Compound Interest Continuously on Ti 84 Calculator

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One of the key variables in choosing any investment is the expected rate of return. We try to find assets that have the best combination of risk and return. In this section we will see how to calculate the rate of return on a bond investment. If you are comfortable using the TVM Solver, then this will be a simple task. If not, then you should first work through my TI 83/TI 83 Plus or TI 84 Plus tutorial.

The expected rate of return on a bond can be described using any (or all) of three measures:

• Current Yield
• Yield to Maturity
• Yield to Call

We will discuss each of these in turn below. In the bond valuation tutorial, we used an example bond that we will use again here. The bond has a face value of $1,000, a coupon rate of 8% per year paid semiannually, and three years to maturity. We found that the current value of the bond is $961.63. For the sake of simplicity, we will assume that the current market price of the bond is the same as the value. (You should be aware that intrinsic value and market price are two different, though related, concepts.)

The Current Yield

The current yield is a measure of the income provided by the bond as a percentage of the current price:

There is no built-in function to calculate the current yield, so you must use this formula. For the example bond, the current yield is 8.32%:

Note that the current yield only takes into account the expected interest payments. It completely ignores expected price changes (capital gains or losses). Therefore, it is a useful return measure primarily for those who are most concerned with earning income from their portfolio. It is not a good measure of return for those looking for capital gains. Furthermore, the current yield is a useless statistic for zero-coupon bonds.

The Yield to Maturity

Unlike the current yield, the yield to maturity (YTM) measures both current income and expected capital gains or losses. The YTM is the internal rate of return of the bond, so it measures the expected compound average annual rate of return if the bond is purchased at the current market price and is held to maturity.

In the case of our example bond, the current yield understates the total expected return for the bond. As we saw in the bond valuation tutorial, bonds selling at a discount to their face value must increase in price as the maturity date approaches. The YTM takes into account both the interest income and this capital gain over the life of the bond.

There is no formula that can be used to calculate the exact yield to maturity for a bond (except for trivial cases). Instead, the calculation must be done on a trial-and-error basis. This can be tedious to do by hand. Fortunately, the TI 83 Plus and TI 84 Plus have the TVM Solver, which can do the calculation quite easily. Technically, you could also use the IRR function, but there is no need to do that when the TVM Solver is easier and will give the same answer.

To calculate the YTM, go to the Finance menu and bring up the TVM Solver. We can find the YTM by solving for I%. Enter 6 into N, -961.63 into PV, 40 into PMT, and 1,000 into FV. Now, scroll up to I% and then press ALPHA ENTER. You should find that the YTM is 4.75%.

But wait a minute! That just doesn't make any sense. We know that the bond carries a coupon rate of 8% per year, and the bond is selling for less than its face value. Therefore, we know that the YTM must be greater than 8% per year. You need to remember that the bond pays interest semiannually, and we entered N as the number of semiannual periods (6) and PMT as the semiannual payment amount (40). So, when you solve for I% the answer is a semiannual yield. Since the YTM is always stated as an annual rate, we need to double this answer. In this case, then, the YTM is 9.50% per year.

So, always remember to adjust the answer you get for I% back to an annual YTM by multiplying by the number of payment periods per year.

The Yield to Call

Many bonds (but certainly not all), whether Treasury bonds, corporate bonds, or municipal bonds are callable. That is, the issuer has the right to force the redemption of the bonds before they mature. This is similar to the way that a homeowner might choose to refinance (call) a mortgage when interest rates decline.

Given a choice of callable or otherwise equivalent non-callable bonds, investors would choose the non-callable bonds because they offer more certainty and potentially higher returns if interest rates decline. Therefore, bond issuers usually offer a sweetener, in the form of a call premium, to make callable bonds more attractive to investors. A call premium is an extra amount in excess of the face value that must be paid in the event that the bond is called.

The picture below is a screen shot (from the FINRA TRACE Web site on 8/17/2007) of the detailed information on a bond issued by Union Electric Company. Notice that the call schedule shows that the bond is callable once per year, and that the call premium declines as each call date passes without a call. If the bond is called after 12/15/2015 then it will be called at its face value (no call premium).

It should be obvious that if the bond is called then the investor's rate of return will be different than the promised YTM. That is why we calculate the yield to call (YTC) for callable bonds.

The yield to call is identical, in concept, to the yield to maturity, except that we assume that the bond will be called at the next call date, and we add the call premium to the face value. Let's return to our example:

Assume that the bond may be called in one year with a call premium of 3% of the face value. What is the YTC for the bond?

Return to the TVM Solver. In this case, the bond has 2 periods before the next call date, so enter 2 into N. The current price is the same as before, so enter -961.63 into PV. The payment hasn't changed, so enter 40 into PMT. We need to add the call premium to the face value, so enter 1,030 into FV. Solve for I% and you will find that the YTC is 7.58% per semiannual period. Remember that we must double this result, so the yield to call on this bond is 15.17% per year.

Now, ask yourself which is more advantageous to the issuer: 1) Continuing to pay interest at a yield of 9.50% per year; or 2) Call the bond and pay an annual rate of 15.17%. Obviously, it doesn't make sense to expect that the bond will be called as of now since it is cheaper for the company to pay the current interest rate.

Make-Whole Call Provisions

The above discussion of callable bonds assumes the old-fashioned type of call. However, for the last 15 years or so, corporations have typically used a "make-whole" type of call. To learn about those, please see my tutorial for make-whole call provisions.

I hope that you have found this tutorial to be helpful.

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Source: http://www.tvmcalcs.com/index.php/calculators/apps/ti_83_bond_yields

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