investments notesLec5 revised0 investments.doc

Lecture 5
Note: Section in Chapter 5 in too advanced for this course. Please ignore it.
1. Computing returns
Holding period return (HPR)
Let Wt = final wealth at the end of year t
W0 = initial wealth level now
T = holding period
Holding period return (HPR) Note: HPR is a multi-year return unless the holding period T=1.
(1) HPR = (WT – W0)/W0
(2) HPR = WT/W0 -1
(3) 1 + HPR = WT/W0
. Find the average annual return (when the holding
period T is longer than a year)
Let Rt = periodic return for year t
(4) Rt = (Wt – Wt-1)/Wt-1
= Wt/Wt-1 -1
(5) 1 + Rt = Wt/Wt-1
(6) Wt = Wt-1 (1+Rt)
How to find the average annual return from these T periodic returns?
Arithmetic mean return AMR
(7) AMR = (R1+ R2+R3+….RT)/T
Or AMR = S Rt/T
(b) Geometric mean return GMR Also known as time-weighted average return
(8) GMR = [(1+R1)(1+R2)…(1+RT)]1/T - 1
Or GMR = [P (1+Rt)] 1/T - 1
GMR is in fact the effective annual rate EAR:
Since W0 (1 + EAR)T = WT
(9) (1+EAR)T = WT/W0
(1+EAR)T = (W1/W0) (W2/W1) (W3/W2)…. (WT/WT-1)
(1+EAR)T = (1+R1) (1+R2)(1+R3).......(1+RT)
EAR = [(1+R1)(1+R2)…(1+RT)]1/T - 1
Hence, EAR is a geometric average of individual periodic returns.
From (9),
(10) EAR = (1+ HPR)1/T – 1
Example in your book
T
25
1/T

W0
$
WT
$100
EAR
(100/) – 1 = (6%)
Note:
AMR is upward biased relative to GMR. The bias increases with the standard deviation of the periodic ratesreturns.
AMR = GMR only if all the periodic returns are the same.
. Annualize a HPR when the holding period is shorter than a year (T < 1)
Let RP = HPR of the period
N = number of holding period per year
Arithmetic annualizationannualization method
(11) APR = RP x N
where APR = annual percentage rate
Geometric (pound interest) annualizationannualization method
(12) EAR = (1+RP)N – 1
2. Risk and Return
We can find the expected return and risk of an asset from the probability distribution of its returns
Expect