MONEY & BOND MARKETS AN INTRODUCTION TO MONETARY ECONOMICS Interest Rate consists of 3 components: 1) inflation 1) inflation 2) reward for postponing consumption 2) reward for postponing consumption 3) risk 3) risk R ≈ E(e) + time preference + risk premium Real = Inflation-adjusted R R ≈ R − e R R = [(1+R)/(1+e)] − 1 R R = [(1+R)/(1+e)] − 1 So, real returns reflect (time preference + risk).

DEFINITIONS: DEFINITIONS: In the money market, we use the concept of interest rate; in risky assets, we use the concept of return. Return: R t = ( V t / V t-1 ) – 1 (% change in value ) Return: R t = ( V t / V t-1 ) – 1 (% change in value ) Log Return = ln (V t / V t-1 ) Log Return = ln (V t / V t-1 ) Interest Rate: i or r r s : annualized simple interest rate r c : annualized compound interest rate r p : periodic interest rate ( the interest to be accrued between t 0 and T, the holding period ) In money market transactions, bankers always quote r s, however the effective annual rate of return is r c.

MONEY MARKETS – INTEREST CALCULATIONS Calculating r hp : r hp = (1+r s /n) k – 1 k is the number of compoundings over the holding period (how many times the initial money is reinvested), r s is the interest rate per unit period of time (typically 1 year). If the interest is not reinvested, you do not need to compound (to take power), rather you should simply multiply: r p = r s * k  Converting r s into r c : r c = (1+r s /n) n –1 where n is the number of compounding (reinvestment) per year. As the compounding frequency (n) increases, the effective annual rate grows. The limit is continuous compounding: r c = e rs r p = e rst

With compounding (reinvesting at the same rate), money grows exponentially (geometrically). Example: (1+0.07) 50 = (that is, if you invest $1000 at 7% for 50 years reinvesting the interest, you end up with $29,460 at the end of 50th year) (1+0.09) 50 = (at 9% you end up with $74,360; so a small difference in rate makes a huge difference) Money Markets: Fed Funds Market, LIBOR, Repo, Money Market Mutual Funds Bank discount: P = F(1-r p )

Present Value of a Future Monetary Value: PV 0 (C t ) = C t / (1+r hp ) (1+r) is called gross return General Formula for Valuation of All Financial Assets: Σ n t=1 PV 0 (CF t ) = PV 0 (C 1 ) + PV 0 (C 2 ) +….+ PV 0 (C n ) = Σ n t=1 [C t / (1+r) t ] = Σ n t=1 [C t / (1+r) t ] General Rule: The value of any asset should equal to the sum of the PV’s of its all cash flows. Under specific assumptions about the cash flow pattern, this calculation is tractable: Perpetuity: A constant periodic cash flow C forever. PV = C / r Growing Perpetuity: A periodic cash flow growing at a constant rate g, forever. PV = C / (r-g) (r>g) Annuity: A constant periodic cash flow C until a specified date T in the future.

Perpetuity (Consol) A constant stream of cash flows that lasts forever 0 … 1 C 2 C 3 C

Growing Perpetuity A growing stream of cash flows that lasts forever 0 … 1 C 2 C×(1+g) 3 C ×(1+g) 2

BONDS Two main types of Bonds: Two main types of Bonds: A) Pure Discount Bills: no interim payments, one single repayment at the maturity B) Coupon Bonds: Pays coupons C at regular intervals + principal at the maturity T.

Pure Discount Bills  Make no periodic interest payments (coupon rate = 0%)  The entire yield to maturity comes from the difference between the purchase price and the par value.  Cannot sell for more than par value  Treasury Bills and principal-only Treasury strips are good examples.  generally used for short term borrowing

Pure Discount Bills: N: par value days to maturity= T-t r s : simple annualized rate r c : compound annualized rate n = 365/(T-t) i.e.: number of compounding per year

Coupon Bonds: B t = coupon rate (C%): indicates the amount of annual coupon payments. Current yield= C / B t ytm: indicates the annual market interest rate that applies to the bond

Yield to Maturity: Bonds Coupon Bond (Coupon rate = 10% = C/F) Consol: Fixed coupon payments of $C forever

Relationship Between Price and Yield to Maturity  Three interesting facts in Table When bond is at par, yield equals coupon rate 2.Price and yield are negatively related 3.Yield greater than coupon rate when bond price is below par value

YTM and Bond Value Discount Rate Bond Value 6 3/8 When the YTM < coupon, the bond trades at a premium. When the YTM = coupon, the bond trades at par. When the YTM > coupon, the bond trades at a discount.

The returns from holding a bond for less than until its maturity consists of two components: 1. Interest gains 2. Capital gains or losses (the price of a bond is inversely related to ytm, i.e. market interest rates). If you hold a bond until its maturity, you receive a certain return (which equals to its interest rate, ytm), hence fixed income.

Yield Curve = Term Structure of Interest Rates A Graph showing the effective annualized interest rates of different terms. Duration: The weighted average maturity terms of cash flows of a bond. It indicates the responsiveness of the bond price to changes in ytm. D = (∂B/B) / ∂r. As such, it is a measure of the riskiness of a bond. Eurobonds Callable Bond: Convertible Bond: Inflation-linked bonds: e.g. TIPS

Duration  Key facts about duration 1.All else equal, when the maturity of a bond lengthens, the duration rises as well 2.All else equal, when interest rates rise, the duration of a coupon bond fall

Risk Structure of Long Bonds in the U.S.

Bond Ratings

Liquidity Premium Theory

22 Market Predictions of Future Short Rates

Case: Interpreting Yield Curves, 1980–2008