[Note:  This is an unusually short set of notes, since we had an exam this week and since I canceled Tuesday's class, and about 35 minutes' worth of Wednesday's, on account of illness.  Eventually I'll hand out some makeup lecture notes.]

[These lectures go with Chapter 4 and the beginning of Chapter 5 of Mishkin's textbook. Some of the material in these lectures, notably the PDV shortcuts, is not in the textbook, however.  For maximum mileage you should use them in conjunction with the handout, "Everything You Always Wanted to Know About PDV But Were Afraid to Ask," which was distributed in class last week.]

* This week:
I.   PDV shortcuts and applications
II.  Yield to maturity

I.  PRESENT-DAY VALUE (PDV): SHORTCUTS AND APPLICATIONS

For something that pays a fixed amount over a finite time span -- like, say, a 10-year, $10-million-a-year NBA contract -- finding the PDV the standard way would involve adding up 10 separate terms and would be very time-consuming. There is, however, a shortcut, which makes use of that consol formula. Let us denote those fixed yearly payments as FP, for fixed payment. The PDV of a payment of $FP over n years is:

PDV = FP/(1+i) + FP/(1+i)² + ... + FP/(1+i)ⁿ
-- (formula (2) on PDV handout)

That's the time-consuming way to compute it. We could, however, just calculate the PDV of an n-year series of fixed payments (FP) as the difference between the PDV of a consol that pays $FP per year forever and a consol that pays $FP per year starting n+1 years from now.

A consol that makes a fixed payment (FP) forever is worth

A consol that pays FP forever, starting n years from now, is worth

Take the difference of those two and you've got the PDV of receiving $FP every year for the next n years:

Note: Many spreadsheet programs, like Excel and Quattro Pro, will calculate PDV for you. Ditto for fancy financial calculators. Be careful, however, when using them, because it's easy to make mistakes. In solving PDV problems, it's better to write out the correct formula and plug the numbers ($FP, i, and n) into it -- i.e., do it "manually" -- or at least to check your spreadsheet-calculated PDV against a "manual" estimate.)

Ex. A college hoops star signs a 10-year, $10-million-a-year NBA contract, starting one year from now. Total payments add up to $100 million, but PDV will be considerably less, since all of those payments are in the future. What is the PDV (if i=.05)?

PDV = ($FP)/i * [ 1 - 1/(1+i)ⁿ]
         = $10/.05 * [ 1 - 1/(1+.05)¹⁰] (in millions)
         = $200 * [ 1 - 1/1.6289] (in millions)
         = $200 * (1 - .6139) (in millions)
         = $200 * .3861 (in millions)
         = $77.2 million

PDV Application #1: BONDS

In comparing long-term and short-term bonds, we already know that long-term bonds are less liquid. They also have greater risk, because the price of a long-term bond is more volatile than the price of a short-term bond, because its PDV is more affected by changes in i (has greater interest-rate risk).

-- Ex.: Compare two discount bonds, both with a face value of $1000, but the first a 1-year bond and the second a 30-year bond. Let the current interest rate be i = 10% = .10.

The price of each bond is its PDV, which is equal to (Face Value)/[(1+i)ⁿ]:

PDV of 1-year bond = $1000/[1.10¹] = $1000/1.1 = $909.09
PDV of 30-year bond = $1000/[1.10³⁰] = $1000/17.4494 = $57.31

Now suppose that after you've bought one of each, the interest rate suddenly doubles, to i = 20% = .20. See how the PDV's, or the resale prices, of the two bonds change:

PDV of 1-year bond = $1000/[1.20¹] = $1000/1.20 = $833.33
--> the 1-year 10% bond is now worth about 92% ($833.33/$909.09) of its original value

PDV of 30-year bond = $1000/[1.20³⁰] = $1000/237.3763 = $4.21
--> the 30-year 10% bond is now worth only about 7% ($4.21/$57.31) of its original value

Q: WHY is the loss so much greater on a long-term bond than on a short-term bond, when i rises?
A: You lock in that low, original interest rate for a much longer period of time. Not only are your annual interest payments lower, but all of your forgone interest could have been compounding over that time.

PDV Application #2: STOCKS

If stocks and bonds carry equal risk, then the PDV of any stock is the present-day value of all future dividend payments associated with the stock. According to fundamental analysis of stock prices (a technique that Warren Buffett says he swears by), the proper price for a stock is its estimated PDV (just as the proper price of a bond is the PDV of all the payments it will be making). The PDV of a stock's entire future dividends is often called the stock's intrinsic value.
-- Many of today's hottest stocks do not currently pay any dividends. If people expected them never to pay any dividends, not even in the distant future, then the PDV's, and hence the "proper" prices, of those stocks would be zero. The fact that stocks like Dell and Microsoft, which do not pay dividends, command high prices on the market suggests that people expect them to start paying dividends sometime in the future. (And it's up to the shareholders, on the company's Board of Directors, to determine when that will be.)

-- For a stock whose yearly dividend is expected to stay the same forever (a very stable company that is already paying dividends - e.g., a utility company), we can find the PDV by applying the formula for the PDV of a consol bond (formula [3] on the PDV handout):
                           PDV_stock = (Annual dividend payment)/i
--> (PDV_stock )/(Dividend) = 1/i (i.e., price-dividend ratio is 1/i)

-- For a stock with projected dividends that are expected to grow over time, the intrinsic value (PDV) of the stock would be greater than (Dividend)/i, and hence the price-dividend ratio would be greater than 1/i.

Q: Why might the PDV/ intrinsic value approach not be such a good way to price stocks?
A: Because nobody knows for sure what a company's future earnings and future dividends will be. The PDV of a share of a stock, or of any asset whose future returns are not known with precision, is just an estimate, and so the intrinsic value approach is only useful if you can accurately predict the company's future earnings. Most of us can't do that very well ("I don't have a crystal ball.")
-- Additional caveats about the PDV approach to stock pricing:
---- Stocks and bonds do not carry equal risk.  Stocks are riskier, because the amount of the future dividend payments is not fixed (unlike the interest and principal payments on a bond) and because in the event of bankruptcy the firm must pay off its bondholders and other creditors before it can divide up its assets among its stockholders.
---- Even if you know a stock's price is way in excess of its PDV (and hence the stock is overpriced), there is still the possibility of selling the stock at a profit, if you can sell it before the stock-market bubble bursts.  That approach to the market is often called noise trading.  (It's potentially the way to get rich the fastest in the stock market, but it's also very risky, and many more people get burned by this strategy than get rich by it.  A "buy-and-hold" strategy is much safer.)

II. YIELD TO MATURITY

Mishkin's textbook says the most important measure of the interest rate is the yield to maturity. The book defines the yield to maturity on a bond as the interest rate that equates the present value of payments received from a bond from its price today. In plain English, it's the effective interest rate on a bond as determined by the bond's future payment schedule and its price. Returning to those PDV formulas, the yield to maturity is what you'd get if you already knew the bond's PDV (or price) and the entire schedule of future payments and then solved for i.

In the simplest cases, the yield to maturity and the interest rate are the same thing. These cases include:
* new discount bonds
* new consol bonds
* new coupon bonds that sell at their face value.

Looking at discount bonds we can see how PDV and yield to maturity are opposite sides of the same coin. A discount bond that pays $FV in n years should sell for its PDV, so:

To find the bond's yield to maturity when you already know its price and face value, just solve for i:

This is the same as the formula for the interest rate on an n-year discount bond, which we saw in an earlier lecture. Conceivably the bond's yield to maturity could be higher or lower than the market interest rate.

To generalize: the YIELD TO MATURITY on an asset is the value of i that would equate the price of the asset with the PDV of that asset's future stream of payments.
-- A bond that is priced properly, so that the price equals the PDV of the payment stream, will have a yield to maturity equal to the market interest rate.

In the case of a coupon bond, if the bond is sold for less than or more than its face value, or if the payments are irregular, then the yield to maturity and the interest rate won't be the same thing.
-- If, based on the market interest rate, a bond is selling for less than its PDV (which means the bond is underpriced or a bargain), then

In such a case, finding the yield to maturity is more complicated. Again you would need to solve for i, but you'd need the help of a computer (a spreadsheet, maybe) to do it.