9/6/2022
Extra
TVM
Dr. Choi
Discussion Agenda

Return measurement – PV and FV
Dr. Choi
1
9/6/2022
Measurement of return -1

You invested $90 in a stock 6-month
ago. Today you’ve just received $1
dividend and then sold it for $95.
Q1) Find the holding period rate of return.
Q2) Per-period effective rate?
Q3) Find the gross rate of return.
Dr. Choi
Measurement of return -2
Q1) Find the holding period rate of return.
HPR 
(95  90)  1  0.0667
90

6.67%
Q) What is its holding period? Ans) 6 months
Q2) What is its gross rate of return over 6
months?
10 6.67%
Just add 100% of principal.
Dr. Choi
2
9/6/2022
Measurement of return -3
Q) Annualize the HPR.
Annualized HPR  6.67% 
12 months
 13.34%
6 months
Q) Did you effectively earn 13.34%?
Ans) No. Effectively 6.67% over 6 months.
Formally we say that 6.67% is 6-month effective rate.
If you make 6.67% again for the following 6 month, you would
earn Effectively more than 13.34%.
Q) Then what is 13.34% called in this case?
Ans) annual rate compounded semiannually.
Or semiannually compounding annual rate.
Dr. Choi
Measurement of return -4
Annual rate compounded m times.
Per year
Compounding period can be daily, weekly, monthly, quarterly,
semi-annual, or annual.
Example 1) Monthly compounding annual rate of
12%. This implies that the monthly effective rate is
12%
 1%
12
Example 2) Daily compounding annual rate of
12%. This implies that the daily effective rate is
12%
 0.03287%
365
Dr. Choi
3
9/6/2022
Measurement of return -5
Example 3) Annual rate of 12% compounded m
times per year. This implies that the per-period
effective rate is
12%
m
Example 4) Continuously compounding annual
rate. This implies that the per-period effective
rate is
12%

Can this be zero?
Dr. Choi
PV, FV and Effective rate of
return
Suppose r(m) is an annual rate compounded m times per year.
mT
Then
 r ( m) 
FVT  PV0 1 

m 

where T = number of years.
Example) You invest $100 in an asset yielding annual rate of 12%
compounded monthly. Find the future value at the end of 2 years.
122
 .12   126.97
FV2  100 1 

 12 
The future value at the end of 467 days?
Dr. Choi
FV467
365
 467 
12

 365 
 .12 
 100 1 

 12 
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9/6/2022
PV, FV and Effective rate of
return
 r ( m) 
FVT  PV0  1 

m 

Important formula :
mT
m
What happens to the quantity,  1  r (m)  , when m approaches infinity?

m
Ans)

Thus



m 
e rc
 r ( m) 
lim  1 
 
m 
m 

 r () 
FVT  PV0   1 

 

T
where rc  r    .
T
  r ( )   
 PV0  e rc T
 PV0  1 
 
  

Dr. Choi
Continued..
m


Verify
 r ( m) 
lim  1 
 
m 
m 

e rc
Suppose we invest $1 at an annual rate of 12%
compounded 1,000,000 times per year. Find its
future value at the end of year 1.
Note that r 1 mil   rc .
1000000
0.12 

 1 

 1000000 
 e0.12 .
In excel, =(1+0.12/1000000)^1000000
In excel, =exp(0.12)
Dr. Choi
5
9/6/2022
How to calculate the Continuously
compounding annual rate.
T-year
Gross rate of return
Aside :
FVT  PV0  e rc T
ln e x  =x
FVT
 e rc T
PV0
 FV 
ln  T   ln e rc T
 PV0 

Thus, C.C. annual rate : rc 

 rc  T
1  FVT 
ln 

T  PV0 
Continuously compounding
T-year rate.
Dr. Choi
Example.
Suppose you invest $100 today.
If your total dollar return become $120 in 270 days,
What is the C.C. Rate of return for 270 days?
FVT

PV0
 FV
ln  T
 PV0



What is the C.C. annual Rate of return?
1  FVT 
T=
ln 

T  PV0 
Dr. Choi
6
9/6/2022
Review


FVT
PV0
T-year gross rate of return:
Continuously compounding 1
 F VT 
ln 
annual rate of return:

T

 P V0 
When r is given as the annual rate compounded m times:
T-year gross rate of return:

 r ( m) 
1 

m 

mT
When r is given as the c.c. annual rate:
T-year gross rate of return:
e r T
Dr. Choi
Wyh C.C.R?
Dr. Choi
7