Bonds: Determining Interest & Yields

Overview

This module delves a little deeper into bonds and specifically outlines how to calculate interest – simple and compounded – as well as yields – basic and implied – on your investment.

It also briefly discusses present and future value calculations.

Understanding the Time Value of Money

It is beneficial to understand the process of earning interest as “putting resources” to work for us. It is also important to understand that if earning interest on interest is a good thing, then the following two scenarios will be even better:
• Earning a higher rate of interest
• The more frequently our interest gets allocated to us, the better

In essence, we need to emphasise the importance of having your “resources” work as hard as possible (higher interest rate) and for as long as possible (interest capitalisation frequency).

The table on the left shows just how the actual amount of interest earned changes over a 12 month investment made with R1,000 at 10%. In each case, the actual interest earned on the deposit is capitalised at different intervals.

Investment 1 000.00
Rate 10%
Days per month 30

The table above clearly highlights the importance of the interest frequency. Even when starting out with the same amount and the same interest rate, the compounding frequency makes a significant difference. The more frequent the capitalisation, the more beneficial.

A sharp eye would have picked up the fact that one cannot just refer to an interest rate as being 10%, without also stating the compounding frequency. A deposit yielding 10% with interest capitalised monthly is in fact equivalent to a higher effective interest rate should we wish to compare it to an investment that only capitalises interest once a year.

Recall from the table above, that the rate of 10%, when capitalised annually only yielded anamount of R98.63 in interest. However when interest is capitalised monthly it made R103.21 so to compare these two investments, the equivalent annual rate on our monthly capitalisation will be higher when shown as an effective rate for interest earned over the entire year.

In order to clarify this issue, when interest rates are quoted, it is customary to include the compounding frequency as one of the following:
• Nominal Annual Compounded Monthly (NACM)
• Nominal Annual Compounded Quarterly (NACQ)
• Nominal Annual Compounded Semi Annually (NACS)
• Nominal Annual Compounded Annually (NACA)

Simple Interest Rate Calculations

There is a standard procedure to follow when we work out interest rates. The formula to calculate interest due = Investment Amount x Interest Rate**

To adjust the interest to accurately reflect the interest due for a specific period, one would have to divide the interest rate by the Day Base, and then multiply it by the amount of days you would wish to hold the investment for.

The Day Base refers to how many days there are in a year. For the Rand market we use 365 days, but for some instruments and in some international markets the day base of 360 is used.

It makes life easier when the following pertinent steps are kept in mind when dealing in interest rate calculations:

When you apply the interest rate to an amount, the first answer you will get is an amount of interest which will be paid on the investment if held for a FULL YEAR.
Once you have this amount, we can work out how much interest is earned per day
Once we have the daily interest earned, we can now accurately calculate the interest due for any period you wish to calculate for.

** The interest percentage needs to be adjusted for the specific period/length of the investment

Let’s look at an example.
Example 1: DEPOSIT
Investment amount of R1, 000
Simple Interest Rate = 10%
Investment Period = 30 Days
How much interest will this deposit earn?

Interest due = Investment Amount x Interest Rate**
** The interest percentage needs to be adjusted for the specific period/length of the investment

Interest due = 1,000 x 10% ÷ 365 x 30
Following our steps highlighted above:
Step 1: Interest Per Year Step 2: Interest Per Day
= 1000 x 10% = 100 ÷ 365
= 1000 x 0.1 = 0.27397 per day
= 100

Step 3: How many days?
= 0.27397 x 30 days
= 8.22

By following the three steps it becomes very easy to calculate the interest that is earned over a period of 30 days. It would have been equally easy to work out if the amount of days were to change to 48 or 67 days as all we would do is multiply the interest per day (0.27397) with the correct amount of days in our deposit.

Compounding Interest Rate Calculations

As indicated, calculating simple interest is in fact very easy. So why is it so tricky to consider compounding interest? What has changed? In essence the answer lies in the following statement.

The calculation of the actual amount of interest due does not change at all, and is in fact exactly the same as what we did when calculating simple interest. The ONLY difference is that the amount on which we calculate the interest changes with every compounding.

In other words, nothing changes other than having to adjust for a changing initial investment value as the interest gets added:

Interest Earned on Investment 1 = Investment Amount x Interest Rate**
Interest Earned on Investment 2 = [Investment 1 plus interest earned] x Interest Rate**
Interest Earned on Investment 3 = [Investment 2 plus interest earned] x Interest Rate**

** In all cases the interest percentage needs to be adjusted for the specific period/length of that particular investment in the exact same way we have done before

We once again will follow the 3 steps shown earlier, with step 4 added with every compounding date:

1. When you apply the interest rate to an amount, the first answer you will get is an amount of interest which will be paid on the investment if held for a FULL YEAR.
2. Once you have this amount, we can work out how much interest is earned per day
3. Once we have the daily interest earned, we can now accurately calculate the interest due for any period you wish to calculate for.
4. Add the interest earned for that period to the amount invested, this new amount now will become the amount earning interest

It should now be very easy to note that compounding can be compared to rolling over of an investment when it matures. Shown in our example below will be an investment for 2 months (60 days) where interest compounds monthly (after 30 days). This is in fact exactly the same as saying, make an investment for 30 days, and at maturity roll it over by a month. In other words, place the full maturing principal plus interest on deposit again for another month.

Let’s look at this compounding example.
Example 2: DEPOSIT FOR 2 MONTHS WHERE INTEREST CAPITALISES MONTHLY
Investment amount of R1, 000
Simple Interest Rate = 10% NACM
Investment Period = 60 Days

How much interest will this deposit earn?
Interest Earned Month 1 = Investment Amount x Interest Rate**
Interest Earned Month 2 = [Investment 1 plus interest earned] x Interest Rate**

Following our steps highlighted above:
Month 1: Interest Per Year Month 2: Interest Per Year
= 1000 x 10% = 1008.22 x 10%
= 1000 x 0.1 = 1008.22 x 0.1
= 100 = 100.822

Step 2: Interest Per Year Step 2: Interest Per Year
= 100 ÷ 365 = 100.822 ÷ 365
= 0.27397 per day = 0.27622 per day

Step 3: How many days? Step 3: How many days?
= 0.27397 x 30 days = 0.27622 x 30 days
= 8.22 = 8.29
So after month two the maturing deposit will be worth R1, 008.22 + 8.29

At this point, note that even though month 2 also had thirty days, and our deposit was still also earning 10%, it earned more interest than in month 1. This due to the fact that by us capitalizing the interest from month 1, or new deposit amount on which we are earning interest at 10% has increased to 1,008.22 which results in more interest. This is the power of compounding interest.
It now becomes evident that when evaluating future values one need to be 100% clear about the compounding frequency. As seen above, by using the incorrect frequency our future value will indeed be incorrect.
This is also true when trying to figure out the present value of a strip of future cash flows.

Future Value Calculations

Future value is when you would have a known value of funds today already, and you wish to know by how much this value will grow, given the interest rate and period involved.
Working out the future value of an investment is surprisingly easy.
Examples would be:
• Making an investment of R1000 into a 6 month fixed deposit and figuring out how much it will be worth when it matures.
• Borrowing R1000 for a year, and working out how much you need to save up to repay this loan.
The simplest way to calculate the future value of an instrument is by using the following formula.
Future value = Present Value + Interest

Remember that interest rate can be expressed in a number of ways such as simple or compounded.

For the purposes of this module we will only consider SIMPLE INTEREST, in other words, we will not be looking at any compounding interest calculations for now.

In the below example we will look at how much an investment will be worth in two months from now. We can work out the future amount using the following formula.
Example 1:
When Interest is added at the end of the period only:
Future Value = present Value + (Present Value x Interest x Day / Day Base)
FV = PV + (PV x i x Day / Day Base)

Example:
Investment today (PV) = R1 000
Interest rate (i) = 10%
Day = 60
Day Base = 365

FV = 1,000 + (1,000 x 0.10 x 60÷365)
FV = 1,016.44
We can summarise the answer by stating, R1, 000 invested today at an interest rate of 10% effective annual interest, equates to an amount of R1, 016.44 maturing in two months from now.

Present Value Calculations

The Present Value of a string of cash flows is the current value as if you were to receive all the cash flows today. In order to obtain this value, one would have to discount all the future values by a desired rate of return.
Having worked out the future value in the example above, we can just as easily start at the other end and work our way backwards. We can easily rephrase the question as follows: What is the current value of a deposit maturing in two months from now at a maturing amount of R1, 016.44 when effective annual interest rates are set at 10%?

Example 2:
Present Value using Simple Interest:
Present Value = [Future Value] ÷ [1 + (Interest x Day / Day Base)]
Example:
Investment Future Maturing Value (FV) = R1, 016.44
Interest rate (i) = 10%
Day = 60
Day Base = 365
Present Value = [1 016.44] ÷ [1 + (0.1 x 60 / 365)]
Present Value = 1,000

When looking at these calculations it becomes easy to see the link between future and present values. In our example above we proved that given an interest rate of 10%, you will be completely indifferent to having R1, 000 in your bank account today, or for you to receive an amount of R1,016.44 in two months from now.

It once again needs to be highlighted that these examples do not take into effect the fact that for some instruments we would need to account for the compounding effect of interest rates. In which case, the formulas would look a little different. The overarching concept however remains the same. As long as we compare apples to apples, we will be able to draw a conclusion of how much any amount is worth today vs its implied future value.

Basic Yield Calculations

A yield can be defined as the income (return) one would make on an investment. In other words, the yield will express the return as a percentage of the amount invested. This is important to understand, any investment return received needs to be compared to how much was needed in order to generate the return.

In other words, assuming we have two completely different investment products to choose from. Comparing a return of R100 earned on each of the two separate investments is of little use, without knowing how much we have to invest in each.

Comparing R500 invested in investment 1 to R1, 000 invested in the other, quickly indicates to us that investment number 1 yielded the highest return.

How did we arrive to this conclusion?

Simple, by taking our amount of income generated (our return) and dividing it by the amount we invested will show the percentage return made by each investment.

Investment 1 Investment 2
Amount Invested: R500 Amount Invested: R1, 000
Interest Earned: R100 Interest Earned: R100
Yield = 100 ÷ 500 Yield = 100 ÷ 1,000
YIELD = 20% YIELD = 10%

We therefore can compare one investment to the next by comparing its yield, this way we know immediately how much we can expect to return on our own individual amount we choose to invest.

As with most interest-bearing instruments calculating the future return when the yield is known is as simple as stating the following:
Maturing Value = Amount Invested x Yield x [Day Count]**
**The [Day Count] is typically reflected as the amount of days invested (d) ÷ the amount of days per year (DB)

Example: 1 (Calculating the future value using Yield Rates)
An investment of R1000
At a yield of 5%
For 45 Days

Interest = 1000 x 0.05 x (45 ÷ 365)
Interest = 6.16

Future Value = Present Value + Interest
= 1,000 + 6.16
= 1,006.16

Implied Yield Calculations

Understanding how yields work, we will also now be able to calculate the yield which is implied by two cash flows.

Keeping with our previous example, we could have also rephrased the question as follows:

If I have an investment of R1000 matures to the value of R1, 006.16 after a period of 45 days, what is the implied yield of this investment?

How would we calculate the yield now? Do you recall our steps to follow which were highlighted at the start of this module? Let’s recap those again:

When dealing in interest rate calculations:
1. When you apply the interest rate to an amount, the first answer you will get is an amount of interest which will be paid on the investment if held for a FULL YEAR.
2. Once you have this amount, we can work out how much interest is earned per day
3. Once we have the daily interest earned, we can now accurately calculate the interest due for any period you wish to calculate for.
So knowing how the process works in general, when working out implied yields all that is required from us is to start at point 3 and work our way back. In other words:
1. Establish the amount of interest for the period of the investment, and work out how much interest is earned per day
2. Once we know how much was earned each day, we can multiply the answer by 365 to getthe amount of interest per year
3. Once we have the interest per year, we can divide this amount by our initial investment to show us the implied yield.

Let’s see if this works:
Example: 2 (Implied Yield)
An investment of R1000 turns into R1, 006.16
Over an investment period of 45 Days
What is the implied yield?

How much Interest earned = 1,006.16 - 1000
Interest = 6.16

Step 1: Interest Per Day
= 6.16 ÷ 45
= 0.1369 per day

Step 2: Interest Per Year
= 0.1369 x 365
= 49.96
= 49.96 ÷ 1,000
= 0.04996
Thus rounded = 4.996% i.e. 5%

The calculation above showed just how easy it is to work out the yields, or alternatively the implied yield when we know the beginning and ending values of an investment.

The key point though is that our calculations already used the interest rate which captures our exact return. In other words, the yield is in fact our actual return

We indicated earlier that some instruments are quoted as a yield and some as discounts. Let’s explore how the yield calculations are done when we invest in instruments quoted at a discount.