At the end of this session you should be able to:
- calculate simple interest
- apply the discount formula
- understand compound interest
- understand the broad concepts in bond pricing
- apply and use the RBA bond formula
This session begins with the relatively straightforward simple interest calculation then builds toward the more complex bond pricing formula by examining the concepts that lie behind it.
Simple Interest
Simple interest is generally used for investments that have a term less than 12 months. It is an easy calculation where interest is calculated as the Amount Invested * Rate * Term.
Mathematically the formula for calculating Simple Interest is as follows:
- i Interest rate per annum
- PV Present Value, or the Amount Invested
- FV Future Value
- t Term in years. If you have whole years then this would be 1, 2, 3 etc. If you have half a year this would be 0.5 or 6/12 (6 months out of 12). If you are investing for 5 months then this would be 5/12 and so on. This can also be expressed as n/365, where n is the number of days
Example 1
An investor has $100 to invest for one year. The interest rate is 5%. How much does the investor have when the investment matures?
i = 0.05
PV = $100
t = 1 year
Using timelines can make understanding financial mathematics easier. Here is a timeline for the example above.
Example 2
If the investment period was for 60 days, how much would the investor have when the investment matures?
i = 0.05
PV = $100
t = 60/365 (note 365 days is the day count convention used in Australia)
i = 0.05
PV = $100
t = 60/365 (note 365 days is the day count convention used in Australia)
Interest=PV*i*n/365
Interest=100*0.05*60/365=0.821917808
FV=PV*(1+i*t)
FV=100*(1+0.05*60/365)=100.821917808=$100.82
The discount formula
As mentioned above, the simple interest formula is normally used for investments with a term of less than 12 months. The examples above showed you how to calculate how much an investor would have at maturity. This is referred to as the Future Value.
The discount formula is the simple interest equation re-arranged, so that you can work out the present value. It is used for bank bills, promissory notes, treasury notes, NCDs and all commercial paper issues. It is also used for a bond price calculation (the Australian standard) when a bond is in its final coupon period. Note that in the final period bond price formula, the calculation does not simply use the face value of the bond for the Future Value. As investors receive a coupon payment on maturity of a bond, it adds the face value of the bond to the final coupon payment amount and then applies the discount formula.
We know from above that 
This can be rearranged as follows 
Where
i = interest rate per annum
PV = Present Value, or the Amount Deposited
FV = Future Value
t = Term in years = n/365. Some countries use 360 days in this calculation, however the Australian standard is to use 365 days.
Example 3
How much does an investor need to invest today to receive $100.82 in 60 days time, if the interest rate is 5%? What is the discount factor?
i = 0.05
FV = $100.82
n = 60
The Discount Factor is 0.991847826087.
Example 4
How much does an investor need to invest today to receive $100,000 in 90 days time, if the interest rate is 7%?
i = 0.07
FV = $100,000
n = 90
The discount formula presented graphically
Looking at the discount formula graphically the maturity amount is $100,000, the investment amount is $98,303.26 and the interest earned (if held to maturity) in this case is $1,696.74 which is the discount amount.
Compound interest
Compound interest is when you earn interest on interest. It is common practice for investments to have interest compounding more frequently than annually, for example monthly, quarterly or semi-annually. Referring back to Example 1, how would the interest earned change for the investor if the interest rate was 5% compounding quarterly instead of annually? What if it compounded monthly? To answer these questions, we will firstly derive the compound interest formula.
Annual Compounding: Investment period = 12 months
Semi-annual Compounding: Investment period = 12 months
In this example, the investor would have $102.50 after 6 months. This is then reinvested for another 6 months and the investor would have $105.06 after 12 months.
Mathematically this is derived as follows:
Where
FV = Future Value
PV=Present Value
i=Interest rate per annum
f = Compounding frequency per year (1= annually, 2 = semi-annually, 4 = quarterly, 12 = monthly)
t=Term of investment in years
Monthly Compounding: Investment period=12 months
i=0.05
f=12 (monthly)
t=1 year
Given the same annual interest rate, the more frequently the compounding occurs, the more interest the investor will earn.
Activity - Reproduce the above table using your calculator
Example 5
What is the Future Value of a $100 deposit, invested at 7% compounding quarterly for 100 years?
PV = 100
i = 0.07
f = 4
t = 100 years
As with the simple interest calculation that was discussed earlier, the compound interest formula can be re-arranged to determine what the Present Value is when you know the Future Value. We know that the Future Value is calculated as follows:
Re-arranging this formula, the Present Value is calculated as
This can also be expressed as
Example 6
How much do you need to invest to receive $500,000 in two years time if invested at 8% compounding monthly? What is the Discount Factor? FV = $500,000
i = 0.08
f = 12
t = 2 years
The Discount Factor is 0.852596376
How do interest rates affect prices?
How do interest rates affect prices? There is an inverse relationship between prices and yields. That is, the higher the interest rate the lower the price and vice versa. This makes sense as if you are earning a higher rate of interest, you would expect to invest less money to receive the same amount at maturity when compared to another investment that has a lower rate of interest.
Below is a graph of the price of a 90 day discount security for various interest rates.
Example 7
See if you can replicate the following table for the 90 day investment above where n = 90 days
Securities with more than one cashflow
Up to now we have been discussing securities that have one cashflow only. That is, the investor receives one payment upon maturity of the investment. There are many securites that have more than one cashflow over the life. Bonds are a good example of this type of security.
Bonds
The features of bonds were discussed in Session 3. We will use the financial mathematics principles discussed above to determine how to calculate the price of a bond.
A bond is a series of cashflows, whereby a set coupon payment is made on a regular basis (typically semi-annually) and the Face Value is paid at maturity along with the final coupon payment. The price you pay for the bond may be more or less than the Face Value you receive at maturity depending on the interest rate at the time that you buy the bond. This interest rate is referred to as the ‘yield to maturity’.
The price of a bond is determined by simply calculating the present value of each cashflow (coupon payments and face value) and adding them together. To do this you need to use the yield to maturity as the interest rate and the compound interest discount factor discussed earlier. Note that when bonds are in their last coupon period the price calculation uses the simple interest discount factor instead of the compound interest discount factor.
Bonds can have lengthy maturity dates so it can be time consuming to calculate the present value of each individual coupon payment and face value to work out the price of the bond. Applying some algebra means that the calculation can be simplified. We won’t go through the derivation of the RBA bond formula, however, the result of the following calculation will give you the same answer as if you went and calculated the present value of each individual cashflow and added them together.
RBA Bond Formula
You can read more about the RBA bond pricing formula on its website www.rba.gov.au. This formula is for a $100 FV bond with semi-annual coupon payments.
Where P = Price of the bond per $100 Face Value (FV)
i = yield to maturity of the bond/2 (ie the rate per semi-annum)
f = Next Coupon Date – Settlement Date (ie number of days to the next coupon date)
d = Number of days in the half year to the Next Coupon Date (ie Next Coupon Date – Previous Coupon Date)
g = the semi-annual coupon amount paid per $100 Face Value. For example, if the bond had an 8% coupon rate paid semi-annually then g = $4
n = the number of half years from the next coupon date to maturity. (ie the number of coupon payments remaining, including the final one paid at maturity with the Face Value)
x = 0 if the bond is ex-interest; 1 if the bond is cum-interest.
Bonds trade ‘ex-interest’ for the last 7 days leading up to a coupon payment. In this instance, the coupon payment is made to the previous bond holder. When a bond is not trading ex-interest then it is trading ‘cum-interest’.
Accrued Interest and Capital Price
Accrued Interest represents how much of the next coupon payment has accrued. The formula is:
Example 8
NSW Treasury Corporation bonds with a semi-annual coupon of 7% and maturing on 1/12/2019 are trading on 28 October 2009 at a yield of 6.5% with an assumed same day settlement date. What is the current price of the bonds? What is the capital price and what is the accrued interest?
First, let’s look at the RBA formula and see what information we need
Interest rates and bond prices
In Example 8 above we calculated the price of a bond that had a coupon rate of 7% and yield of 6.5%. How does the bond price change when yields move in the market?
Remember that there is an inverse relationship between price and yield, meaning that as interest rates increase, the bond price decreases and vice versa.
The following graph and table shows how the price of this bond changes for various levels of interest rates. See if you can reproduce these figures using the RBA formula. Note that this graph is not a straight line and that it has a slight curve.
A bond is made up of set coupon payments and the Face Value on maturity. The coupon rate is set at the time of the issue and does not change over the life of the bond. The yield (and therefore price), however, moves in the market in the same way that share prices fluctuate each day. Remember that the yield is the return that you will earn on your investment. The Coupon rate is not reflective of the total interest that you earn over the life of the bond. Rather, the coupon rate sets the cashflows for the bond. It is common for the coupon and yield to be the same rate when the bond is first issued, however, changes in the market place will cause the yield to go up and down depending on what is going on in the financial markets.
When the coupon rate and the yield are the same rate, the clean price (the price excluding accrued interest) of a $100 Face Value bond is $100.
When the coupon rate is higher than the yield (C > i), the clean price of a $100 Face Value bond is more than $100. This is because the coupon payments that you receive over the term of the bond are higher than they would be if you were receiving a payment based on the yield when you purchased the bond, so you need to pay more for the cashflow stream. In this situation the bond is trading at a premium.
When the coupon rate is lower than the yield (C < i), the clean price of a $100 Face Value bond is less than $100. This is because the coupon payments that you receive over the term of the bond are lower than they would be if you were receiving a payment based on the yield when you purchased the bond, so you need to pay less for the cashflow stream. In this situation the bond is trading at a discount.
Conclusion
Understanding the concepts of simple and compound interest and the discount formula help to understand the bond pricing formula. It is more important to understand the broad concepts behind the bond formula and how to apply it rather than getting the calculation right. A fixed interest broker has calculators that will perform the calculations and they can supply the bond prices to investors on request.
Review questions
1. In the equation 
are:
- Face Value and Present Value
- Face Value and Prior Value
- Future Value and Prior Value
- Future Value and Prior Value
2. Using the same equation, calculate the Face Value of a bond where
PV = 100, i= 7.0%, f = quarterly coupon payments so is 4, t = 3
- 44.402
- 76.28592
- 81.20579
- 81.62979
3. What is the relationship between interest rates and bond prices?
- Direct, that is if interest rates rise, bond prices rise
- Inverse, that is if interest rates fall, bond prices rise