Investment Decisions – Financial Concepts

1 Interest calculations

In the previous chapter frequent reference was made to interest rates. We now have to look in just a little more detail about how interest is calculated and how the timing of receipts and payments affects their values.

Simple interest

If an interest rate is 5% and an amount of $1,000 is deposited in the bank (known as the principal) then, assuming no withdrawals are made, the amount of cash in the investor's account will be as follows:

The 5% interest rate is always applied to the principal of the loan and not to the whole account balance ie it is not applied to any interest previously earned.

Exactly the same calculation would be applied if we had borrowed $1,000 at 5% simple interest. After 10 years, $1,500 would be owed to the bank.

Compound interest

With compound interest, the interest rate is applied to not only the original principal but also to any interest added. Assuming no withdrawals are made the balances at the end of each year would be:

You will see that with compound interest the bank balance rises much more sharply than with simple interest

The formula for the future value of the bank deposit (or loan) is:

S = P(1 + r)ⁿ

  where

  P = the principal

  r = the annual interest rate as a decimal ie 15% = 0.15

  n = the number of years

2 Discounting

2.1 Single receipts or payments

Interest calculations project forward to show how much will be in a bank account after a number of years. Discounting does the reverse: it says something like "If I want to end up with $1,000 in a bank account after 3 years and the interest rate is 5%, how much do I have to deposit now?

The compound interest formula can be used:

S = P(1 + r)ⁿ

where S is the amount in the account after n years. So:

P = S x 1/(1 + r)ⁿ

1,000 = P (1.05)³

So, P = 1,000/(1.05)³ = $863.84

(You can check this by working out the final amount after three years if $863.84 is invested at 5%).

$863.84 is known as the present value of $1,000 received after 3 years at a 5% discount rate.

Similarly, if $1,000 were received after 10 years at a discount rate of 10%, the present value would be:

P = 1,000/(1.10)¹⁰ = $385.54

To make the calculation easier, discount tables are used and provided in the exam. Here's an extract:

Present value tables

For the last example, go down the left hand column until you get to 10, the number of years hence that the flow occurs, then go rightwards until the discount rate is 10%. The table figure is 0.386 (rounded to 3 decimal places) so the present value of any amount received after 10 years at a 10% discount rate is 0.386 times the amount, here $386.

Discounting is required for you to properly evaluate cash flows as not just the raw amount needs to be considered but also when it is received. For example, say you were offered the following choice:

(i)   Receive $10,000 now; or

(ii)   Receive $10,000 in 4 years time

Interest rate = 7%

We would instinctively choose option (i): if we receive the money now we can invest it now at 7% whereas that possibility is only available after 4 years in option (ii). The present value quantifies the benefit from earlier receipts:

(i)   Receiving $10,000 means you have $10,000 now

(ii)   Receiving $10,000 in 4 years at 7% interest rate is equivalent to receiving $10,000 x   0.763 = 7,630 now (0.763 is the 4 year 7% table figure).

You might not have needed discounting to make the right decision here, but what about this choice:

(i)   Receive $6,000 in 2 years; or

(ii)   Receive $7,400 in 5 years

Interest rate = 8%

Both amounts need to be turned into their present values, eliminating timing differences and allowing us to concentrate on the amounts.

(i)   PV of receiving $6,000 in 2 years at an interest rate of 8% = 6,000 x 0.857 = $5,142

(ii)   PV of receiving $7,400 in 5 years at an interest rate of 8% = 7,400 x 0.681 = $5,039

So option (i) is better.

Discounting takes into account that:

• Later receipts suffer greater risk and uncertainty
  

• Later receipts have their values eroded by inflation
  

• Earlier receipts can be reinvested earlier to produce more profits.
  

2.2 Annuities

An annuity is a constant amount of money paid or received for a number of years. This can be a relatively common pattern of cash flow, for example, annual rents, insurance premiums or pension receipts.

The present value of annuities can be worked out using the method we have seen above, but another set of tables is available that makes the calculation much faster. These tables are known as annuity tables or cumulative discount tables.

Annuity tables

2.3 Perpetuities

A perpetuity is a constant annual amount, starting at time 1 and going on for ever. Once you get up to around an annuity for a period of 50 years, the inflows can effectively be regarded as a perpetuity because flows after 50 years are discounted so heavily that they can usually be ignored.

The cumulative discount factor for a perpetuity = 1/Discount rate as a decimal

3 Chapter 11 questions

Question 1

How much will be in a bank account at the end of the following investments?

(a)   $1,000 invested for 5 years at 4% simple interest

(b)   $500 invested for 4 years at 6% compound interest

(c)   $600 invested for 5 years. For the first two years 3% compound interest was earned and 5% compound interest was then earned for the final three years.

Question 2

What is the present value of $345 received after 4 years if the discount rate to be used is 7%?

Question 3

What is the present value of $250 received every year from time 1 to time 10 if the discount rate is 4%?

Question 4

What is the present value of $1,200 received in perpetuity, starting at time 1, if the discount rate is 5%?

Question 5

What is the present value of receiving $1,000 for years 4 - 8 inclusive if the discount rate is 5%?

Question 6

What is the present value of a perpetuity of $500 pa that starts at time 3 if the discount rate is 7%