125.350: FINANCIAL RISK MANAGEMENT
Week 1 Problems
SIMPLE VS GEOMETRIC RETURNS
Geometric return:
Geometric mean return provides more accurate measurement of the true return by considering year-over-year compounding
If ‘A’ dollar invested for 1 year at r rate of interest, the value of investment in 1 year,
If ‘A’ dollar invested for 2 years at r rate of interest, the value of investment in 2 years,
If ‘A’ dollar invested for n years at r rate of interest, the value of investment in n years,
If A = $1, FVn = (1 + r)n
What is the average or mean future value?
Mean of future value =
Mean return = (subtracting 1 since $1 is the original investment)
Now consider r is different for each year ( r1
, r2
, r3
,…..,rn
) , the mean future value of $1 investment would be,
Mean return = => Geometric mean
COMPOUNDING & DISCOUNTING
The opposite of compounding is discounting
In Future Value (FV) calculations, we use compounding
In Present Value (PV) calculations, we use discounting
If ‘A’ dollar is received in 1 year at r rate of interest, the value of the investment at present,
If ‘A’ dollar is received in 2 year at r rate of interest, the value of the investment at present,
If ‘A’ dollar is received in n year at r rate of interest, the value of the investment at present,
If $1 dollar is received in n year at r rate of interest, the value of the investment at present,
So far we have assumed: interest rate compounds annually
If the interest rate compound semi-annually per year at a rate r,
If the interest rate compound quarterly per year at a rate r,
If the interest rate compound monthly per year at a rate r,
If the interest rate compound daily per year at a rate r,
If the interest rate compound m times per year at a rate r,
If the interest rate compound m times per year for t years at a rate r,
CONTINUOUS COMPOUNDING & DISCOUNTING
In the limit as we compound more and more frequently, we obtain continuously compounded interest rates
In the limit as we discount more and more frequently, we obtain continuously discounted interest rates
If we invest $100 today (time 0) at a continuously compounded rate r for time T, it will grow to $100e rT in T years
If we receive $100 at time T it is discounted to $100e −rT today (time zero) when the continuously compounded discount rate is r
What does e stand for?
e is a constant which can be defined as an infinite series:
Using the first 4 terms, we get:
Using the first 6 terms, we get:
Using the first 6 terms, we get:
As we keep increasing the terms the value of e increases at a decreasing rate => incremental effect becomes very small
In EXCEL, if we type: =exp(1) we get the value of e = 2.718282 => e represent a continuous case
where, m tends to go infinite
➢ If r compounds continuously (m tends to go infinite) in a year, the FV of $1 =
If r compounds continuously in a year, then we can write r as an exponential function =
The opposite of compounding is the discounting => if r is continuously compounded, the discount rate =
➢ If r compounds continuously (m tends to go infinite) in T years, the FV of $1 =
If r compounds continuously in T years, then we can write r as an exponential function =
The opposite of compounding is the discounting => if r compounds continuously in T years, the discount rate =