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I'm Larry Walther.
This is principlesofaccounting.com Chapter 24, and in this module we're going to look
at compound interest and present value calculations.
You may have seen these concepts in other classes, a business math class,
earlier in the financial accounting course, or perhaps in a finance class.
It starts with a basic awareness that a dollar to be received today
is worth more than a dollar to be received in the future.
The reason being that dollar can be reinvested and earn returns.
Very simply, if Alpha and investment returns $100 a year for 5 years and
Beta returns $50 a year for 10 years, which is preferable?
Well, Alpha is preferable, because you get the money back sooner and
you're able to reinvest it and earn additional returns elsewhere.
Now we can actually start to quantify the value of these cash flows.
First, though, we need to think about compound interest.
Compound interest allows for interest to be earned on previously accrued interest.
Annual interest becomes larger each year.
This is also referred to as future value concepts.
And so for example, if one invests $1 at 10% for one year,
you can see that it would grow to a dollar and a dime after one year, $1.10.
If that $1.10 is reinvested for the next successive year,
it would earn $0.11 of interest and grow to $1.21, and so
it would go in time, as shown by this graphic.
$1 would grow to $1.10, $1.10 would grow to $1.21,
$1.21 would grow to $1.331, and so it would go.
And there's a way to quantify or capture this mathematics by formula.
It's important to recognize (1+i) to the n,
where i is the interest rate per period and n is the number of periods.
And so, we can apply this and say if $1 is invested for
25 years, how much will it grow to at the end of 25 years?
And the answer is $10.83.
And the way we can calculate that is 1.1 to the 25th power would
give us a factor of 10.83.
Now, there are tables that have calculated these values for us.
They're called future value tables.
We'll see in a moment some present value tables.
But they're useful, for example, in a calculation such as the following.
If $5,000 is invested for 10 periods at 5%, it'll grow to $8,144.45.
That's calculated as $5000 times 1.62889, which is found in the table,
the future value of a lump sum table, in the 5% column 10 period row.
And so these tables are included on the website and
in the appendix to your managerial accounting textbook.
In looking at the tables,
it's important to note that the interest rate is the interest rate per period.
A period can be a year, a quarter, or a month.
To show by example, if we have a 12% annual interest rate, but
we're compounding interest every month rather than every year,
then we're no longer dealing with a 12% interest rate per period.
And we're no long dealing with, if it's a 2-year period, 2 periods.
We're instead dealing with 1% per period for a total of 24 periods.
So we would want to look at the 1% 24-period sale in the table.
Turning our attention to annuities, annuities are level streams of payments.
Each payment is the same and it occurs on a regular interval.
For example, some retirement accounts or investment accounts, leases, insurance
settlements, tax-related calculations are often based on the concept of annuities.
An annuity due, also known as an annuity in advance,
is one where the payments are made at the beginning of each period.
For example, if one invests $5,000 at the beginning of each year for
the next 5 years at 10%, how much will they have?
We're looking for how much that future value will grow to.
And this graphic shows very clearly how that happens.
The first 5,000 is invested for a full 5 years.
So we multiply it times 1.1 to the 5th.
The next $5,000 payment at the beginning of year 2 is only in the pool or in
the investment pool for 4 years, so we're going to compound it times 1.1 to the 4th.
And so it would go totaling up the value of each of the individual $5,000
payments gives us a $33,578 as the future value of that annuity.
As you suspect, there's a table that'll show that.
The future value and annuity table would show in the 10% column 5-period row,
a factor 6.71561 that we could multiply times the $5,000
amount to find the future value of that annuity due.
An ordinary annuity is just the opposite.
The payments are at the end of each period.
And so here I'm just moving the $5,000 payments from the beginning to the end of
each year, repeating the sequence of calculations except noting that the first
payment's only invested for 4 years, the next one for 3 years and so it would go.
The last payment of $5,000 is only worth $5,000 since it occurs
on the very last day.
It has no opportunity to grow with interest, in other words.
Present value is just the opposite of future value.
It's also known as discounting.
It determines the current worth of cash to be received in the future.
And it's really reciprocal mathematics.
For example, how much would one take today in lieu
of $1 to be received in one year if the interest rate is 10%?
And the answer is $0.909090, or $0.909.
And the mathematics is the reciprocal, the factor,
the 0.909090, that would be one over 1.1 to the first power, for example.
We can extend this.
A $25,000 lump sum to be received at the end of 10 years.
If 8% is the annual interest rate, and we're compounding semiannually,
it would have a present value of 11,410.
We would find that by multiplying the 25,000 times, 1 over 1.04 to the 20th.
The 1.04 reflects the 4% semiannual interest rate, 8% annual, 4% semiannual.
And the 20 periods reflects 10 years.
There's 20 semiannual periods in a 10-year window.
And here's showing how we can calculate those amounts for an annuity, a stream of
payments for $5,000 will be received at the beginning of the first year,
second year, third year, fourth year and fifth year.
Discounting back by the formulations would give us the present
value of the annuity due in this case.
Or, if we go to the next slide, the present value of an ordinary annuity.
I've moved the payments to the end of the period.
And it's got a slightly lower present value of 18,954.
Electronic spreadsheets usually include functions for calculating present values.
Here I'm replicating one of the pop up windows that you might have for
a particular cell where we're entering the 10% interest rate per period, 5 periods,
the payment amount $5,000.
So we've got five $5,000 payments, and
it returns a result of $18,954 to the cell to which that is attached.
So that's a very nice tool for both present and
future values that is in most electronic spreadsheet software.
Let's look at a fairly complex example to close this video.
So we've got Markum Real Estate is considering buying an office building.
The building will be vacant for two years while it's renovated.
And then it will produce rents of $100,000 at the beginning of year three.
100,00 at the beginning of year four and 100,000 at the beginning of year five.
And then we're going to sell the building at the end of the fifth year for 700,000.
And Markum desires to know the present value of the cash flow assuming a 5%
interest rate.
Well, here's a graphic that shows how we could do that.
It looks fairly complex, but when you break down into individual components,
it's somewhat simplified.
Here's our cash inflow across the top of the screen.
At the beginning of Year 3, we get 100,000, and the beginning of Year 4 and
at the beginning of Year 5.
And then we get our $700,000 sales price at the end of Year 5.
First of all this involves an annuity and a lump sum.
The present value of the annuity at $100,000 is
285,941 as of the beginning of Year 3.
But I need to discount that back for 2 additional years to find
the present value of that stream of payment as of the beginning of Year 1.
So the present value of the rents as of the beginning of Year 1 is 259,357.
The $700,000 terminal payment has a present value of $548,471.
And we total the two components to find that the present value of
the cash inflows is $807,828, assuming a 5% interest rate.
We'll revisit this example in the next module and show how we can enhance our
decision-making process through this type of analysis.
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