[SOLVED] 125350 FINANCIAL RISK MANAGEMENT Prolog

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 =