The Concept of Present Value


 Aubrie Bell
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1 The Concept of Present Value If you could have $100 today or $100 next week which would you choose? Of course you would choose the $100 today. Why? Hopefully you said because you could invest it and make more money. Therefore $100 today will be worth more to you than $100 next week. Let s say that you invested that $100 for 1 year at 7% interest. How much would you have at the end of the first year? Obviously $107. But let s look at that the opposite way. If I wanted to have $1,000 in one year and I knew that I could earn 10% interest, how much would I need to invest today for it to grow to $1,000 in a year? We could work this out algebraically: X +.10X = $1, X = $1,000 X= So if I invested $ today and I could earn 10% interest, it would grow to $1000 in one year. This is the concept of present value. What if instead, we wanted to receive $1000 at the end of two years? How much would we need to invest? In order to figure this one out, you would need to compute the present value at the end of the first year (we did that) and then take that amount and divide it again by 1.10 (step two in the above algebraic equation). It would be $ (909.09/1.10). How would this change if it were three years. You would then take the $ and divide by 1.10.As you can see this can get very complicated. That is why accountants have come up with tables to help us out with this. In appendix A of your textbook, you will find several present value tables. We will be using these tables throughout the rest of this module. What these tables represent is compounding interest. Turn to Appendix A at the back of your book and take a look at the present value tables. After taking a look at those tables you are probably asking why we would even want to know about present value, much less learn how to use all of those numbers. Actually the present value concept is used everywhere. Actuaries use it to determine how much money must be invested into a retirement plan in order to have a certain benefit payout when you retire. The lottery uses it also. Think about how lotteries operate. If a lottery representative ended up at your doorstep tomorrow, he would probably ask you
2 if you wanted to take the $10,000,000 today or if you would rather have $1,250,000 every year for the next 10 years. They know that if you chose to take the payments, they can invest that $10,000,000 over the next 10 years and make more money than the $1,250,000 that they are paying you per year. That s why they give you that option. The main reason we are considering it in this class, however, is that present value is used to compute the selling price of bonds. When a corporation issues a bond, the price that buyers are willing to pay is based upon three factors: 1. The face amount of the bonds. 2. The contract rate of interest (this is the interest rate that is found on the face of the bonds, the interest rate the issuer is promising to pay the buyer). 3. The market rate of interest (the interest rate that investors can currently get on other investments in the marketplace). This is also called the effective rate of interest. Computing the Price of a Bond Computing the present value of a lump sum: For starters, let s get used to working with the present value tables at the back of your book. If you look at the first table, the heading should say "Present Value of $1". This is the table that you will use when you want to figure the present value of a lump sum. For example, what is the present value of receiving $100,000 in 10 years from now at 7% interest? First look at how to read the tables. Across the top you will see interest rates starting with 5% and going up to 14%. Down the left hand column you will see the numbers These represent the number of periods. So to compute the present value of the above example, find the factor that corresponds to 10 periods at 7% interest. You should come up with In order to get the present value, you will take that factor and multiply it by $100,000. This should compute to $50,835 ($100,000 x.50835). That is how much you would need to invest today for it to grow into $100,000 in 10 years. Computing the present value of an annuity: So far we have been using what is called a lump sum, a single amount of money to be received all at one time. In order to compute the selling price of a bond we must also look at something called an annuity. An annuity is a series of equal payments at equal intervals, such as benefits from a pension plan or interest paid on a bond. Present values can also be computed on an annuity, but you must use a different table than the one used for a lump sum. Go to the tables in your book and look for the table with the heading "Present Value of Ordinary
3 Annuity". Anytime you want to find the present value of a series of payments, you will use this table to compute it. Let s look at an example of computing the present value of an annuity. Assume you won the lottery with a $5 million prize. You can take a lump sum now or have your winnings paid out at $500,000 per year for 10 years. If you take the lump sum, you can only have $3 million. If market interest rate is 11%, which option would you choose? In order to figure this out you need to put the annual payments on the same playing field as the lump sum. As it stands now, we would be comparing apples to oranges. The way to do this is to figure out what the present value of the annual payments is. Go to the Present Value of Ordinary Annuity table and locate the factor that corresponds to 11% and 10 periods. You should have found the factor Multiply that factor times the annual payment of $500,000 and the answer is $2,944,615. So, if you were given the option, you should take the lump sum of $3 million because the annual payments are really only worth $2,944,615 today. How would the above scenario change if you could only earn 6% interest? Try it and see what you come up with. Answer: $3,680,045. So in this case, you should take the annual payments because they are worth more than $3,000,000. Bond Pricing The reason that we have looked at both a lump sum and an annuity is because both of these will affect the price that a buyer is willing to pay for a bond. The selling price of a bond is made up of two computations: (1) the present value of the face amount and (2) the present value of the periodic interest payments. This should look familiar because (1) represents the lump sum and (2) represents the annuity. Both of these added together will determine the price at which a bond will sell. Premiums and Discounts Let s go back and look at interest rates. A bond always carries a specific interest rate called the contract rate of interest. This is the percentage that a bond issuer would use to compute the interest payments that will be made to the bondholder. That contract rate of interest may be different than what the current market rate for interest. When this occurs, the bond will sell at either a premium or a discount. Let s look at an example. If Tidal Inc. issues a bond that carries a contract rate of interest of 10% and the current market rate of interest is 12%, then the bond will sell at a discount. Why is this? If an investor can make 12% interest on an investment elsewhere in the market, the investor is not going to be
4 willing to buy the 10% bond unless some incentive is offered. That incentive comes in the form of a discount from the face value of the bond. For example, if the bond has a face value of $1,000, then the investor might be willing to pay $940 for it since the interest rate is lower. The same thing happens with a premium. If the bond issuer has a bond that carries an interest rate of 10% and the current market rate of interest is 8%, then investors are going to want to purchase the 10% bond. They will also be willing to pay more than its face value. Why? Because the investor can earn 10% on the bond, but can only earn 8% in the marketplace. With a face value of $1,000, the investor might be willing to pay $1,050 for the bond. What happens in the event that the contract rate of interest is the same as the market rate of interest? The bond sells at its face value and there is no premium or discount. Putting it all together computing the selling price of a Bond. There are several steps that you will take to figure the selling price of a bond. The steps are as follows: 1. Determine the number of periods and interest rate that you will be using to find the factor in the present value tables. This is where it gets confusing, so hang in there. The number of periods you will use depends upon how often interest is paid on the bond. When using the present value tables, you are looking up how many interest periods there will be over the term of the bond. Most interest is paid semiannually on bonds. What this means is that there are two interest payments every year. So if the bond that you are pricing has a 10 year term and pays interest semiannually, then there will actually be 20 interest periods (10 years X 2 interest payments per year). So when looking up the factor in the present value tables, look for 20 periods. The interest rate can also be confusing. You have two different interest rates to contend with the contract rate of interest and the market rate of interest. Which interest rate do you use to look up the factor in the present value table? Always use the market rate of interest when looking up the factors. We will use the contract rate of interest later. Okay, now to confuse you even further. Because the interest payments are paid semiannually, you will need to divide the market rate of interest in half. Why? Look at it this way. If you are paying interest payments at 10% interest but you are making two payments per year, then each payment will be 5%. So, if you were given this information: You are issuing a 10 year, $1,000 bond with a contract rate of interest of 10%, payable semiannually and the market rate of interest is 12%. What number of periods and what interest rate would you look for to find the factor in the present value tables? 10 year bond X 2 payment per year = 20 periods 12% market rate of interest / 2 = 6% So, the factor that you would look for on the tables would correspond to 20 periods at 6% interest. 2. Go to the present value of $1 table and find the factor that corresponds to the appropriate periods and interest rate.
5 3. Multiply that factor times the face value of the bond. This will give you the present value of the face amount. 4. Determine the periodic interest to be paid on the bond. This is where you will use the contract rate of interest. This is the only time you will use this interest rate. Multiply the face amount of the bond by the contract rate of interest. This will give you the annual interest payment. But remember, you are paying interest twice a year, so you will need to divide the annual interest payment by Go to the present value of an ordinary annuity and find the factor that corresponds to the interest rate and periods that you figured in step 1. Make sure you are using the correct table. 6. Multiply this factor by the interest payment that you calculated in step 4. This will give you the present value of the interest payment. 7. Compute the selling price of the bond. Add the present value of the face amount from step 3 to the present value of the interest payment in step 6 and you have the selling price of the bond. It seems very confusing, but it really does get a lot easier once you ve done a few of them, so let s do some examples. A. Compute the selling price of a $100,000 bond due in 5 years, paying 12% interest when the market rate of interest is 14%, interest paid semiannually. Steps: 1. The number of periods will be 10 (5 years X 2 interest payments per year) and the interest rate to use on the tables will be 7% (14% market rate / 2). 2. The factor from the present value of $1 table is The present value of the face amount will be $100,000 x = $50, The annual interest payment will be 100,000 x 12% = $12,000 $12,000 / 2 = 6,000 semiannual interest payment 5. The factor from the present value of an ordinary annuity is The present value of the interest payment is 6,000 X = $42, The selling price of the bond is $50,835 (step 3) + $42,141 (step 6) = $92,976 An easy way to check your answer is to determine whether the bond should sell for a discount or a premium. Remember, if the market rate of interest is greater than the contract rate of interest, the bond will sell at a discount. In this example, the market rate of interest is 14% and the contract rate is 12%. This means that in order to entice investors into buying this bond, it will have to be sold at a discount, or less than $100,000. We know that we are at least close to the selling
6 price because we determined that the price is lower than $100,000 just as it should be. Let s do another example: B. Compute the selling price of a $100,000, 4 year bond, paying 12% interest when the market rate of interest is 11%, interest paid semiannually. The first thing to do is to determine if the bond will be selling at a premium or a discount. If the market rate of interest is lower (11%) than the contract rate of interest (12%) than an investor would be willing to pay more than face value for the bond. In this scenario, the selling price will be greater than $100,000. Steps: 1. The number of periods will be 8 (4 years X 2) and the interest rate will be 5.5% (11% market rate / 2). 2. The factor that corresponds to 8 periods and 5.5% interest is Multiply the face value of $100,000 times and you get $65, Compute the periodic interest payments to be paid every six months. 100,000 X 12% contract rate of interest = $12,000. But it will be paid twice a year so each payment will be $6, The factor that corresponds to 8 periods and 5.5% interest in the present value of an ordinary annuity table is Multiply the interest payment of $6,000 times and you get $38, The selling price for the bond will be $65,160 (from step 3) plus 38,007 (from step 6) which equals $103,167, which is the selling price of the bond. Check yourself by determining if the bond sold for a premium as we discussed earlier. The selling price is greater than the face value of $100,000 so it did sell at a premium. Journal Entries for Bonds Payable Once the selling price of the bond has been determined, a journal entry must be made in the books of the issuing corporation. Let s use the above bond problem II to prepare a journal entry. The journal entry will consist of a debit for the amount of the selling price, which will be cash. It will also include a credit for Bonds Payable, which will be the face amount (the amount that the corporation has promised to pay back to the bondholder). If the bond sold for a premium or a discount, this must also be reflected in the journal entry. If the bond sold at a discount, you will make a debit entry to "Discount on Bonds Payable". If the bond sold at a premium, you will make a credit entry to Premium on Bonds Payable".
7 Using bond problems above, the journal entry to record the issuance of the bond that sold for $103,167 would be: Cash 103,167 Premium on bonds payable 3,167 Bonds Payable 100,000 What would the journal entry be to record the issuance (sale) of a $100,000 bond that sold for $96,220? Cash 96,220 Discount on Bonds Payable 3,780 Bonds Payable 100,000 (Hint, Cash will always be debited for the selling price and Bonds Payable will always be credited for the face value of the bond, the difference between the two will be a discount or a premium) Journal Entry for Periodic Interest Payment After the bond has been issued, interest must be paid every six months to the bondholder. When you determined the selling price of the bond, you already determined how much interest you are going to pay to the bondholders every six months (it was step 4). But if the bond sold for a discount or a premium, that must also be accounted for somehow every six months. There is a reason for this. When a corporation sells a bond at a discount, they are receiving less cash than they will need to pay back at the end of the bonds term. This discount amount is really just additional interest that has been paid up front to the buyer of the bond. A premium on the other hand is where the corporation selling the bond receives more cash than they will need to pay back at the end of the bonds term. This premium is considered a reduction in the amount of interest that the corporation will be paying. Bond discounts and premiums must be amortized over the life of the bond. There are two methods that can be used to do this: (1) effective interest rate method, and (2) straightline method. We will be using the straightline method. Amortization by the straightline method is like depreciation by the straightline method. You take the total amount of the discount or premium and divide that by the number of interest periods in the bond s term.
8 For example, if there was a discount of 10,000 on a 5 year bond, paying interest semiannually, there would be 10 interest periods. So every 6 months 1/10 of the discount or 1,000 would be amortized. This amortization is included in the journal entry to record the semiannual interest payment. Using bond problem above for selling a bond at a premium, we calculated that our semiannual interest payment was $6,000. We also determined that $3,167 (rounded) was the premium. Because there are 8 interest payments, we will divide the $3,167 by 8 = 396 (rounded). So every six months over the life of the bond, a journal entry will be made to record the interest payment and the amortization of the premium. The journal entry will be: Interest Expense (6, ) 5604 Premium on Bonds Payable 396 Cash 6,000 The easiest way to prepare this entry is to work from the bottom up. We know for sure that every six months we will be sending a cash interest payment of $6,000 to the bondholder. We also know how much of the premium we will be amortizing every 6 months. Notice that the "Premium on Bonds Payable" is now debited. When we made the initial entry for the Premium when we sold the bond, it was a credit entry. It is now debited for two reasons: (1) it is reducing the amount of interest expense, and (2) by amortizing the premium every six months we are reducing the credit balance. By the end of the bond s life, the balance in the premium account will be zero. The interest expense debit will be the amount you need to make the journal entry balance. The above entry will be made 8 times over the life of the bond. Using the problem above for selling at a discount, the journal entry for interest payments would be: Interest Expense (6, ) 6,378 Discount on Bonds Payable ($3,780 / 10 periods) 378 Cash 6,000
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