2Interest Rates, Compounding, and Present Value In economics, an interest rate is known as the yield to maturity.Compounding is the process that gives us the value of a sum invested over time at a positive rate of interest.Present value is the process that tells us how much an expected future payment is worth today.
3CompoundingAssume you have $1 which you place in an account paying 10% annually.How much will you have in one year, two years, etc?An amount of $1 at 10% interestYear n$ $ $ $1(1 + i)nFormula: FV = PV(1 + i)
4Compounding over Time Extending the formula over 2 years 3 years FV = PV(1 + i) (1 + i) or FV = PV(1 + i)23 yearsFV = PV(1 + i) (1 + i) (1 + i) = PV(1 + i)3n yearsFV = PV(1 + i)n
5Present ValuePresent value tells us how much an expected future payment is worth today.Alternatively, it tells us how much we should be willing to pay today to receive some amount in the future.For example, if the present value of $1.10 at an interest rate of 10% is $1, we should be willing to spend $1 today to get $1.10 next year.
6Present Value FormulaThe formula for present value can be found by rearranging the compounding formula.FV = PV(1 + i) solve for PVFV/(1 + i) = PV
7Present Value over Time Extending the formula over 2 yearsFV = PV(1 + i)2PV = FV/(1 + i)23 yearsFV = PV(1 + i)3PV = FV/(1 + i)3n yearsFV = PV(1 + i)nPV = FV/(1 + i)n
8Things to NoticeAn increase in the interest rate causes present value to fall.Higher rates of interest mean smaller amounts can grow to equal some fixed amount during a specified period of time.A decrease in the interest rate causes present value to rise.Lower rates of interest mean larger amounts are needed to reach some fixed amount during a specified period of time.
9Example:How much must I invest today to get $10,000 in five years if interestrates are 10%?PV = FV/(1 + i)nPV = $10,000/( )5 = $10,000/ = $6,209.2rates are 5%?PV = $10,000/( )5 = $10,000/ $7,835.15
10More Things to Notice Present value is always less than future value. (1 + i)n is positive so FV/(1 + i)n < FVIn addition, PV4 < PV3 < PV2 < PV1(1 + i)1 < (1 + i)2The longer an amount has to grow to some fixed future amount, the smaller the initial amount needs to be.
11Time Value of MoneyThe longer the time to maturity, the less we need to set aside today. This is the principal lesson of present value. It is often referred to as the “time value of money.”
12Example:If I want to receive $10,000 in 5 years, how much do I have to investnow if interest rates are 10%?$10,000 = PV( )5$10,000/ = $If I want to receive $10,000 in 20 years, how much do I have to invest$10,000 = PV( )20$10,000/6.7275= $
13Yield to MaturityYield to maturity is the interest rate that equates the present value of payments received from a debt instrument with its value today.Yield to maturity can be calculated using the present value formula.PV = FV/(1 + i)i = FV - PV/PV
15Relationship between Yield to Maturity and Price Yields to maturity on a 10% coupon rate bond with a face valueof $1000 maturing in 10 yearsPrice of Bond Yield to Maturity
16Relationship between Yield to Maturity and Price Three interesting facts:Price and yield are negatively related.When the bond is at par, yield equals coupon rate.Yield is greater (less than) than the coupon rate when the bond price is below (above) par value.
17Current YieldIn more complicated cases, yield to maturity can be difficult to calculate. Tables are available that can be used. And, of course, calculators do a fine job.There are also simple formulas that can approximate yield to maturity such as current yield.
18Current YieldCurrent yield is an approximation for yield to maturity that is used to calculate the interest rate on a bond quickly.Formula:Current yield = Coupon/Bond Price
19Inverse RelationshipWe can use the current yield formula to see clearly the inverse relationship between interest rates and bond prices.Current yield = Coupon/Bond PriceThe coupon is a fixed payment, it does not change. Therefore, if yields rise, bond prices must fall, and if yields fall, bond prices must rise.
20IntuitionAssume you buy a $1,000 bond today with a fixed coupon of $100. You are receiving a 10% return. Let a year pass, and you find you want to sell you bond. You call your broker and say, “Sell!” Your broker sighs and tells you that bonds just like yours now yield 12%. What price can you expect to receive?
21Example Use the current yield formula: 0.12 = $100/PB0.12PB = $100PB = $100/.12 = $833.33You must reduce your price until $100 represents a 12% rate of return.
22The Behavior of Interest Rates The Bond Market Model
23Understanding Interest Rates Economists use three different models to explain how interest rates are determined.The bond market modelThe money demand/money supply modelThe loanable funds model
24The Bond Market ModelThe bond market model is useful because of the issues that can be considered within its framework.The impact of changes in----WealthExpected interest rates or expected returnExpected inflationRiskiness of bonds relative to other assetsLiquidity of bonds relative to other assets
25The Bond Market ModelThe bond market can be modeled using the concepts of demand and supply.The demand for bonds is determined by individuals and institutions who wish to hold their wealth in bonds.The supply of bonds is provided by institutions that issue bonds to raise funds.
26The Demand for BondsThe demand for bonds comes from savers, people who have funds in excess of their spending needs.They are willing to hold bonds for two reasons:Interest earnedPotential capital gains
27Bond Demand Rate of return According to the asset theory of demand, people compare one asset relative to another and choose the one that best suits their needs.As the opportunity cost of an asset increases, people find it increasingly unattractive.
28Opportunity CostThe opportunity cost of an asset is defined as the difference between the rate of return received by the asset and the rate of return on an alternative asset.When bond yields are high, people prefer bonds because the opportunity cost of holding other assets is high.When bond yields are low, people prefer other assets because the opportunity cost of holding bonds is high.
29Bond DemandInvestors who demand bonds based on opportunity cost considerations prefer to buy when interest rates are high and sell when interest rates are low.
30Bond Demand Speculation When choosing an asset, investors also consider risk.Interest rate risk occurs when the market value of a bond falls because interest rates rise.As we have seen, the existence of interest rate risk means investors face the possibility of capital losses when interest rates rise and capital gains when interest rates fall.
31SpeculationInvestors who speculate in the bond market prefer to buy when interest rates are high and sell when interest rates are low.When interest rates are high, people expect them to fall. As they fall, bond prices rise, yielding a capital gain.When interest rates are low, people expect them to rise. As they rise, bond prices fall, diminishing capital gains or yielding a capital loss.
32Bond DemandBoth the opportunity cost motive and the speculative motive result in investors demanding bonds when interest rates are high and selling bonds when interest rates are low.
33The Demand Curve for Bonds Let r = RET = (F - P)/PIf F = $1,000 and P = $950, r = 5.26%If F = $1,000 and P = $900, r = 11.1%High bond prices are associated with low interest rates.Low bond prices are associated with high interest rates.
34The Demand Curve for Bonds PriceInterestRateWhen bond prices are high,interest rates are low, andbond demand is low.When bond prices are low,interest rates are high, andbond demand is high.ilowPBhighPBlowihighDemandQDlowQDhigh
35Bond SupplyThe supply of bonds comes from institutions, governments (domestic and foreign), and businesses.The quantity of bonds supplied depends in part on the interest rate bond suppliers must pay to attract funds.As interest rates increase, the quantity supplied falls.As interest rates decrease, the quantity supplied rises.
36The Supply Curve for Bonds PriceInterestRateSupplyAs bond prices rise, bond yieldsfall, and quantity supplied rises.As bond prices fall, bond yieldsrise, and quantity supplied falls.ilowPBhighPBlowihighQSlowQShigh
37EquilibriumEquilibrium is a state of rest. Either there are no forces causing change or there are equal opposing forces.In the bond market, equilibrium occurs when the quantity of bonds demanded just equals the quantity of bonds supplied.
38Equilibrium & Disequilibrium BondPriceInterestRateSExcess supply occurs whenbond prices are high and interestrates are low.Excess demand occurs whenbond prices are low and interestrates are high.ABilowPBhighEPBeqieqihighPBlowFGD
39Disequilibrium Excess Supply Excess Demand More people want to sell bonds than want to buy them.Bond prices fall and interest rates rise.Excess DemandMore people want to buy bonds than want to sell them.Bond prices rise and interest rates fall.
40Mechanics of an Increase in Demand BondPriceInterestRateSbIncreases in bond demand causebond prices to rise and bondyields to fall.P2i1aP1i2D2D1Q1 Q2
41Mechanics of a Decrease in Demand BondPriceInterestRateSDecreases in bond demand causebond prices to fall and bondyields to rise.aP1i2bP2i1D1D2Q2 Q1
42Shifts in the Demand for Bonds According to the asset theory of demand, changes in bond demand are caused by changes in---WealthExpected return on bonds relative to expected returns on other assetsExpected riskiness of bonds relative to other assetsLiquidity of bonds relative to other assets.
43Bond Demand and WealthWealth is defined as a stock of assets that produce income. Wealth is not income.In a business cycle expansion, wealth grows, causing the demand for bonds to rise and the demand curve to shift to the right.In a business cycle contraction, wealth shrinks, causing the demand for bonds to fall and the demand curve to shift to the left.
44Bond Demand and Expected Returns: Bonds Higher expected interest rates in the future decrease the demand for long-term bonds and shift the demand curve to the left.Lower expected interest rates in the future increase the demand for long-term bonds and shift the demand curve to the right.
45Returns on Different Maturity 10% Coupon Rate Bonds Term Initial i Initial P New i New P30 10% %20 10% %10 10% %5 10% %1 10% %
46Bond Demand and Expected Returns: Other Assets Higher expected returns on other assets relative to bonds cause bonds to become less attractive and the bond demand curve shifts left.Lower expected returns on other assets relative to bonds cause bonds to become more attractive and the bond demand curve shifts right.
47Bond Demand and Expected Returns: Inflation An increase in the expected rate of inflation will cause the demand for bonds to decline and the demand curve to shift to the left.A decrease in the expected rate of inflation will cause the demand for bonds to increase and the demand curve to shift to the right.
48Bond Demand and RiskAn increase in the riskiness of bonds causes the demand for bonds to fall and the demand curve to shift to the left.An increase in the riskiness of other assets causes the demand for bonds to rise and the demand curve to shift to the right.
49Bond Demand and Liquidity Increased liquidity of bonds results in an increased demand for bonds and the demand curve shifts right.Increased liquidity of other assets results in a decreased demand for bonds and the demand curve shifts left.
50Bond Demand and Liquidity Increased liquidity of bonds results in an increased demand for bonds and the demand curve shifts right.Increased liquidity of other assets results in a decreased demand for bonds and the demand curve shifts left.
51Mechanics of an Increase in Supply BondPriceInterestRateS1S2An increase in the supply of bondscauses bond prices to fall and bondyields to rise.ai2P1bP2i1DQ1 Q2
52Mechanics of an Decrease in Supply BondPriceInterestRateS1S2A decrease in the supply of bondscauses bond prices to rise and bondyields to fall.bi1P2aP1i2DQ2 Q1
53Shifts in SupplyShifts in the supply curve for bonds are caused by changes in….The expected profitability of investment opportunitiesExpected inflationGovernment activities
54Shifts in the Supply of Bonds Expected Profitability of Investment OpportunitiesWhen an economy is growing rapidly, there are many profitable investment opportunities. The supply of bonds increases and the supply curve shifts to the right.When an economy is contracting, there are fewer profitable opportunities. The supply of bonds decreases and the supply curve shifts left.
55Shifts in the Supply of Bonds Expected InflationAn increase in expected inflation causes the supply of bonds to increase and the supply curve to shift right.Inflation causes the real cost of borrowing to fall.
56Shifts in the Supply of Bonds Government ActivitiesHigher government deficits increase the supply of bonds, causing the supply curve to shift right.Reductions in government deficits decrease the supply of bonds, causing the supply curve to shift left.
57Real and Nominal Interest Rates Nominal interest rate is the rate of interest that makes no allowance for inflation.The real interest rate is the rate of interest that is adjusted for expected changes in the price level.It more accurately reflects the true cost of borrowing and lending.
58Fisher EquationThe Fisher equation states that the nominal interest rate equals the real interest rate plus the expected rate of inflationin = ir + pRearranging terms we find:ir = in - p
59Logic behind the Inflation Premium Lenders want to be compensated for the loss in buying power due to inflation.Buyers understand that they will be repaying debt with dollars that buy less.The interest rate must reflect these facts.
60The Fisher Effect When expected inflation increases: Bond supply increases and the supply curve shifts right.Bond demand decreases and the demand curve shifts left.As a result, bond prices fall and interest rates rise.When expected inflation rises, interest rates rise. This is the Fisher Effect.