Ch3 returns
Returns
1
RETURNS
Prices and returns
Let Pt be the price of an asset at time t.
Assuming no dividends the net return is
Pt
Pt − Pt−1
−1=
Rt =
Pt−1
Pt−1
The simple gross return is
Pt
= 1 + Rt
Pt−1
Returns
2
Example: If Pt−1 = 2 and Pt = 2.1 then
2.1
Pt
1 + Rt =
=
= 1.05 and Rt = 0.05
Pt−1
2
Returns
3
The gross return over k periods (t − k to t) is
1 + Rt (k) :=
Pt−1
Pt−k+1
Pt
Pt
···
=
Pt−k
Pt−1
Pt−2
Pt−k
= (1 + Rt ) · · · (1 + Rt−k+1 )
Returns are
• scale-free, meaning that they do not depend on monetary units (dollars, cents, etc.)
• not unit-less — unit is time; they depend on the units of t (hour, day, etc.)
Returns
4
Example:
Time
t−2
t−1
t
t+1
P
200
210
206
212
1.05
.981
1.03
1.03
1.01
1+R
1 + R(2)
1 + R(3)
1+R
1.05 = 210/200
.981 = 206/210
1.03 = 212/206
1.06
Returns
5
Example:
Time
t−2
t−1
t
t+1
P
200
210
206
212
1.05
.981
1.03
1.03
1.01
1+R
1 + R(2)
1 + R(3)
1+R(2)
1.03 = 206/200
1.01 = 212/210
1+R(3)
1.06 = 212/200
1.06
Returns
6
Log returns log prices: pt := log(Pt ) log(x) = the natural logarithm of x
Continuously compounded or log returns are logarithms of gross returns: rt := log(1 + Rt ) = log
Pt
Pt−1
where pt := log(Pt )
= pt − pt−1
Returns
Example: Suppose Pt−1 = 2.0 and Pt = 2.06. Then
1 + Rt = 1.03, Rt = .03, and rt = log(1.03) = .0296 ≈ .03
7
Returns
8
Advantage — simplicity of multiperiod returns rt (k) := log{1 + Rt (k)}
= log {(1 + Rt ) · · · (1 + Rt−k+1 )}
= log(1 + Rt ) + · · · + log(1 + Rt−k+1 )
= rt + rt−1 + · · · + rt−k+1
Returns
9
Log returns are approximately equal to net returns: • x small ⇒ log(1 + x) ≈ x
• therefore, rt = log(1 + Rt ) ≈ Rt
• Examples:
* log(1 + .05) = .0488
* log(1 − .05) = −.0513
• see Figure
Returns
10
0.2
0.1
0 log(1+x) x
−0.1
−0.2
−0.2
−0.1
0 x 0.1
0.2
Comparison of functions log(1 + x) and x.
Returns
11
Behavior of returns
What can we say about returns?
• cannot be perfectly predicted — are random.
Returns
12
• ancient
1
RETURNS
Prices and returns
Let Pt be the price of an asset at time t.
Assuming no dividends the net return is
Pt
Pt − Pt−1
−1=
Rt =
Pt−1
Pt−1
The simple gross return is
Pt
= 1 + Rt
Pt−1
Returns
2
Example: If Pt−1 = 2 and Pt = 2.1 then
2.1
Pt
1 + Rt =
=
= 1.05 and Rt = 0.05
Pt−1
2
Returns
3
The gross return over k periods (t − k to t) is
1 + Rt (k) :=
Pt−1
Pt−k+1
Pt
Pt
···
=
Pt−k
Pt−1
Pt−2
Pt−k
= (1 + Rt ) · · · (1 + Rt−k+1 )
Returns are
• scale-free, meaning that they do not depend on monetary units (dollars, cents, etc.)
• not unit-less — unit is time; they depend on the units of t (hour, day, etc.)
Returns
4
Example:
Time
t−2
t−1
t
t+1
P
200
210
206
212
1.05
.981
1.03
1.03
1.01
1+R
1 + R(2)
1 + R(3)
1+R
1.05 = 210/200
.981 = 206/210
1.03 = 212/206
1.06
Returns
5
Example:
Time
t−2
t−1
t
t+1
P
200
210
206
212
1.05
.981
1.03
1.03
1.01
1+R
1 + R(2)
1 + R(3)
1+R(2)
1.03 = 206/200
1.01 = 212/210
1+R(3)
1.06 = 212/200
1.06
Returns
6
Log returns log prices: pt := log(Pt ) log(x) = the natural logarithm of x
Continuously compounded or log returns are logarithms of gross returns: rt := log(1 + Rt ) = log
Pt
Pt−1
where pt := log(Pt )
= pt − pt−1
Returns
Example: Suppose Pt−1 = 2.0 and Pt = 2.06. Then
1 + Rt = 1.03, Rt = .03, and rt = log(1.03) = .0296 ≈ .03
7
Returns
8
Advantage — simplicity of multiperiod returns rt (k) := log{1 + Rt (k)}
= log {(1 + Rt ) · · · (1 + Rt−k+1 )}
= log(1 + Rt ) + · · · + log(1 + Rt−k+1 )
= rt + rt−1 + · · · + rt−k+1
Returns
9
Log returns are approximately equal to net returns: • x small ⇒ log(1 + x) ≈ x
• therefore, rt = log(1 + Rt ) ≈ Rt
• Examples:
* log(1 + .05) = .0488
* log(1 − .05) = −.0513
• see Figure
Returns
10
0.2
0.1
0 log(1+x) x
−0.1
−0.2
−0.2
−0.1
0 x 0.1
0.2
Comparison of functions log(1 + x) and x.
Returns
11
Behavior of returns
What can we say about returns?
• cannot be perfectly predicted — are random.
Returns
12
• ancient