present value of an annuity

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.

  1. You are not receiving anything for giving up your money for 10 years.
  2. $40,000 will probably be worth less than $40,000 in 10 years because of inflation.
  3. 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.

Example #1:

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.

pv1

We now have the two components:

pv2

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.

pv3

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.

Example #2:

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.

pv4

The bonds will sell for $229,760. The premium on the bonds is $29,760.

Final Thoughts

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.

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