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2. Try the worked example below

– Calculating compound interest

For the example introduced above you would probably start to evaluate this by looking at the interest you could earn on an investment of 10,000 and comparing that with the 11,600 you have been offered in three years’ time.

There might be an offer available of 5% interest per annum which you could earn if you took the 10,000 now and invested it.

In summary you can earn 5% each year if you take the 10,000 now or receive 11,600 in three years’ time.

For the “cash now” option you would have Balance * (1 + Rate/100) = 10,000 x 1.05 = 10,500 after one year.

The interest for the second year takes into account the fact that you have received the 500 of interest into your bank account so that the formula for interest for the second year is (new) Balance * (1 + Rate/100) = 10,500 x 1.05 = 11,025 after two years and 11,025 x 1.05 = 11,576.25 after three years.

Note that the compound interest formula for “n” years is Balance * (1 + Rate/100)ⁿ which in our example is 10,000 * (1 + 5/100)3 = 10,000 * 1.1576 = 11,576.25.

From this you can see that the 11,600 offer is a better option since it offers a higher return than 5%.

This very simple example illustrates the concept of TVM – the Time Value of Money. Most such decisions are more difficult to evaluate and that’s where Juki can help.

Juki can help you evaluate almost any investment decision. Loans and leasing are more complex especially when there are repayments which are not simply at the end of the loan or lease term but happen during the course of the lease, usually monthly or quarterly.

For example, if you wanted to borrow 10,000 for three years and had the options to repay it in three years’ time with either a single repayment of 11,600 or three equal repayments of 3,650 starting one year from now, which option would you take?

We have already seen that a single repayment of 11,576.25 represents an annual interest rate of 5% so the 11,600 on offer is a little higher than that.

So what is that rate?

This is where Juki comes in. Take a look at the following screen which shows that the annual interest rate is 5.07176% for the 11,600 payment after three years.

You can see the cash flow which proves the Funding rate and shows the interest calculated each year in the following screenshot:

For example, Balance * (1 + Rate/100) = 10,000 * 5.07176 / 100 = 507.18 rounded to two decimal places.

If we now adjust the payment profile to be three payments of 3,650 starting one year from now and click the Go button the cash flow below indicates that the effective interest rate is 4.67871%.

APR stands for annualised percentage rate and is the effective annual interest rate. In our example we are compounding annually so the annualised rate is the same as the interest rate calculated and displayed in the “Funding rate” box:

Since the rate for three annual payments (4.67871%) is lower than the rate for the single payment of 11,600 payment in three years’ time (5.07176%) we can see that three payments of 3,650 in arrears is a better option.

– Discounting

So far we have looked at calculating the interest received in the future.

Another equivalent way of evaluating TVM decisions is using discounting.

Discounting is the reverse process of calculating interest whereby we work out what amount invested now would generate a cash flow in the future.

In the example above we had that New Balance (Future Value) after n years = Balance now (Present Value) * (1 + Rate/100)n

With discounting we focus on Present Value and this is evaluated by rearranging the interest formula to have Present Value=Future Value / (1 + Rate/100)n

For example we know from above that 11,600 received in three years’ time is equivalent to 11,600 / (1 + 5.07176/100)3 = 10,000.

For the three payments of 3,650 starting one year from now each cash flow has to be discounted. If we use the rate of 4.67871% calculated by Juki we have:

3,650 / (1 + 4.67871/100) + 3,650 / (1 + 4.67871/100)2 + + 3,650 / (1 + 4.67871/100)3 = 3,486.86 + 3,331.01 + 3,182.13 = 10,000

The values (1 + 4.67871/100), (1 + 4.67871/100)2 and (1 + 4.67871/100)3 are called discount factors.

So we can see by choosing the discount rate (or having Juki calculate this rate) then the 10,000 now has the same value as the future payments discounted based on when they occur.

The rate that makes the discounted future payments equal the initial balance (investment) is called the internal rate of return (IRR) or yield.

From the above we can see that the further into the future a cash flow occurs the less its value is today (Present Value). The higher the discount rate the smaller the Present Value.

– More Complex Examples

Leases and loans are more likely to be repayable monthly than annually so we will now move to a more realistic example of 36 monthly payments and use Juki to calculate what they should be to achieve exactly a 5% Funding rate.

This is the Juki screen which shows that the monthly payments are each 299.71:

In this screen you will see that the APR is 5.11619%. The reason that this is higher than the 5% we entered for Funding rate is that we are compounding interest monthly in line with the payment profile.

At the end of each month this interest is added to the balance outstanding and hence has interest on interest calculated in the following month.

5% compounded monthly is a monthly rate of 5%/12 months = 0.41667% per month which over a year gives an APR of (1 + Rate/100)12 = (1 + 0.41667/100)12 = 1.0511619 which is the interest factor (1 + Rate/100) for 5.11619%, our APR

These simple examples show how quickly things can get complicated. However, the key to understanding each cash flow is that it has a different value depending on the date it is received.

A cash flow received in the future is worth less than one received now and can only be compared with the value at today’s date by choosing a suitable discount rate.

In the next article we will explore other calculations and issues involved in asset finance calculations such as how to split a payment into interest and capital.