Most interest calculations that you will encounter are simple interest calculations. In a simple interest calculation, interest is calculated for a defined period of time based on the outstanding balance. Simple interest is used for savings accounts, notes receivable, notes payable, bonds, student loans and lots of other applications. We will discuss how simple interest calculations apply to debt, but the methodology is the same for other applications.
The amount of interest charged on a loan is based on three factors: principal, interest rate and time.
Principal is the outstanding balance on a loan. As a loan is paid down, the principal balance decreases. Therefore the interest on the loan also decreases. If the monthly payment on the loan is an equal amount each month, over time, less of the payment will go to interest and more to the principal balance.
The interest rate is the amount of interest charged on the loan. Typically, interest is expressed as an annual percentage rate, also called APR. Although interest is expressed as an annual rate, most loans charge interest monthly. To calculate the monthly rate, divide the annual interest rate by 12.
Time is the duration over which the interest is accruing. If interest is charged monthly, typically we would use the number of days the month divided by 360. Yes, I know there are 365 days in a year, but before calculators and computers, it was much easier to calculate based on 360 days. This became the tradition even after the invention of calculators because banks found they would earn more interest on outstanding debt using 360. Pretty sneaky, huh?
To calculate the amount of interest on a loan, we use this formula:
Interest = P*R*T or Principal * Rate * Time
On February 1, Technorama borrows $10,000 from the bank on a 8%, 90-day note with interest due at the time of repayment. How much cash will Technorama need to pay off the note when it comes due?
First, we need to identify our PRT. Principal is the amount borrowed, $10,000. The rate is 8%. Remember that rates are expressed as an annual rate even though the loan is only for 90 days. The duration of the loan, time, is 90 days. Now we can set up our formula.
Interest = $10,000 * 8% * 90/360
Interest = $200
The question asks how much cash will be required to pay off the note. $200 is not the answer. To pay off the note, Technorama must pay the interest and the principal. Therefore, the cash required is $10,200.
When doing simple interest calculations, just remember PRT. Always use the annual rate and multiply it by the amount of time for which you are calculating the interest.
Let me ask you a question:
Would you give someone $40,000 today to receive $40,000 back in 10 years?
I imagine that your answer is most likely NO. Why not? There are probably a number of reasons.
- You are not receiving anything for giving up your money for 10 years.
- $40,000 will probably be worth less than $40,000 in 10 years because of inflation.
- It would make more money sitting in your savings account at 0.1%
We all look for our money to work for us. When evaluating investments, we typically look at rate of return. Essentially, we are asking “How much would I be willing to pay today to have X tomorrow?” A lot of us do this with retirement. How much would I need to put away today to have $1,000,000 at retirement? When we ask that question, we are asking how much $1,000,000 in 40 years is worth today.
That is a present value calculation. We are asking “What is the present value of $1,000,000 forty years from today?” In order to calculate how much we need to put away, we would need to look at the rate of return our money would get in an available investment. The higher the rate of return we can earn on the money, the less we must invest to reach the $1,000,000 desired.
There are two types of present value calculations: present value of $1 and present value of an annuity. Present value of $1 is for lump sum payments, like the $1,000,000 described above. An annuity is a series of payments made over time. With an annuity, the question asked is “How much would I need to invest today to receive $5,000 a month for the next 40 years?” You are not asking about a single payment. You want to receive multiple payments over the next 40 years.
You are probably wondering why I am rambling on about present value of $1 and annuities when I should be discussing bonds. Put yourself in the shoes of a bond purchaser. What exactly are you purchasing when you purchase $20,000 worth of 10 year, 8% semiannual bonds?
You are purchasing the right to receive $20,000 in 10 years when the bond matures. We need to calculate how much that is worth in today’s dollars by calculating the present value of the lump sum payment.
There is also something else you would receive: interest payments. Over the life of the bond, you would receive 20 payments of $800 ($20,000 x 8% x 6/12). Sounds like an annuity, doesn’t it? The interest payments are considered an annuity and the present value of those payments must also be considered when calculating the present value of the bond.
When trying to figure out how much to pay for a bond, we must calculate the present value of all the payments that will be received. That includes the $20,000 received 10 years from now and the 20 payments of $800 received over the next 10 years. The present value is determined by the market value.
As discussed in the introduction to bonds, when the market rate is higher than the face rate of a bond, a bond will sell at a discount. Let’s see how that discount is calculated.
Using Present Value Tables
We must calculate the present value of each of the components. Luckily for us, there are tables that can be used to calculate this quite easily. There are two tables used in bond calculations. The first is called Present Value of 1. This table is used to calculate the present value of single lump sum payments, like the single repayment received when a bond matures. The second table is called Present Value of Annuity. This table is used when there is a series of payments, like the interest payments received over the life of the bond.
Both of the tables have the number of periods and the interest rates. The number of periods is the number of payments made over the course of the life of the bond. For the number of periods, use the number of interest payments that will be made over the life of the bond. The interest rate is the market rate used for each interest payment. For example, if the company sells a 6%, semiannual bond when the market rate is 8%, the interest rate used would by half the market rate. We would use 4% because the market rate determines the present value of the bond and because the interest payments are for half of a year’s worth of interest (8% * 6/12 = 4%).
Once we have that information, we can look up the discount factor in each of the tables and complete the calculation.
On January 1, 2014, Earnings Management Inc. sells $200,000 worth of 10-year, 8% semiannual bonds when the market rate is 10%. How much will the bonds sell for?
In order to calculate the present value of the bond, we must first figure out what the company will pay out over the life of the bond. There are two components: the principal payment of $200,000 at the end of ten years and the interest paid semiannually.
For the interest, we will use I=PRT.
We now have the two components:
Since the market rate is 10% and the interest is paid semiannually, we will use 5% to calculate the present value. There are two interest payments per year for 10 years for a total of 20 payments.
When we look in the Present Value of $1 table, the discount factor is 0.3769. Another way to look at this is to say that the $200,000 repayment after 20 periods is 37.69% of the face value. This is because we could get a 10% rate of return on alternative investments. We are only willing to pay the amount today that would achieve $200,000 in 10 years at 10%, even though this bond will only pay 8%. To entice the buyer, the company must discount the bond.
$200,000 * 0.3769 = $75,380
The buyer is only willing to pay $75,380 today to receive $200,000 in 10 years. That is a steep discount, but remember this is only one piece of the calculation. We must also factor in the present value of the interest payments.
Now look at the Present Value of an Ordinary Annuity table. What is the discount factor for our interest payments? The answer is 12.4622. You should have looked up 20 periods at 5%. Remember to use the market rate, not the face rate.
Notice that the discount factor is greater than 1. The Ordinary Annuity table already accounts for the fact that 20 payments will be made. Therefore, when we do the calculation, we only need to multiply the factor by a single interest payment.
$8,000 * 12.4622 = $99,698 (rounded to the nearest whole dollar)
The present value of all the interest payments is $99,698. This is significantly less than the $160,000 in actual interest payments that will be made over the life of the bond, but remember that it has been discounted to the value of the investment today.
Add the two components together to calculate the present value of the bond.
$75,380 + $99,698 = $175,078
The present value of the bond is $175,078. If the face value of the bond is $200,000, the discount would be $24,922.
What Would Happen If You Used the Wrong Interest Rate?
When doing these calculations, I like to determine if the bond will sell for par, discount or premium before doing the calculation. If I do this first, I know what my answer should look like. What would happen if we used the wrong interest rate, the face rate, in the problem above? Would we still get a discount? Let’s see what happens.
Rather than using 5% for 20 periods in the calculation, use the face rate of 4%. The factor for the present value of $1 would be 0.4564. The factor for the interest payments would be 13.5903.
This is essentially par value off a bit due to rounding in the calculation. Remember what we said previously. If face rate and market rate are the same, the bond will sell for par. If you know you should get a discount because the market rate is higher than the face rate and you get par, you have used the wrong interest rate. Redo the calculation with the market rate. Taking a moment to make sure your calculation makes sense can save you valuable points on an exam.
On January 1, 2014, Earnings Management Inc. sells $200,000 worth of 10-year, 8% semiannual bonds when the market rate is 6%. How much will the bonds sell for?
The first step is to determine if the bonds will sell for par, a discount or a premium. Since the face rate is 8% and the market rate is 6%, these bonds will sell for more than face value or a premium.
Interest paid is based on the face rate of the bonds. Since the face rate of the bond is the same as the previous example, the interest payments are still $8,000 every six months. The only thing that is different is the discount factor that will be used.
In the Present Value of $1 table, we will look up 20 payments at 3% to calculate the present value of the $200,000 payment. The factor we will use is 0.5537.
In the Present Value of an Ordinary Annuity table, we will also look up 20 payments at 3% to calculate the present value of the $8,000 interest payments. The factor used for the interest payments is 14.8775.
Now we have all the information needed to finish the problem.
The bonds will sell for $229,760. The premium on the bonds is $29,760.
By taking a moment to figure out what our answer should look like (par, discount or premium) before doing the calculation, we can determine if our answer is reasonable when calculated.
Remember, the face rate is used to calculate the amount of interest paid and the market rate is used to calculate the present value of the payments. With a bond, there are two payments. The first is the face value of the bond that will be paid once at the end of the life of the bond. For this calculation, use the Present Value of $1 table. The second is the value of the interest payments received. Use the Present Value of an Ordinary Annuity for this calculation. When calculating the present value of the interest payments, remember to multiply a single interest payment by the factor. Add the present value of the bond to the present value of the interest payments to calculate how much the bond will sell for.
Breaking down a bond issue problem
Bond issue price calculations with changing market rate
A bond is a liability companies use when a large amount of cash is needed. Rather than go to a bank or other lender, a company will issue bonds and sell them to the public. By selling bonds on the open market, the company has more control over the terms of the liability, such as interest rate and duration. Market forces still play a part. For example, if the interest rate offered by the company is too low, the public may not be interested in buying the bonds. If the market believes that the company may not pay back the bonds, the market will demand a higher interest rate.
Bonds can be traded, similar to publically traded stocks. It is not uncommon for a bond to have multiple owners before it matures because bonds typically have long maturity periods. According to the Securities Industry and Financial Markets Association, the average maturity of a corporate bond issued in December 2013 was 15 years. Typically, bonds are issued in denominations of $1,000, $5,000 or $10,000. The company determines the total amount of cash it needs to raise with the issuance. Bond certificates are printed and sold to an investment firm, also called an underwriter. The underwriter then sells the bonds to the public. Bonds are subject to the same changes in market value that stocks experience. The market value of a bond relates to the interest rate the bond is paying compared to the rate people can get on other similar investments. The market value can also fluctuate based on the market’s perception of the company’s ability to repay the bond.
A bond certificate will contain the face value of the bond. Face value is the amount that will be received at maturity. This is also called par value. It will also have the stated interest rate and the maturity date. The maturity date is the date the bonds will be repaid unless the company has the option and elects to repay them early.
The face value of a bond is not repaid until the maturity date of the bond unless the company that issues the bond chooses to repay the bond sooner. Only interest payments are made during the life of the bond. At maturity, the bond holder or buyer will receive the face value of the bond.
When a company issues bonds, it must record the amount of cash received and the corresponding liability. Recording the liability is the easiest part because the liability is always equal to the face value of the bond. To determine how much cash will be received, we need to know if the bond will sell for par value.
A bond will sell for par value if the stated interest rate is equal to the market rate. If that is the case, the company will receive cash equal to the face value of the bond.
Hill and Valley, Inc. issues $400,000 worth of 10-year, semiannual, 8% bonds on December 1. The market rate at the time of issuance is also 8%. Record a journal entry for the issuance of the bonds.
Since the stated interest rate and the market rate are the same, these bonds will be sold at face value. The journal entry for a par value bond, like this one, is fairly simple. The accounts will be Cash, to record the increase in cash, and the liability will be called Bonds Payable. The amount of the entry is the face value of the bond.
When the market rate is not the same as the stated or contract rate, the bond payable and cash will not be the same. If the market rate is higher than the stated rate, that means people are not willing to pay as much for the bonds. Either there is risk associated with the company or there are better investments elsewhere. In order to entice the public to buy the bonds, the company must offer a discount on the bonds. The company will receive less cash than face value. The difference between the face value of the bond and the cash received is called the bond discount or discount on bonds payable.
Hill and Valley, Inc. issues $400,000 worth of 10-year, semiannual, 8% bonds on December 1. The market rate at the time of issuance is 10%; therefore, the bonds will only bring $350,152. Record the journal entry for the issuance of the bonds.
In this case, the market rate is higher than the stated rate which means that the bonds will sell for less than face value.If the public can get 10% elsewhere, why would they pay full price to only receive 8%? They wouldn’t. So while the bond will pay $400,000 at the end of the 10-year term, the bond is only worth $350,152 right now (we will discuss how you calculate that number later in the material).
The difference between the amount of cash received and the liability is called Discount on Bonds Payable. This is a contra-liability, linked to Bonds Payable. Since Discount on Bonds Payable is a contra-liability, the normal balance is a debit. This makes sense because we need something to add to Cash on the debit side to balance out the $400,000 Bond Payable.
When a company offers a bond at a higher interest rate than the market expects, the public is willing to pay more for the bonds. This causes more cash to come in than the amount of the liability. In cases like this, we say that the bond sells for a premium.
Why would a company offer a bond at a premium? This can occur when the company offers a slightly higher interest rate than the market rate or when the company is so stable that it is almost certain that the creditors will be repaid. In today’s record low interest rate environment, the public is willing to spend a bit more money up front to get a better interest rate.
When a bond sells for a premium, the amount of cash generated from the sale is higher than the liability. In order to balance the journal entry, we create an account called Premium on Bonds Payable. This is an additional liability that attaches to Bonds Payable, just like a contra-account would. However, because the normal balance in Premium on Bonds Payable is a credit balance, it is not considered a contra-liability.
Hill and Valley, Inc. issues $400,000 worth of 10-year, semiannual, 8% bonds on December 1. The market rate at the time of issuance is 6%; therefore, the bonds will bring $459,512. Record the journal entry for the issuance of the bonds.
Because more cash is generated from the sale than the amount of the outstanding liability, the bonds are selling at a premium. The company will receive $459,512 in Cash but the Bond Payable is only $400,000. The amount of the premium is $59,512 (we will discuss how to calculate the premium later in the material). Cash is increasing, the Bond Payable is increasing and the Premium on Bonds Payable is increasing.
Recording Interest Payments
Most bonds pay interest on a recurring basis, typically annually or semiannually. Bonds that do not pay interest, called zero coupon bonds, are heavily discounted because the current value of a bond is based on the combined value of the interest and principal payment to be received. Since there are no interest payments, buyers look for a return on investment when they purchase the bonds. In order to get that return on investment, the bonds are heavily discounted.
Recording the interest payment on a bond is similar to the calculation used in other types of debt, except when there is a discount or premium. When there is a discount or premium, that amount must be divided up amongst all the interest payments; this is called amortization. On the date the bond matures, the amount of the discount or premium must be fully amortized, meaning that the balance in those accounts must be zero. Each time interest payment is made, a portion of the discount or premium must be included in the entry.
Recording Interest for Par (Face) Value Bonds
The cash payment for interest is calculated based on the principal balance of the bond, the face rate of the bond and the amount of time each interest payment covers. Many times you will see this referred to as:
Since the outstanding principal of a bond is not paid until maturity, the interest payment is always the same.
Hill and Valley, Inc. issues $400,000 worth of 10-year, semiannual, 8% bonds on December 31. The market rate at the time of issuance is also 8%. Record a journal entry for the first interest payment on June 30.
Because this bond was issued at par value, the interest calculation is simple. Just use I = PRT. In this case, principal is $400,000. The interest rate is 8%. Time should be expressed as a fraction of months covered by the payment over the number of months in the year. Since these are semiannual interest payments, each payment is for six months’ worth of interest. You can also look at it from the perspective that there are two payments each year, so therefore, we need to cut the annual interest rate in half.
No matter how we look at it, the time portion of the calculation is the same. Now let’s plug the information into the formula and calculate the cash payment.
Now, we must write the journal entry. The company is going to pay the interest on June 30, so we know that cash is one of the accounts. Interest is a cost of the bond, therefore it is an expense. Interest expense is the other account. Cash is decreasing and the expense is increasing.
Adjusting for interest accrued but not paid
When working with par value bonds the calculation and resulting journal entry are fairly simple. There is one catch, though. What if the bond was issued on December 1, rather than December 31?
The matching principle states that we must match revenue and expenses. Because the bond was issued on December 1, there is one month of interest that must be accrued at the end of the year. We must do an adjusting entry to record the one month worth of interest expense. Because the interest will not be paid until June 1, this also creates a payable: Interest Payable.
Hill and Valley, Inc. issues $400,000 worth of 10-year, semiannual, 8% bonds on December 1, 2013. The market rate at the time of issuance is also 8%. Record all entries related to the first interest payment on June 1, 2014
This question is a bit more open-ended than the last, because there are actually two different ways we could handle this. Both involve an adjusting entry and the entry for the payment, but one method requires a reversing entry. If the reversing entry is not done, the entry for the June 1 payment is a bit more complicated. We will run through both versions.
The cash payment on June 1 is still $16,000 because we are still discussing a $400,000, 8% semiannual bond.
The only difference is the timing of the interest expense. One month if interest falls into 2013; five months fall into 2014. The first thing we need to do is figure out the monthly interest. Because this is a six-month payment, we can divide $16,000 by six. For simplicity, we will round to the nearest whole dollar. The monthly interest is $2,667.
If the bonds were repaid on December 31, 2013, the company would be required to repay the bonds plus $2,667 in interest. To ensure the financial statements are complete and accurately reflect all activity, the company must record the $2,667 in Interest Expense. The amount will not be paid until June 1, 2014 so we will record the amount as a liability. It is due to the bondholders, which is why it is a liability.
This entry will be done whether you do the reversing entry or not. The purpose of a reversing entry is to undo an adjusting entry. Why would you undo an adjusting entry? In order to make someone’s job easier! Let’s look at this example without the reversing entry.
On June 30, we need to record the payment of $16,000 to the bondholders. Fairly simple, right? We record cash decreasing by $16,000. We also record $16,000 of interest expense — or do we? Wait, the interest expense is not $16,000 because $2,667 of interest expense was recorded on December 31. The interest expense from January 1 to June 1 is $13,333. So what about the other $2,667 needed to balance the entry? That is in Interest Payable. We need to debit the liability to show that it has been paid off.
Using reversing entries with interest payable
Now if you are the bookkeeper, are you going to remember to record the decrease in the payable? Most likely, the bookkeeper will record the entire $16,000 to interest expense which will require someone to do an adjusting entry later on to fix the error. How can we make the bookkeeper’s life easier? Use a reversing entry!
After the December 31 entry has been completed, we can do a second entry dated January 1 to undo the adjustment. Notice that the adjusting entry is done in the new year. To undo the entry, debit the payable and credit the expense.
How does this solve our problem? Well, I’m sure you can see that the reversing entry clears the payable, bring the balance to zero. What will the entry do to our expense? The expense now has a $2,667 credit balance. On June 1, the bookkeeper records the entry to record interest expense and the payment of the interest.
Let’s look at the T-account for Interest Expense.
By completing the reversing entry, we simplify the entry on June 1! Either method is fine as long as we are consistent.
Recording Interest on a Discounted Bond
With a discounted bond, there are three items that need to be handled when we do the entry for interest payments.
- Calculate how much cash will be paid. The amount of cash required is the same for all bonds with the same face rate and denomination. A $400,000, semiannual 8% bond will require the same amount of cash for the interest payment whether it is sold at par, a discount or a premium. Only the interest expense is affected by the discount or premium.
- Calculate the amount of amortized bond discount. Over the life of the bond, we must amortize or phase out the bond discount. The discount is phased out by using a straight-line approach, similar to amortization for intangible assets.
- Compute the interest expense. The interest expense for a discounted bond is equal to the cash needed for payment plus the amount of amortized bond discount.
Interest expense = Cash + reduction in bond discount
Let’s look at an example to help solidify this concept.
Hill and Valley, Inc. issues $400,000 worth of 10-year, semiannual, 8% bonds on December 31. The market rate at the time of issuance is 10%; therefore, the bonds will only bring $350,152. Record the journal entry for the first interest payment on June 30 assuming the company uses straight-line amortization.
A $400,000 bond that brings $350,152 in cash was discounted $49,848.
First, we will figure out the cash payment. This is an 8% bond where the interest is paid twice a year. Interest is calculated off the face value of the bond. Remember PRT!
This discount must be amortized over the life of the bond. Since it is a 10-year bond with semiannual payments, there are 20 interest payments over the life of the bond. We can take $49,848 divided by 20 payments or $2,492.40. This is the amount of amortization each time an interest payment is made.
Finally, we need to calculate the interest expense. Essentially, the interest expense is pulled into the journal entry. We stated earlier that interest expense is the amount of cash plus the amount the bond discount is reduced.
The three accounts are Cash, Discount on Bonds Payable and Interest Expense. Cash is decreasing so we credit the account. The normal balance in Discount on Bonds Payable is a debit (contra liability), so to reduce the account we will credit the account. Interest Expense is an expense account, so we debit the account. Now we have all the information we need to construct the journal entry.
Notice that the Interest Expense is just plugged into the entry. You cannot calculate the interest expense in a conventional way by multiplying by a percentage rate. Even if you were to look at the market rate, that would not help. The interest at the market rate would be $20,000 ($400,000 * 10% * 6/12). This is actually why companies are willing to sell bonds at a discount. The interest at market rate would be higher than the interest expense at a lower face rate plus the amortized discount.
Recording Interest on a Premium Bond
When a company sells a bond at a premium, the purchasers pay more than face value for the bonds. The premium helps to offset some of the cost of the bonds, lowering the interest expense of the bonds. The amount of cash required for all 8% bonds is the same.
The method for dealing with a bond premium is exactly the same as a bond discount.
- Calculate the amount of cash required using PRT.
- Calculate the amount of bond premium to amortize.
- Compute the interest expense. In this case, the bond premium will reduce the interest expense.
Why does the premium reduce interest expense? The amount of cash is based on the face rate of the bond. Because the bond purchasers paid extra for the bond, the company more money than the face value of the bond. That additional cash helps to offset the amount the company pays in effective interest. A portion of each cash payment is a return of the premium to the purchasers. This lowers the interest expense to the company.
Let’s run through some numbers.
Hill and Valley, Inc. issues $400,000 worth of 10-year, semiannual, 8% bonds on December 31. The market rate at the time of issuance is 6%; therefore, the bonds will bring $459,512. Record the journal entry for the first interest payment on June 30 assuming the company uses straight-line amortization.
If bonds with a face value of $400,000 bring $459,512 in cash, there is a premium on the bonds. The premium is $59,512.
Step 1 is to calculate the amount of cash required. We have a $400,000, 8% semiannual bond.
For each interest payment, Cash will decrease or be credited $16,000.
Step 2 is to calculate the amount of bond premium to be amortized. Since the company uses straight-line amortization, we will record the same amount of amortization each time interest is paid. On a 10-year semiannual bond, there will be 20 payments.
$59,512 / 20 = $2,975.60
Each time an interest payment is recorded, we will amortize $2,975.60 of premium. The normal balance in Premium on Bonds Payable is a credit. Therefore, in order to amortize or reduce the amount of the account, we must debit the account.
The last step is to compute the amount of interest expense. Interest expense is $16,000 less the amount of the amortized premium. When bond purchasers pay a premium it is as though they are offsetting some of the interest. For each payment made, $2,975.60 of the premium is returned to the purchasers which lowers the amount of interest expense for the company. The amount of interest expense is $13,024.40.
We have all the information we need to write the entry.
When working with bonds, remember that a par value bond sells for face value. If the market interest rate is higher than the face rate, the bond will sell for less than face value. The bond will be discounted. If the market interest rate is lower than the face rate, the bond will sell for more than face value. The bond will be sold for a premium.
Discounts and premiums do not affect the amount of cash paid for interest. These items do affect the amount of interest expense recorded by the company. Discounts and premiums must be amortized over the life of the bond, each time an interest payment is made. By the time the bond matures, the discount or premium should have a zero balance. A discount increases the amount of interest expense recorded by the company. A premium reduces interest expense.
Introduction to Bonds
Journal Entries for Bond Issuance