Determinants of Interest Rates Chapter Two Determinants of Interest Rates

Interest Rate Fundamentals Nominal interest rates: the interest rates actually observed in financial markets Used to determine fair present value and prices of securities Two types of components Opportunity cost on competing investments Adjustments for individual security characteristics Opportunity cost: All securities’ rates are dependent on rates available on competing investments. These rates will be a function of the underlying supply and demand of funds available. As indicated later, adjustments for individual security characteristics would include default risk, maturity, liquidity risk and payment terms. Because these characteristics are different for different securities, we have different interest rates on each.

Real Interest Rates Additional purchasing power required to forego current consumption What causes differences in nominal and real interest rates? If you wish to earn a 3% real return and prices are expected to increase by 2%, what rate must you charge? Irving Fisher first postulated that interest rates contain a premium for expected inflation. (1+R) = (1+r)(1+h), where R is nominal, r is real, h is inflation The answers are inflation and about 5% respectively.

Loanable Funds Theory Loanable funds theory explains interest rates and interest rate movements Views level of interest rates in financial markets as a result of the supply and demand for loanable funds Domestic and foreign households, businesses, and governments all supply and demand loanable funds Ignore inflation for the moment

Supply and Demand of Loanable Funds Interest Rate Quantity of Loanable Funds Supplied and Demanded

Determinants of Household Savings Interest rates – the higher the greater saving Tax policy – the higher, the less saving Income and wealth - the greater the wealth or income, the greater the amount saved Attitudes about saving versus borrowing Credit availability - the greater the amount of easily obtainable consumer credit the lower the need to save Job security and belief in soundness of entitlements

Determinants of Foreign Funds Invested in the U.S. Relative interest rates and returns on global investments Expected exchange rate changes Safe haven status of U.S. investments Foreign central bank investments in the U.S. Why is so much money coming to the US from abroad right now? Foreign funds suppliers examine the same factors as U.S. suppliers except that they must also factor in expected changes in currency values, global interest rates, different tax rates and sovereign risk. There is typically some built in demand for U.S. investments however because the U.S. is considered a safe haven, i.e., a country with relatively low political and economic risk and a stable currency. High levels of reserves are indicative of foreign central bank activity to limit the growth in the value of their currencies against the dollar. This may be done to stimulate their export sectors. The dollars are often reinvested in the U.S., typically in Treasuries. This provides an additional source of financing to the U.S. and helps remove a market discipline from U.S. borrowers.

Federal Government Demand for Funds (a.k.a. Deficit Financing) Source: CBO 2011 report, http://www.cbo.gov/ftpdocs/74xx/doc7492/08-17-BudgetUpdate.pdf Note: Government demand for funds is expected to remain high over the next 10 years.

Shifts in Supply and Demand Curves change Equilibrium Interest Rates Increased supply of loanable funds Increased demand for loanable funds Interest Rate DD* Interest Rate SS SS DD DD SS* i** E* E i* E i* E* i** Q* Q** Q* Q** Quantity of Funds Supplied Quantity of Funds Demanded

Factors that Cause Supply and Demand Curves to Shift

Factors that Cause Supply and Demand Curves to Shift

Factors that Cause Supply and Demand Curves to Shift

Determinants of Interest Rates for Individual Securities ij* = f(IP, RIR, DRPj, LRPj, SCPj, MPj) IP - inflation premium RIR – real risk-free rate DRP – default risk premium LRP – liquidity risk premium SCP – special feature premium MP maturity premium Inflation (IP) IP = [(CPIt+1) – (CPIt)]/(CPIt) x (100/1) Real Interest Rate (RIR) and the Fisher effect RIR = i – Expected (IP) Define CPI as the consumer price index. ij* = equilibrium nominal interest rate for a given security

Determinants of Interest Rates for Individual Securities (cont’d) Default Risk Premium (DRP) DRPj = ijt – iTt ijt = interest rate on security j at time t iTt = interest rate on similar maturity U.S. Treasury security at time t Liquidity Risk (LRP) Special Provisions (SCP) Term to Maturity (MP)

Term Structure of Interest Rates: the Yield Curve (a) Upward sloping (b) Inverted or downward sloping Flat Recessions tend to preceded by inverted yield curves Yield to Maturity (a) (c) (b) Time to Maturity

Unbiased Expectations Theory Long-term interest rates are geometric averages of current and expected future short-term interest rates 1RN = actual N-period rate today N = term to maturity, N = 1, 2, …, 4, … 1R1 = actual current one-year rate today E(ir1) = expected one-year rates for years, i = 1 to N

Liquidity Premium Theory Long-term interest rates are geometric averages of current and expected future short-term interest rates plus liquidity risk premiums that increase with maturity Lt = liquidity premium for period t L2 < L3 < …<LN

Market Segmentation Theory Individual investors have specific maturity preferences Interest rates are determined by distinct supply and demand conditions within many maturity segments Investors and borrowers deviate from their preferred maturity segment only when adequately compensated to do so Also known as Preferred Habitat Theory

Implied Forward Rates A forward rate ( f ) is an expected rate on a short-term security that is to be originated at some point in the future The one-year forward rate for any year N in the future is:

Question The 2 year rate is 2% per year The 1 year rate is 1.5% What is the implied forward rate for year 2? Answer Then 2f1 = (1.02)2 / (1.015) – 1 = .025 or 2.5% next year

Time Value of Money and Interest Rates The time value of money is based on the notion that a dollar received today is worth more than a dollar received at some future date Simple interest: interest earned on an investment is not reinvested Compound interest: interest earned on an investment is reinvested

Present Value of a Lump Sum Discount future payments using current interest rates to find the present value (PV) PV = FVt[1/(1 + r)]t = FVt(PVIFr,t) PV = present value of cash flow FVt = future value of cash flow (lump sum) received in t periods r = interest rate per period t = number of years in investment horizon PVIFr,t = present value interest factor of a lump sum

Future Value of a Lump Sum The future value (FV) of a lump sum received at the beginning of an investment horizon FVt = PV (1 + r)t = PV(FVIFr,t) FVIFr,t = future value interest factor of a lump sum

Relation between Interest Rates and Present and Future Values (PV) Future Value (FV) Interest Rate Interest Rate

Present Value of an Annuity The present value of a finite series of equal cash flows received on the last day of equal intervals throughout the investment horizon PMT = periodic annuity payment PVIFAr,t = present value interest factor of an annuity NOTE: PMT / i is a perpetuity, which lasts only t-periods Note that 1 / (1+i)N = (1+i)-N

Future Value of an Annuity The future value of a finite series of equal cash flows received on the last day of equal intervals throughout the investment horizon FVIFAr,t = future value interest factor of an annuity

Effective Annual Return Effective or equivalent annual return (EAR) is the return earned or paid over a 12-month period taking compounding into account EAR = (1 + rper period)c – 1 c = the number of compounding periods per year

Financial Calculators Setting up a financial calculator Number of digits shown after decimal point Number of compounding periods per year Key inputs/outputs (solve for one of five) N = number of compounding periods I/Y = annual interest rate PV = present value (i.e., current price) PMT = a constant payment every period FV = future value (i.e., future price) Chapter 2 Problems: On page 55, try 20, 22, and all of page 56.

Interest Rates and Security Valuation Chapter Three Interest Rates and Security Valuation

Various Interest Rate Measures Coupon rate periodic cash flow a bond issuer contractually promises to pay a bond holder Required rate of return (r) rates used by individual market participants to calculate fair present values (PV) Expected rate of return or E(r) rates participants would earn by buying securities at current market prices (P) Realized rate of return ( r ) rate actually earned on investments

Required Rate of Return The fair present value (PV) of a security is determined using the required rate of return (r) as the discount rate CF1 = cash flow in period t (t = 1, …, n) ~ = indicates the projected cash flow is uncertain n = number of periods in the investment horizon

Expected Rate of Return The current market price (P) of a security is determined using the expected rate of return or E(r) as the discount rate CF1 = cash flow in period t (t = 1, …, n) ~ = indicates the projected cash flow is uncertain n = number of periods in the investment horizon

Realized Rate of Return The realized rate of return ( r ) is the discount rate that just equates the actual purchase price ( ) to the present value of the realized cash flows (RCFt) t (t = 1, …, n)

Bond Valuation - PV of coupon payments and PV of Par or Face Value. The present value of a bond (Vb) can be written as: Par = the par or face value of the bond, usually $1,000 INT = the annual interest (or coupon) payment T = the number of years until the bond matures r = the annual interest rate (often called yield to maturity (ytm))

Bond Valuation A premium bond has a coupon rate (INT) greater than the required rate of return (r) and the fair present value of the bond (Vb) is greater than the face or par value (Par) Premium bond: If INT > r; then Vb > Par Discount bond: if INT < r, then Vb < Par Par bond: if INT = r, then Vb = Par

Equity Valuation The present value of a stock (Pt) assuming zero growth in dividends can be written as: D = dividend paid at end of every year Pt = the stock’s price at the end of year t rs = the interest rate used to discount future cash flows Note: This is a Perpetuity Formula or a Consol. We find this perfect for preferred stock, which have fixed dividends.

Equity Valuation The Dividend Growth Model or Gordon Model The present value of a stock (Pt) assuming constant growth in dividends can be written as: D0 = current value of dividends Dt = value of dividends at time t = 1, 2, …, ∞ g = the constant dividend growth rate

Equity Valuation The return on a stock with zero dividend growth, if purchased at current price P0, can be written as: The return on a stock with constant dividend growth, if purchased at price P0, can be written as: Dividend yield + capital gain yield

Relation between Interest Rates and Bond Values 12% 10% 8% 874.50 1,000 1,152.47

Impact of maturity on Price Volatility in the face of interest rate changes Absolute Value of Percent Change in a Bond’s Price for a Given Change in Interest Rates Time to Maturity

Impact of Maturity on Price Volatility with increase in yields Absolute value of percentage price change when interest rates increase from 7% to 7.5% for bonds with a 6% coupon and different maturities. Note that volatility increases at a decreasing rate.

Impact of Coupon Rates on Price Volatility Bond Value High-Coupon Bond Low-Coupon Bond Interest Rate

Impact of Coupon on Price Volatility with decrease in yields Absolute value of percentage price change when interest rates decrease from 7% to 6.5% for bonds with various coupon but all with a 10 year maturity.

Impact of r on Price Volatility Bond Price Interest Rate How does volatility change with interest rates? Price volatility is inversely related to the level of the initial interest rate You may wish to point out that actual price changes are curvilinear, duration based predicted price changes are linear with r. Volatility varies along line: prices are a nonlinear function of interest rates, Blue line is actual price change, green line is predicted price change. Concept: At Higher interest rates, volatility is lower: Reason is that discounting far out cash flows more heavily to begin with at higher interest rates, this increases the near term percentage PV weights in relation to the long term weights. The four variables that affect volatility are coupon and maturity (which are captured by duration), change in ytm or change in r and the starting ytm or the starting r. (r = ytm) r

Duration Duration is the weighted-average time to maturity (measured in years) on a financial security Duration measures the sensitivity (or elasticity) of a fixed-income security’s price to small interest rate changes Duration captures the coupon and maturity effects on volatility. So the only other two variables needed to predict volatility are r and the change in r.

Duration – weighted average of coupon and Par value Duration (Dur) for a fixed-income security that pays interest annually can be written as: P0= Current price of the security t = 1 to T, the period in which a cash flow is received T = the number of years to maturity CFt = cash flow received at end of period t r = yield to maturity or required rate of return PVt = present value of cash flow received at end of period t Note that if the security makes semiannual or monthly payments then the cash flow, the interest rate and the number of periods must be adjusted to reflect the payment frequency.

Duration and Volatility 9% Coupon, 4 year maturity annual payment bond with a 8% ytm Dur = 3.5396 years Duration = 3.5396 years What is the Duration if zero coupon 4 year bond?

Duration Duration (Dur) (measured in years) for a fixed-income security, in general, can be written as: m = the number of times per year interest is paid, the sum term is incremented in m units

Closed form duration equation: P0 = Price INT= Periodic cash flow in dollars, normally the semiannual coupon on a bond or the periodic monthly payment on a loan. r = periodic interest rate = APR / m, where m = # compounding periods per year N = Number of compounding or payment periods (or the number of years * m) Dur = Duration = # Compounding or payment periods; Durationperiod is what you actually get from the formula This version is in closed form, no summation needed. It is convenient for longer term securities. If you divide by Price*m you get the duration in years. m= # of compounding or payment periods per year

Duration Duration and coupon interest Duration and yield to maturity the higher the coupon payment, the lower the bond’s duration Duration and yield to maturity the higher the yield to maturity, the lower the bond’s duration Duration and maturity duration increases with maturity but at a decreasing rate

Duration and Modified Duration Given an interest rate change, the estimated percentage change in a (annual coupon paying) bond’s price given by Note: -Dur is the price elasticity of interest rates. This really doesn’t demonstrate the usage of modified duration (MD). MD is used for non-annual payment securities.

Duration and Modified Duration Modified duration (DurMod) can be used to predict price changes for non-annual payment loans or securities: It is found as: where rperiod = APR/m Using modified duration to predict price changes: Note that the annual change in interest rates is plugged into the prediction model. An example calculation is provided at the end of this file.

Duration Based Prediction Errors Duration is an accurate predictor of price changes only for very small interest rate changes. For day to day fluctuations duration works quite well but when interest rates move significantly, such as when the Fed makes an announcement of a rate change, the predicted pricing errors can become significant. The prediction errors arise because bond prices are not linear with respect to interest rates.

Convexity Convexity (CX) measures the change in slope of the price-yield curve around interest rate level R Convexity incorporates the curvature of the price-yield curve into the estimated percentage price change of a bond given an interest rate change:

Duration Practice Problem Using Modified Duration Dur Annual = 4.3774 years. Predicted Price Change Using Modified Duration Try Page 87-89: 7, 11, 18, 26, 32, 40, and 41.