all
income is received at end of a year). Calculate the
PV of Bill’s lifetime earnings (assume r = 4%).
AFM 271 - Time Value of Money
Slide 27
Cont’d
infrequent annuity: payments occur less often than
once per year
this also involves a two step calculation; e.g. you
plan to purchase a new $35,000 car every 5 years
for next 30 years starting in 5 years. Calculate the
PV of your car purchases (assume r = 6.5% per
annum).
AFM 271 - Time Value of Money
Slide 28
Cont’d
growing annuity: like an ordinary annuity, but payments grow at
a rate of g per period
0
1
2
3
0
C
C(1 + g)
C(1 + g)2

PV = C × 
1−
h
n
n+1
C(1 + g)n−1
0
(1+g)
(1+r)
r−g
in 

e.g. XYZ Fund will pay out distributions over next 10 years.
The first distribution, $80 per unit, will be paid out a year
from today. Subsequent distributions will grow at a rate of
8%. No further cash flow is expected once the ten
distributions have been paid out. Find the PV of one fund
unit (assume r = 11%).
AFM 271 - Time Value of Money
Slide 29
Mortgages
many varieties, but these are basically annuities with monthly
payments, semi-annual compounding, a 25 year maturity
(usually), and a term (typically 5 years) less than the maturity
e.g. find the monthly payment on a $225,000 mortgage,
assuming the interest rate on the initial 5 year term is 6%
find the monthly payment after the initial 5 year term, if the
interest rate changes to 8%
AFM 271 - Time Value of Money
Slide 30
Amortization Schedules
applicable to loans being paid off over a number of periods,
with constant monthly payments (e.g. mortgages, leases)
the loan is said to be “amortized” over the loan period
does not apply to debt where only interest is paid
periodically, with principal repaid as a lump sum at maturity
(e.g. corporate bonds)
see spreadsheet handout (try to recreate it yourself)
note that as time passes:
principal balance decreases (eventually to zero when the
last payment is made)
interest portion of each payment becomes smaller
principal portion of each payment becomes larger
AFM 271 - Time Value of Money
Slide 31