All Terrain Thinking

A Compendium of things I think are Important

Algebra: Calculating why it's not in your wallet

In today's world it is very important to understand some fundamentals of personal finance. More specifically, it is important to know what financial instruments are available when you are want to borrow money and when you have some money to save. Stocks and Bonds are two very important financial instruments in the US economy. To understand bonds, it is essential that you first understand present value analysis. For this reason I have provided the following simple examples. Although there are many different types of bonds, they all can be considered as simply IOUs and therefore the examples which we will be looking at could easily be expanded to consider more realistic examples.

For most of you, the bond that you are most likely to be familiar with is the US Savings Bond. In fact, many of you may have some $25 bonds which were gifts at those important times in your life. The reason that this bond is such a great gift is that the bond says that it is worth $25 dollars, but in truth the aunt who gave it to you paid much less than $25 and you will not be able to get the $25 until sometime in the future.

Example 1: Savings Bond

To see how this works consider the case of a ten year bond. Also assume that the interest rates are 8%. This bond is issued by the US government as a way of raising money to pay its bills today, while promising to pay you $25 in 10 years. The question is: How much should you pay them for the guarantee of $25 in ten years? Stated somewhat differently, how much do you need to put away today and invest at the given interest rate (8%) so that in ten years you will have $25? This is obviously a PV problem as you are given the values for T(10), g(.08), and FV(25). The solution to the problem is:

You should pay no more than $11.58 for the promise to receive $25 ten years from now because this is what you would have to invest today to accumulate $25 ten years from now.

Example 2: Treasury Note

For a more realistic example, consider a three year, $1,000 bond at an interest rate of 10%. You would give the government your $1,000 today and it would pay you the $100 (10% interest) at the end of this year and the next two years. At the end of the third year it would also give you back the initial $1,000. In other words, you would be paying $1,000 to receive $100, $100, and $1,100 at the end of the next three years. My question to you is: Why is the price of this bond equal to $1,000?

The reason that the bond's price is $1,000, is that the $1,000 is the present value of the stream of future payment ($100, $100, $1100). Because a $ today is not worth the same as a dollar next year (would you trade me a $100 today for a $100 one year from today?), you need to calculate the present value for each year and then add up these three present value figures for the bond's present value (value today = price today).

What happens to the price of your bond if interest rates change?

To help you understand the relationship between interest rates and bond prices you can think of this bond as a machine which you just purchased that prints money at the rate of 10% a year. This is an asset, just like a home, which has a price. If interest rates never changed then this would be the end of the story, your money machine would retain its value of $1,000.

But interest rates do change. What happens to the price of the bond if interest rates change? To answer this, one need only redo the present value calculations with a different interest rate. As you can see below, there is a negative relationship between the interest rate and the price of the bond. When interest rates fall to 8%, those individuals owning the 10% bonds will see the value of their bond increase. The reason for this is that at an 8% interest rates, money will not grow as fast as at 10%. Your bond pays 10% so you need to put away $1,000 to get $100 interest in one year. For someone earning 8%, meanwhile, they will need to invest more to end the first year with $100. For example, at 10% you needed to invest $90.90 to earn $100 in one year, but if the interest rate falls to 8%, you will need to invest $92.60 to reach $100 in one year. Looked at a little differently, a person buying a bond today for $1,000 would get $80 at the end of the year ($1,000*.08) so they have a money machine that can print money at a rate of 8%. Because your machine prints it at the higher rate of 10%, your machine will be worth more than the new machines. I will leave it for you to explain what happens when interest rates rise.

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