Consider a principal P invested at rate r compounded annually for n years: Compound Interest After the first year: so that the total is now 1

After the second year and now 2

After the third year and so 3

After the n th year, the total will be 4

Example 1 How long will it take for $750 to amount to $1500 at an annual rate of 11% compounded daily? Use 1 year = 365 days. 5

Solution 11% annually Let n be the number of interest periods. Then Note: we will leave this as a fraction to avoid a rounding error. 6

7

8

This is equivalent to 6 years days. So the amount will double in 6 years, 111 days. The number of years corresponding to days is 9

Note well the discussion on page 131 of the study guide regarding how to answer “how long” type questions. 10

Example 2 After 4 years, a $200 investment amounts to $ At what nominal rate compounded monthly was the investment made? 4 years = 48 interest periods. 11

(Note that is exact and so no rounding error will occur). 12

Solution: Method 1 13

Solution: Method 2 14

= 1.04% is the monthly interest rate. The nominal rate (convert to annual rate) will be 15

Comparing Compound Interest rates 10% compounded monthly is “better” than 10% compounded quarterly but not as good as 10% compounded daily. The more often interest is paid the sooner it can begin to earn interest. 16

What about these rates? 11% compounded quarterly 10.5% compounded monthly 9.75% compounded daily By converting each to a rate compounded annually, a direct comparison can be made. Which is best? 17

This rate compounded annually is called the effective rate and is given by 18

Hence, 11% compounded quarterly is an effective rate of 10.5% compounded monthly is an effective rate of 19

9.75% compounded daily is an effective rate of Clearly then 11% compounded quarterly is effectively the best of the three. 20

How is this formula derived? Consider an investment of $1 at 11% compounded quarterly. Then after 1 year, it would amount to 21

Interest earned = S − P = − 1 = This is equivalent to an annual rate of 11.46% 22

Present Value This formula tells us how much should be invested now to produce S after n interest periods at rate r per period. 23

Example I know that in 2 years I will need $1500 to fund a certain project. How much should I invest now at 12% compounded quarterly to meet this goal? $ should be invested. 24

Equations of Value A person must make a payment of $6000 in 5 years time to clear a debt. She proposes to pay $500 now, $2500 in 2 years time and a final payment in 3 years to pay out the loan. What is the final payment? (Assume money is worth 10% compounded quarterly) x x 25

At the end of the 3rd year $500 will amount to $2500 will amount to The $6000 has a value of 26

The debt will be cleared with a payment of $ at the end of the 3rd year. 27

The equation which states the value of the different monies at one particular time is called an equation of value. A payment x now $500 in 2 years $2500 in 4 years Consider a different method of payment for the $6000 due in 5 years time: What should x be? 28

A payment x now $500 in 2 years $2500 in 4 years Method 1: x

The initial payment is $

A payment x now $500 in 2 years $2500 in 4 years Method 2: x

x = $

You should choose a point in time that will minimise the number of calculations to be done (usually at the unknown x) x

Net Present Value (NPV) When an investment returns cash payments (cash flows) after specific time intervals, then: If NPV > 0, the investment is profitable, otherwise it is not profitable. NPV = (Present value of returns) – (Initial investment). 34

Example Suppose a $15,000 investment returns the following cash flows at the indicated times. (Assume a rate of 12% compounded quarterly.) 35

The venture is not profitable - better to invest at 12% compounded quarterly. 36

The decision is not as simple as it appears. For example: a rate of 8% compounded quarterly produces an NPV = showing the venture to be a profitable one. As always care is needed 37