Money and Banking Lecture I: Interest Rates Guoxiong ZHANG, Ph.D. Shanghai Jiao Tong University, Antai September 11th, 2018
Interest Rates Are Important Source: http://www.cartoonistgroup.com
Concept of Interest Rates Interest rate measures the time value of money: people prefer to consume today instead of tomorrow money can be used as capital for production money can help for transaction (liquidity) Interest rates take many forms: bonds: yield to maturity loans and mortgage rates central bank: discount rate, federal funds rate interbank lending rates: LIBOR, SHIBOR
Yield to Maturity Definition: the interest rate (discount rate) that equals the present value of future cash flow payments from a debt instrument with the debt instrument s value today (internal rate of return) debt instruments: simple loan; fixed-payment loan (fully amortized loan); coupon bond; discount bond (zero-coupon bond). Calculation: simple loan: P V = F V (1+i) N fixed payment loan: P V = N n=1 coupon bond: P V = N n=1 discount bond: P V = F V (1+i) N Perpetuity: P V = C i F P (1+i) n C (1+i) n + F V (1+i) N
Facts about Bonds The price of a coupon bond and its yield to maturity is negatively related; If a coupon bond is priced at its face value, then its yield to maturity equals its coupon rate; If a coupon bond s price is higher than its face value, then its yield to maturity is lower than its coupon rate; The more distant a bonds maturity, the greater the size of the percentage price change associated with an interest-rate change.
Interest Rate Risk Source: Mishkin (2013)
Bond Duration Duration for a coupon bond is used to measure bond price s sensitivity to interest rate dp di 1 P = [ C (1 + i) 2C 2 (1 + i)... NC 3 (1 + i) F V N+1 (1 + i) ] 1 N+1 P N n = [C (1 + i) + N n+1 (1 + i) F V ] 1 N+1 P n=1 Macaulay Duration = dp Modified Duration = dp di 1 di P 1. P (1 + i);
Interest Rate and Bond Return One period return rate for a bond R = C Pt+1 Pt + P t P t = current yield + capital gain Usually for a bond, its capital gain dominates its current yield, and therefore its return can be negative if the interest rate rises a lot.
Effective Interest Rate The effective t-period rate r t is related to the effective one period rate r as 1 + r t = (1 + r) t. Often we need to compute the effective annual rate(ear) from effective monthly rates; its also called annual percentage yield (APY). This equation also applies for fractional holding periods (t < 1).
Annual Percentage Rate annual percentage rate (APR ) is no more than a form of interest rate quotation: APR = r 1 M M; where M is the number of compounding periods per year and r 1 M the effective 1 year rate. M APR and EAR are related as EAR = (1 + APR M )M 1. is
Continuous Interest Rates Let ρ be the continuously compounded interest rate for one period with effective interest rate r, then we have e ρ = lim (1 + ρ M M )M = 1 + r ρ = ln(1 + r) The continuous interest rate for any period length t is ρ t = ρ t. additive over time This additivity property of continuous interest rates makes it very popular in option pricing (stochastic differential equations such as Black-Scholes). It has also been gaining its popularity in macro-finance studies. (Yuily Sannnikov, 2016 Clark Award)
Interest Rates Spreads Source: Mishkin (2013)
Risk Structure of Interest Rate Default risk: chances that the bond issuer can not make interest payment or pay off the face value at maturity. U.S. treasury bonds are widely regarded as default free. The difference between interest rates on bonds and interest rates on default free bonds (interest rate spread) are called risk premium. This spread also reflect their difference in liquidity: default free safe asset (fly to quality ) high demand high liquidity even safer asset... (What makes a safe asset, He, Krishnamurthy, and Milbradt (2015)) Therefore its also called risk and liquidity premium. Municipal bonds usually have lower interest rates than treasury bonds because interest payments from municipal bonds are exempt from federal income tax.
Bonds Rating Source: Mishkin (2013)
Term Structure of Interest Rate Yield curve: a plot of the yield on bonds with differing terms to maturity but the same risk, liquidity and tax considerations Upward-sloping: long-term rates are above short-term rates Flat: short- and long-term rates are the same Inverted: long-term rates are below short-term rates Yield curves are used to describe the term structure of interest rates for particular types of bonds. http://finance.yahoo.com/bonds/composite_bond_rates
Three Facts on Yield Curve Interest rates on bonds of different maturities move together over time When short-term interest rates are low, yield curves are more likely to have an upward slope; when short-term rates are high, yield curves are more likely to slope downward and be inverted Yield curves almost always slope upward
Co-movements of interest rates on US treasury bonds Source: Mishkin (2013)
Expectation Theory Assumes that bond holders consider bonds with different maturities to be perfect substitutes; Then holding n-periods bonds should give identical expected interest payment as holding one-period bonds period by period: i n,t = it + ie t+1 +... + i e t+n 1. n Expectation theory can explain the first two facts but not the last one.
Segmented Market Theory Assumes that bond holders consider bonds with different maturities to be not substitutes at all; It also assumes investors generally prefer bonds with shorter maturities that have less interest rate risk; Segmented market theory can explain the last one but not the first two.
Liquidity Premium Theory Liquidity premium (preferred habitat) theory combines the above two theories: investors have a preference for bonds of shorter maturity over ones with longer maturity; But they are willing to buy long maturity bonds if they can generate somewhat higher expected return: i n,t = it + ie t+1 +... + i e t+n 1 n + l n,t }{{}. liquidity premium Liquidity premium theory can explain all the three facts about yield curves.
Money Supply and Interest Rate People always term expansionary monetary policy as either increasing money supply (quantity rule) or reducing interest rate (price rule); Is there really a one-to-one relationship between money supply and interest rate? More specifically does more money supply always cause lower interest rate?
Money Supply and Interest Rates Source: https://fred.stlouisfed.org/series/manmm101usm657s#0