Financial Market Theory Tuesday, September 18, 2018 Professor Edwin T Burton

Calculating Rates of Return, Yields, Etc. Stocks: calculating returns including capital gains and dividends Bonds: calculating yields, annualized yields, yield-to-maturity, duration September 18, 2018

Capital Gains (and Losses) (Must specify a specific period of time) Pt is the price of the stock at date t Pt+1 is the price of the stock at date t+1 Capital Gain (or Loss) = Pt+1 - Pt September 18, 2018

Dividends Many stocks do not pay any dividends (tech stocks, especially) Stocks that pay dividends, pay them quarterly (every three months) Overall dividend yield (annual dividend divided by stock price) is approximately 2 percent in recent years September 18, 2018

Total Annual Return (for stocks) Capital Gain (or Loss) Plus Dividend Pt+1 – Pt + Dividends (during the year) This is usually given as a percentage of (beginning of period) stock price: (Pt+1 – Pt + Dividends) divided by Pt The above formula is known as the “rate of return” for stocks (annual) September 18, 2018

Calculating Treasury Bill Yields Assume that the principal amount of the treasury bill is $ 1 million Begin with the “quote” or discount from principal of the treasury bill Suppose the treasury bill is quoted at 10 percent How long until maturity? If maturity is 365 days away, then the current price of the treasury bill is $ 900,000 If maturity is “X” number of days away, then the current discount is (X/365 times $ 1 million) For a 90 day bill, X = 90 days; Discount is 10 percent times 90/365 times $ 1 million = $ 24,658. Thus price is $ 1,000,000 minus $ 24,658 = $ 975,342 Yield is equal to $ 24,658 divided by $ 975,342 = 2.5169 percent Annualized yield = 2.5169 % times (365/90) = 10.207 percent Note: Mistake in Chapter 6, page 2 in the example. It should read a price of $ 990,000, not a price of $ 98. Dorian Ramirez was kind enough to point out this mistake. September 18, 2018

Calculating Bond and Note Yields Assume bond and note principal is always $ 100,000 Bonds and notes pay “coupons” twice a year. These coupons are fixed and unchanging for the life of the bond or note. For example, the 14s of Aug 11 was originally issued in August of 1981 as a 30 year treasury bond maturing on August 15, 2011. It paid $ 7,000 twice a year starting on February 15, 1982 and continuing on (Feb 15 and Aug 15 dates) for thirty years. The final coupon payment was on the maturity date August 15, 2011. On that date the holder of the bond received both the final coupon payments as well as the $ 100,000 principal payment. A pricing convention: “100” means a current price of $ 100,000. A price of 90 means a price of $ 90,000. A price of 87 1/8 means a price of $ 87,125.00 and so on. The price of the bond fluctuates from minute to minute in the market place once the bonds or notes have been sold (issued) to the public September 18, 2018

Yield Concepts Current Yield Annual coupon payments divided by current price For the 14s of Nov 11, the annual coupon payments are $ 14,000 If the current price is $ 100,000, then this bond’s current yield is 14 percent If the current price is $ 90,000, then this bond’s current yield is 15.56 percent If the current price is $ 120,000, then this bond’s current yield is 11.67 percent Observe that “current yield” is constantly changing When the price of the bond goes up, the yield goes down When the price of the bond goes down, the yield goes up But current yield is not used much in practice, instead yield-to-maturity is more typically used. September 18, 2018

Yield-to-Maturity (Use the 14s of Aug 11 as the example for the calculation) Value of the bond = 7000 1+𝑟 1 2 + 7000 1+𝑟 1 + 7000 1+𝑟 3 2 ……………………….. 107000 1+𝑟 30 Set Current Price equal to the “value of the bond” Current Price of the Bond = 7000 1+𝑟 1 2 + 7000 1+𝑟 1 + 7000 1+𝑟 3 2 ……………………….. 107000 1+𝑟 30 Solve this equation for r. “r” that solves this equation is the “yield-to-maturity” September 18, 2018

In actual practice The yield-to-maturity calculation uses “half years.” So that the exponents in the preceding slide are all doubled, in order to solve for the “yield-to-maturity” This means the r that solve the equation using half years, must then be doubled after solving for r This makes little difference in practice and we shall ignore it in this course. September 18, 2018

The Concept of “Duration” Attempts to provide a measure of risk as overall rates rise and fall. When overall rates rise, then the price of bonds will tend to fall. As overall rates fall, the price of bonds will tend to rise. How much can one expect to see prices fall or rise as rates change. This is what duration attempts to measure. For obvious reasons, maturity may not be a good guide to “interest rate risk” in the US Treasury market. Duration basically “weights” each payments by its current value. Then using those weights, duration is a weighted average of the maturity of each payment, summed over the life of the bond. Duration changes moment to moment as prices changes. Duration risk is highest when rates are low, lowest when rates are high. September 18, 2018

Compounding Means, “the clock starts again” Suppose you make 10 percent return every year on $ 100 investment If no compounding then At the end of first year, you have $ 110 At the end of the second year, you have $ 120 ($ 10 per year) If compounding then But, at the end of the second year, you have $ 121 (because you made 10 percent on $ 110, not 10 percent on $ 100 in the second year. Compounded rates are always lower than average rates, if compounding is occurring