[There are again just two lectures this week, because Fri., Oct. 4 is our first midterm exam.]

This week:
I.   How changes in interest rates affect bond prices
II.  How changes in interest rates affect stock prices
III. Compounded interest rates and annual percentage yields (APYs)
IV.  Real vs. nominal interest rates
V.   Present-day value (PDV) - begin

I.  HOW CHANGES IN INTEREST RATES AFFECT BOND PRICES

Recall from last time that a discount bond's price and its interest rate have an exact inverse relation.  The higher the interest rate the bond pays, then the greater the "discount" (i.e., the lower the price) it must sell at.  This inverse relation between interest rates and bond prices is one of the key things to know about the bond market.

There is a secondary, or resale, market for bonds.  For example, if you buy a 30-year Treasury bond, do you really want to hold onto it for that long? Might need the cash sooner than that. Can sell it to someone else. The price of an old bond will be affected by interest rate on new ones.

IN THE SECONDARY BOND MARKET, WHERE OLD BONDS ARE RE-SOLD, THE PRICES OF ALL OLD BONDS ARE NEGATIVELY RELATED TO THE CURRENT INTEREST RATE.
-- Ex.: Interest rate is 10%. I just bought a $1000 one-year discount bond that pays 10% interest, i.e., the market rate.  I must have paid

------ If the interest rate falls to 5%, then I'm in luck. My bond will still pay $1000 a year from now, for a net gain of $90.91, whereas a new bond will also pay $1000 a year from now but will cost about $950 (for a smaller net gain, of about $50).  So I should be able to sell my old bond, paying 10% interest, for about $950, which would be a profit of about $41 for me.  To be precise, the going price for a new $1000, one-year discount bond paying 5% interest and the going price for my previously issued $1000, one-year discount bond should both be

--> BOND PRICES (ON THE SECONDARY MARKET) ARE NEGATIVELY RELATED TO THE CURRENT INTEREST RATE.
---- If i goes up, old bonds are worth less than before (prices of old bonds fall)
---- If i goes down, old bonds are worth more than before (prices of old bonds rise)
-----> This is why bondholders LOVE falling interest rates.
(It's also why bondholders want the government to reduce the national debt, because that should lead to lower interest rates, as shall be explained in a later unit.)

Summing up, here's how changes in current interest rates affect bond prices:
 

Bond type	Effect of an increase in the current interest rate  on the bond's price
coupon bond (new)	unchanged  (because new coupon bonds sell at face value)
coupon bond (old)	price falls in resale market.
discount bond (new)	price falls
discount bond (old)	price falls in resale market.

II.  HOW CHANGES IN INTEREST RATES AFFECT STOCK PRICES

Q: Given that stocks don't pay interest, why is it that stock prices are strongly affected by changes in interest rates?
A: Because stocks and bonds are substitutes. When people look to invest their money, they look at the expected returns of different assets, including stocks and bonds. If the interest rate (i) goes up, then more people will want to buy (new) bonds that pay those higher interest rates. So more people will buy bonds, and fewer people will buy stocks.
---- Stated more properly: If i goes up, new bonds are more attractive to potential buyers than before --> the relative return on stocks worsens, so the demand for stocks will decrease, and stock prices fall.

Stock prices are also negatively related to the current interest rate.
 

If i goes up ->	more people buy new bonds; demand for stocks falls ->	stock prices fall.
If i goes down ->	fewer people buy new bonds; demand for stocks rises ->	stock prices rise.

Another, more concise way of explaining the inverse relation between interest rates and stock prices (or old-bond prices) is with the concept of PRESENT DISCOUNTED VALUE (coming up soon).

III. COMPOUNDED INTEREST RATES AND ANNUAL PERCENTAGE YIELDS

At your bank, you might notice a listing of the rates they pay on CD's. For each CD, you'll see two rates posted: the interest rate (i) and the annual percentage yield (APY). In a better, simpler world, these two rates would always be the same, but because of the way interest is calculated, they often are not. The annual percentage yield (APY;or APR, for annual percentage rate) is the total value of the interest payments on a financial asset as a percentage of its original purchase price. The APY would be the exact same thing as the interest rate if the bank paid out the interest only at the end of the year; instead, in most cases interest is compounded (paid out) at several times during the year, e.g. monthly. Some banks have continuous compoundingof interest, whereby your account balance earns some tiny amount of interest every second of every day. If a bank compounds interest m times per year, then, instead of paying out i% interest at the end of the year, it pays out (i/m)% interest m times per year.
-- Ex.: If the interest rate on your credit-card balance is 24% and that interest is compounded every month, then, instead of being charged 24% interest on your unpaid balance at the end of the year, you'll be charged 2% (= 24%/12) interest at the end of every month, and your total interest charges will be much higher.
-- Ex.: If the interest rate on savings account is 5% and the bank compounds interest fourtimes per year, then the bank pays out 1.25% interest on those accounts at the end of every third month.
-- The greater the frequency of interest payments or compounding, the larger the total interest payments over the year. For an account holder or creditor (someone to whom money is owed), the more frequent the compounding, the better, and continuous compounding is the best. For a debtor (someone who owes money), the less frequent the compounding, the better, and continuous compounding is the worst.

For an asset with an interest rate of i and payments that are compoundedmtimes per year, the annual percentage yield (APY), or effective interest rate or compounded interest rate, is calculated as follows:

APY = [ (1 + i/m)^m - 1 ] (* 100%)

Ex.: Suppose a CD at Marine Midland Bank has an interest rate of 8%. Let us see how its APY varies with the amount of compounding.
 

i	Frequency of compounding	APY
8% (= .08)	yearly (m=1)			8%
8%	quarterly (m=4)	(1+.08/4)4 - 1 =	1.024 - 1 =	8.243%
8%	monthly (m=12)	(1+.08/12)12 - 1 =	1.0066712 - 1 =	8.300%
8%	daily (m=365)	(1+.08/365)365 - 1 =	1.000219178365 =	8.328%

IV.  ANNUAL PERCENTAGE YIELD (finish)

Compounded interest has been called the most powerful force in the world. That might be exaggerated, but the magic of compounded interest is something you should acquaint yourself with. The main way that "the rich get richer" is through compounded interest or compounded dividends. It's also a way that many middle-class people become rich.
-- Real-life ex.: A former psychology professor at SUNY-Oswego put all his pension fund contributions (TIAA-CREF) into stocks, with all dividends reinvested (works much like compounded interest), and retired a millionaire.

Q: If your boss offered to cut your salary to a penny a day but to double it every day (to 2 cents tomorrow, 4 cents the next day, etc.), would you come out ahead by accepting the offer?
A: Absolutely! That's a 100% daily interest rate. Your daily salary would increase by leaps and bounds, to $0.64 after the first week, $81.92 after the second week, over $10,000 after the third week, over $1 million by the end of the fourth week, and over $1 billion after 38 days. Well before the end of the second month, your salary would be larger than all of world GDP.

Recall from before that the annual percentage yield (APY), or effective interest rate, is a function of the nominal interest rate i and the frequency of compounding m.  In that double-your-salary-every-day example, your salary reaches such huge heights so fast because i is very large (100%) and the compounding is very frequent (daily, or m=365).

If the compounding is continuous (i.e., m = infinity), then the APY is even larger.  To calculate it, we have to find the mathematical limit of (1+i/m)^mas m approaches infinity. You could try plugging larger and larger values of m into that formula and then grinding it out on a calculator or spreadsheet, and seeing what number it approaches. If i = 1 (i.e., 100%), then the values of that formula will approach a number very close to 2.7. Fortunately, mathematicians worked this all out for us a long time ago, and they found that the limit is a number that's equal to about 2.718, which they call e. (By the way, e is the inverse of the natural logarithm; you can find it on your calculator, if you've got a good one.) With a little more math (not shown), we can get a formula for the APY for any interest rate with continuous compounding. It is:

APY = e ⁱ - 1

In the previous example of a CD that pays 8% (nominal) interest, the APY was 8.328% when the compounding was daily (m=365).  If the compounding were continuous, then we could calculate

APY = e^.08 - 1 = 1.08329 - 1 = 8.329% (not all that big a difference, but could translate into a big difference if a large sum of money is involved or if the asset is held for a long period of time)

V. REAL VS. NOMINAL INTEREST RATES

The nominal interest rate i does not account for the impact of inflation.  The REAL INTEREST RATE is the inflation-adjusted interest rate, and it is the one that matters.

NOTATION:
i  = nominal interest rate (the posted interest rate)
i_r = real interest rate
p = inflation rate
p = expected inflation rate

ex ante (or expected) real interest rate =  i_r^e= i - p^e
-- this is the real interest rate that drives investment by firms in new plant and equipment

ex post (after-the-fact) real interest rate = i_r= i - p
-- ir indicates the real (inflation-adjusted) cost of borrowing and lending

Q: What is the real interest rate (ex ante) on a 1-year bond today that pays 6% interest, if the expected inflation rate over the next year is 2%?
A:  i_r^e = 6% - 2% = 4%

Q: What is the ex post real interest rate on that bond if inflation turns out to be 5% over the next year?
A:  i_r = 6% - 5% = 1%

If inflation rises,
* borrowers (debtors) benefit (their debts become a lot easier to pay off, and less burdensome)
* lenders (creditors, savers) lose (the real value of their savings shrinks, and the real interest rates they receive are reduced and could even become negative)
-- "Inflation is when people who have saved for a rainy day get soaked"

Looking only at nominal interest rates can be misleading. In the late 1970s, for example, nominal interest rates were very high, yet real interest rates were actually negative, because inflation was high and accelerating.

Two recent financial innovations, designed to protect lenders from the ravages of inflation, are (1) variable-interest-rate mortgages, on which the interest rate, rather than being fixed (as on a regular mortgage), is adjusted periodically as the inflation rate or other interest rates change; and (2) inflation-indexed Treasury bonds, which guarantee the bondholder a certain positive real interest rate.

VI.  PRESENT-DAY VALUE (PDV) - begin

Present-day value, present value, and present discounted value all mean the same thing:  what some future payment or set of future payments is worth to you today, i.e., right here, right now.  The essence of this very important concept is that a dollar in the future is worth less than a dollar today, because if you had that dollar today, you could invest it and earn interest on it and end up in the future have more than just a dollar.  (This concept is sometimes called the time value of money.)

Breaking it down, word for word:
PRESENT DISCOUNTED VALUE
  |                    |                                  |
right now      "discounting" for            what it's worth
                    the way that interest
                    reduces the value of
                    future payments relative
                    to current payments

Or, in a nutshell:
PRESENT-DAY VALUE = what it's worth now

The present-day value (PDV) of any future payment will depend on three things:
* The size of the payment positively affects its PDV -- a million dollars tomorrow is still better than $100 today.
* The higher the current interest rate, the lower the PDV of any future payment.  The higher the interest rate, the greater the value of $1 today relative to $1 in the future.  If the interest rate is very high, you would forgo a lot of interest if you said, "Sure, I'll give you $100 today, and when you give me $100 four years from me now, that'll be just as good."  (It would be if the interest rate were zero, there were no inflation, and you were very patient, but even if there's no inflation and you have infinite patience, as long as i > 0%, then the PDV of $1 in the future is less than $1.)
* The longer you wait for the payment, the lower its PDV (the more it has to be "discounted").  As long as you have to wait at all for the payment, it's not worth as much as having the same amount of money right now.  (The character named Wimpy in the Popeye cartoons who says "I'd gladly pay you Tuesday for a hamburger today" is a man who understands the concept of PDV.)

The basic present-value formula should look very familiar to you, because it's the same as the formula for the price of a discount bond.  Let FV be the Face Value of the discount bond or the amount of a future payment (i.e., its Future Value), and recall that i is the market interest rate and n is the number of years between now and the date of the payment, and the formula is

If you are trying to calculate the PDV of a set of future payments to be made at different times, then you need to compute this formula for every single payment and then add all those PDV's up.  (See problem #5.a. on "Solutions to Problem Set 3" for an example.)