# Math Review Present value or present discounted value

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```					                                                                               Econ 134. Financial Economics
Prof. Beatriz de Blas                                                                            Spring 2006

Math Review

1     Present value (or present discounted value)
How to value future income now. We can use the real interest rate:
One cookie in period t         !    1 + rt cookies in period t + 1
Therefore,
1
cookies in period t     !    1 cookie in period t + 1
1 + rt
If we save during 2 periods, what we get is

One cookie in t                     !    1 + rt cookies in t + 1                 !    (1 + rt )(1 + rt+1 ) cookies in t + 2
Therefore,
1                                   1
cookies in t   !                 cookie in t + 1            !    1 in t + 2
(1 + rt )(1 + rt+1 )                      (1 + rt+1 )
Example:
20
1 + rt
Example:
500
If the interest rate is rt = rt+1 = 0:1 = 10%; then 500 cookies in t+2 are worth     = 413; 223
1:1
t + 2: Assume that rt = 5% and rt+1 = 6%: What is the present value of all this income ‡ in  ow
terms of cookies at time t?
150      200
P V = 100 +       +             = 100 + 142:857 + 179:695 = 422:552
1:05 (1:06)(1:05)

1.1   In…nite number of periods
In general, if you receive income wt in period t; wt+1 in t + 1; and so on, then the present value
(from the point of view of time t) of all this income is:
wt+1          wt+2                       wt+3
P V = wt +         +                    +                                + :::                   (1)
1 + rt (1 + rt )(1 + rt+1 ) (1 + rt )(1 + rt+1 )(1 + rt+2 )
where rt is the real interest rate between t and t + 1:
But if both income and real interest rates are constant, we can simplify the formula. To do
that, we need to do as follows:

(1    x)(1 + x + x2 + x3 + :::) = 1 + x + x2 + x3 + :::                (x + x2 + x3 + x4 :::) = 1     (2)

and then, we have one of the key formulas
1
1 + x + x2 + x3 + ::: =               :                                  (3)
1       x
Econ 134. Financial Economics
Prof. Beatriz de Blas                                                                    Spring 2006

If the real interest rate r and income w are constant for all the periods, then the present
ow
value of all the income ‡ is
w      w                     1      1                              w
PV = w +        +         + ::: = w 1 +      +         + :::            =             ;       (4)
1 + r (1 + r)2               1 + r (1 + r)2                           1
1
1+r
that is,
1+r
PV =       w:                                            (5)
r

1.2    Finite number of periods
ow
If the real interest rate r is constant, and you receive a constant income ‡ w from period 0
to period T; then the present value of that income ‡   ow, in terms of period 0; is
w      w                w       1+r                   w               w
PV = w +          +       2
+ ::: +        T
=     w                                           :::;    (6)
1 + r (1 + r)          (1 + r)     r                (1 + r)T +1     (1 + r)T +2
that is,
1      1+r
PV =     1                w:                             (7)
(1 + r)T +1  r
Example: if you earn 1000 cookies every period from t = 0 to t = 10; and the real interest rate
ow
is r = 10% = 0:1; the present value of this income ‡ is
1         1:1
PV =     1                        1000 = 7144:567 cookies at time t = 0:                  (8)
1:1111       0:1

1.3    More examples
Example 1: To value at time 2006 a bond that pays X cookies at time 2009
X
(1 + r2006 )(1 + r2007 )(1 + r2008 )
It is worth buying it if the price of the bond is less than the PV.
It is worth selling it if the price of the bond is more than the PV.
We are indi¤erent between buying or selling it if PV=PRICE.
Example 2: Assume that IBM stocks pay (and will always pay) a dividend of 5 cookies a year,
and that the real interest rate is (and will be) 4% per year.Then the present value of an IBM
stock is
5       5               5      1:04
PV = 5 +          +         2
+ ::: =     1 = 0:04 5 = 130 cookies.        (10)
1:04 (1:04)             1 1:04
Then, it is worth buying the stocks when its price is less than 130.
It is worth selling the stocks when its price is more than 130.
We are indi¤erent between selling or buying when its price is equal to 130.
Example 3: We are thinking on whether to buy an appartment that costs 300,000 cookies, or
renting an appartment for 900 cookies/month. The annual interest rate is 6%; an that is why
the monthly interest rate is 0:5% = 0:005: Therefore, the present value of renting is
900    900            1:005
P V = 900 +         +         + ::: =       900 = 180900 cookies.                       (11)
1:005 (1:005)2         0:005
Question: should we buy or rent?
Econ 134. Financial Economics
Prof. Beatriz de Blas                                                                       Spring 2006

2    Some statistics
Given a random variable X; with possible realizaitons fx1 ; x2 ; :::; xn g we can compute some of
the moments related to that variable, for example:

Expected Value

The expected value (or population mean) of a random variable indicates its average or central
s
value. It is a useful summary value (a number) of the variable’ distribution.
The expected value of a random variable X is symbolised by E(X) or .
If X is a discrete random variable with possible values x1 ; x2 ; x3 ; :::; xn , and p(xi ) denotes
P (X = xi ), then the expected value of X is de…ned by:
n
X
= E(X) =           xi p(xi )                               (12)
i=1

where the elements are summed over all values of the random variable X.

If X is a continuous random variable with probability density function f (x), then the
expected value of X is de…ned by:
Z 1
= E(X) =       xf (x)dx:                           (13)
1

Variance

The variance of a random variable is a non-negative number which gives an idea of how
widely spread the values of the random variable are likely to be; the larger the variance, the
more scattered the observations on average. Stating the variance gives an impression of how
closely concentrated round the expected value the distribution is; it is a measure of the ’
of a distribution around its average value.
Variance is symbolized by V (X) or V ar(X) or 2 :
The variance of the random variable X is de…ned to be:

V (X) = E fX         E(X)g2                                  (14)

where E(X) is the expected value of the random variable X.

Covariance

The extent to which two random variables vary together (co-vary) can be measured by their
covariance. Consider the two random variables X and Y , with means E(X) and E(Y ), the
covariance can be de…ned as:

Cov(X; Y ) = E f(X       E(X))(Y            E(Y ))g :                   (15)

The covariance can be positive, negative or zero. If the covariance is positive, then values
of X above the mean are related to values of Y above the mean. If the covariance is negative,
then values of X above the mean are related to vlaues of Y below the mean. Finally, we can
prove that if the variables are independent, then the covariance is zero, but not necessarily the
other way round.

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