Time Value of Money

TVM -

Compounding

$ Today Future $

Discounting

Future Value (FV)

Definition -

»

FV = ?

0 1 2 N

PV=x

FVn = PV(1 + i)n

Future Value Calculations

Suppose you have $10 million and decide to invest it in a security offering an interest rate of 9.2% per annum for six years. At the end of the six years, what is the value of your investment?

What if the (interest) payments were made semi-annually?

Why does semi-annual compounding lead to higher returns?

Future Value of an Annuity (FVA)

Definition -

»

FVA = ?

0 1 2 N

i

iFVA

n

n

1)1(

AA A

Ordinary Annuity vs. Annuity Due

Ordinary Annuity

A AA

0 1 2 N

i%

A A

0 1 2 N

i%

Annuity Due

A

Future Value of an Annuity Examples

Suppose you were to invest $5,000 per year each year for 10 years, at an annual interest rate of 8.5%. After 10 years, how much money would you have?

What if this were an annuity due?

What if you made payments of $2,500 every six-months instead?

Present Value (PV)

Definition -

»

FV = x

0 1 2 N

PV= ?

PV = P0 = FV / (1 + i)n

Present Value Calculations

How much would you pay today for an investment that returns $5 million, seven years from today, with no interim cashflows, assuming the yield on the highest yielding alternative project is 10% per annum?

What if the opportunity cost was 10% compounded semi-annually?

Why does semi-annual compounding lead to lower present values?

Present Value of an Annuity (PVA)

Definition -

»

PVA = ?

0 1 2 N

ii

PVAn)1(

11

AA A

Present Value of an Annuity Examples

How much would you spend for an 8 year, $1,000, annual annuity, assuming the discount rate is 9%?

What if this were an annuity due?

What if you were to receive payments of $500 every six-months instead?

TVM Properties

Future Values

An increase in the discount rate

An increase in the length of time until the CF is received, given a set interest rate,

Present Values

An increase in the discount rate

An increase in the length of time until the CF is received, given a set interest rate,

Note: For this class, assume nominal interest rates can’t be negative!