When the interest rate is lower, the future earnings are discounted by a smaller
factor, and hence their
present
discounted value is higher.
9.3 DISCRETETIME AND CONTINUOUSTIME GROWTH RATES
As Equation M.9.5 suggests, if a variable
X
grows from an initial value
X
0
, at a constant
growth rate
g
(in percent/year),
compounded annually
, its value at time
t
(in years) is
given by the familiar “compound interest” formula,
X
t
=
X
0
(1 +
g
)
t
.
(M.9.8)
Suppose that we set
g
=100%/year, and
t
=1 year. Then we have
X
t
=
X
1
=
X
0
(1 + 1)
1
= 2
X
0
. That is, after one year, the value of
X
will double.
Suppose now instead that we compound
semi
annually, at the same annual rate. In
this case, we halve the growth rate per subperiod, to 50% per 6 months, but now we
compound twice per year. At the end of the year,
X
1
=
X
0
(1 + 1/2)
2
= 2.25
X
0
. More fre
quent compounding at the same annual rate increases the value of
X
at the end of the
year. If we compound quarterly at the same annual rate, we have
X
1
=
X
0
(1 + 1/4)
4
=
2.44140625
X
0
. As we shorten the length of the subperiods and correspondingly increase
the number of subperiods per year, we are moving towards
continuous
compounding.
In the limit, as
n
➝
∞
, the expression (1 + 1/
n
)
n
approaches the value e
<
2.718281828,
the base of the natural logarithms. If the growth rate
g
(expressed in percent/year) is an
instantaneous
growth rate with continuous compounding, then the discreteperiod for
mula of Equation M.9.7,
X
t
=
X
0
(1 +
g
)
t
, becomes
X
t
=
X
0
e
gt
.
(M.9.9)
This expression may look somewhat forbidding, but actually, it can often simplify
our calculations in the analysis of growth rates.
E
XAMPLE M.9.7
In 2010, the Consumer Price Index (CPI) is at 120 in terms of 2006 base year prices
(that is, the CPI for 2006 is 100). In 2025, the CPI is at 268 in terms of 2006 base year
prices. What is the average annual inﬂation rate between 2010 and 2025? Calculate
using the discretetime and continuous time formulas.
DiscreteTime Method:
Setting 2010 as period 0, we have
t
= 15 years, so that
P
15
= 268 =
P
0
(1 +
g
P
)
15
= 120(1 +
g
P
)
15
. Hence, (1 +
g
P
) = (268/120)1/15 = 1.05502699,
and so the average annual rate of inﬂation = 5.50%/year.
ContinuousTime Method:
Again with 2010 as period 0 and
t
= 15 years, we have
P
15
= 268 =
P
0
e
g
P
t
= 120
e
g
P
(15)
. This equation gives us (268/120) =
e
g
P
(15)
, and taking
M96
MATH MODULE 9: GROWTH RATES, INTEREST RATES, AND INFLATION: THE ECONOMICS OF TIME
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View Full Documentthe natural logarithm of both sides and rearranging gives us
g
P
= [
ln
(268/120)]/15 = .053566, or 5.36%/year.
[The difference in the results using the two methods results from the fact that
with continuous compounding, a
lower
average annual growth rate will reach the
same Fnal value. (With only a bit of practice, the continuoustime method is faster
than the discretetime method.)]
The continuoustime method has another computational advantage over the dis
cretetime method, in relating nominal and real interest rates. If compounding is instan
taneous, then no “discrepancy” arises of the sort that requires the correction in Equation
15.5 (page 489) of the text. Hence, for example, if the nominal rate of interest
n
is
6%/year and the rate of inﬂation
q
is 10%/year, then the
real
rate of interest
i
is
exactly
n
–
q
= 0.06 – 0.10 = –4%/year. Similarly, if your nominal income grows at an instanta
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 Fall '12
 Danvo
 Inflation, Interest Rates, Interest, present discounted value

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