Chapter 4Chapter 4Understanding Interest RatesUnderstanding Interest Rates

Learning ObjectivesLearning Objectives

1. Detail terms present value and interest rate1. Detail terms present value and interest rate 2. Discern measurments of interest rates: YTM, 2. Discern measurments of interest rates: YTM,

Current Yield, Yield on Discount basisCurrent Yield, Yield on Discount basis 3. Illustrate inverse relationship of bond prices 3. Illustrate inverse relationship of bond prices

and interest ratesand interest rates 4. Explain difference between nominal and real 4. Explain difference between nominal and real

interest ratesinterest rates 5. Explain difference between interest rates and 5. Explain difference between interest rates and

rates of returnrates of return

ContentsContents

Present Value and Interest RatesPresent Value and Interest Rates Subprime Mortgages and Interest RatesSubprime Mortgages and Interest Rates Measuring Interest RatesMeasuring Interest Rates Bond Prices and Interest RatesBond Prices and Interest Rates International Interest RatesInternational Interest Rates Nominal vs Real Interest RatesNominal vs Real Interest Rates Interest Rates vs Rates of ReturnInterest Rates vs Rates of Return

Present Value and Interest RatesPresent Value and Interest Rates

PV = FV / (1+i) ^ tPV = FV / (1+i) ^ tA dollar paid to you 1 year from now is A dollar paid to you 1 year from now is

worth less than one dollar paid to you worth less than one dollar paid to you todaytoday

Ex. PV of $250 paid in 2 years, i=15%Ex. PV of $250 paid in 2 years, i=15%PV = 250 / (1 + .15) ^ 2PV = 250 / (1 + .15) ^ 2 = $189.04= $189.04

Credit Market InstrumentsCredit Market Instruments

1 – Simple Loan1 – Simple Loan2 – Fixed-Payment Loan2 – Fixed-Payment Loan3 – Coupon Bond3 – Coupon Bond4 – Discount Bond4 – Discount Bond

Simple LoanSimple Loan

Lender provides borrower funds, borrower Lender provides borrower funds, borrower repays at maturity date plus additional repays at maturity date plus additional interest amountinterest amount

$500 loan with 10% interest:$500 loan with 10% interest:1 year: $500 * (1 + 0.10) = $5501 year: $500 * (1 + 0.10) = $5502 years: $550 * (1 + 0.10) = $6052 years: $550 * (1 + 0.10) = $605

P = F (1 + i) ^ tP = F (1 + i) ^ t P = Total paymentP = Total payment F = Face ValueF = Face Value i = interest ratei = interest rate t = number of periodst = number of periods

Coupon BondCoupon Bond

Face value amount is paid to issuer for Face value amount is paid to issuer for ownership of bond, then issuer pays ownership of bond, then issuer pays owner of bond fixed interest payment owner of bond fixed interest payment every period until maturity, then pays face every period until maturity, then pays face value to ownervalue to owner

Three pieces of information: Issuer of Three pieces of information: Issuer of bond, maturity date of bond, coupon rate bond, maturity date of bond, coupon rate of bondof bond

Coupon BondCoupon Bond

P = C * (1 – (1 / (1 + i) ^ t)) / r + F / (1 + i) ^ tP = C * (1 – (1 / (1 + i) ^ t)) / r + F / (1 + i) ^ t P = bond priceP = bond price C = Coupon rateC = Coupon rate i = interest ratei = interest rate t = number of periodst = number of periods F = face valueF = face value

Ex. $1000, 5% coupon, annual bond with Ex. $1000, 5% coupon, annual bond with 15 year maturity. What would the price be 15 year maturity. What would the price be if the interest rate was 6.5%?if the interest rate was 6.5%?

A. $858.96A. $858.96

Discount BondDiscount Bond

Like a coupon bond, but without couponsLike a coupon bond, but without couponsNo interest payments, is issued at a No interest payments, is issued at a

discount of face valuediscount of face valueEx. Face value of $1000 may be bought Ex. Face value of $1000 may be bought

for $900 and in one years time the owner for $900 and in one years time the owner will be paid $1000will be paid $1000

Examples include Canadian government Examples include Canadian government treasury bills and long term zero-coupon treasury bills and long term zero-coupon bondsbonds

Discount BondDiscount Bond

Similar formula as Coupon Bond, just Similar formula as Coupon Bond, just missing the coupon section:missing the coupon section:

P = F / (1 + i) ^ tP = F / (1 + i) ^ t P = price of bondP = price of bond F = face valueF = face value i = interest ratei = interest rate t = number of periodst = number of periods

$10,000, zero-coupon bond maturing in 7 $10,000, zero-coupon bond maturing in 7 years, interest rate is 4%. Selling price?years, interest rate is 4%. Selling price?

A. $7,599.18A. $7,599.18

Fixed Payment LoanFixed Payment Loan

Lender provides borrower funds, borrower Lender provides borrower funds, borrower pays back lender in fixed payments every pays back lender in fixed payments every period until the loan is paid off (ie. period until the loan is paid off (ie. Mortgage or auto loan)Mortgage or auto loan)

http://http://www.tdcanadatrust.comwww.tdcanadatrust.com/mortgages//mortgages/

Subprime LoansSubprime Loans

““A mortgage granted to a borrower A mortgage granted to a borrower considered subprime, that is, a person considered subprime, that is, a person with a less-than-perfect credit report. with a less-than-perfect credit report. Subprime borrowers have either missed Subprime borrowers have either missed payments on a debt or have been late with payments on a debt or have been late with payments. Lenders charge a higher payments. Lenders charge a higher interest rate to compensate for potential interest rate to compensate for potential losses from customers who may run into losses from customers who may run into trouble or default.”trouble or default.”

Subprime MeltdownSubprime Meltdown Background:Background: U.S. housing bubble began in early 2000’sU.S. housing bubble began in early 2000’s Low interest rates and new “mania” for buying houses Low interest rates and new “mania” for buying houses

caused housing prices to skyrocketcaused housing prices to skyrocket Some banks and newly-formed lending institutions took Some banks and newly-formed lending institutions took

advantage of the situation by offering long term advantage of the situation by offering long term mortgages to people who could normally not afford them mortgages to people who could normally not afford them – subprime borrowers– subprime borrowers

These borrowers relied on the capital value of their new These borrowers relied on the capital value of their new homes to pay off the mortgage, rather than their earning homes to pay off the mortgage, rather than their earning powerpower

Subprime Meltdown OutcomeSubprime Meltdown Outcome

The rapid rise in housing prices created a The rapid rise in housing prices created a bubble, which burst in late 2005bubble, which burst in late 2005

Housing prices began to fall and, Housing prices began to fall and, eventually, many subprime borrowers eventually, many subprime borrowers began to default on their loansbegan to default on their loans

This caused the collapse and bankruptcy This caused the collapse and bankruptcy of many of these institutions, and has hurt of many of these institutions, and has hurt the financial industry substantiallythe financial industry substantially

Subprime MeltdownSubprime Meltdown

Subprime Meltdown OutcomeSubprime Meltdown Outcome

This meltdown has depressed the US This meltdown has depressed the US economy, causing the Federal Reserve to economy, causing the Federal Reserve to lower it’s benchmark ratelower it’s benchmark rate

Has caused a weakening of the US dollar, Has caused a weakening of the US dollar, and billions of dollars of losses for and billions of dollars of losses for investment fundsinvestment funds

Subprime Meltdown InternationalSubprime Meltdown International

All of this has caused Canada to be All of this has caused Canada to be affected as wellaffected as well

Canadian dollar has risen against US Canadian dollar has risen against US dollar and is now at paritydollar and is now at parity

Depressed US housing market and high Depressed US housing market and high dollar has resulted in a crisis for the dollar has resulted in a crisis for the forestry industry (ie. Canfor mill closures) forestry industry (ie. Canfor mill closures)

Subprime MeltdownSubprime Meltdown

Initially, it was all caused by extremely low Initially, it was all caused by extremely low interest rates in the early 2000’s in order to interest rates in the early 2000’s in order to stimulate the economy after the dot-com stimulate the economy after the dot-com crashcrash

Measuring Interest RatesMeasuring Interest Rates

Yield to MaturityYield to MaturityCurrent YieldCurrent YieldYield on a Discount BasisYield on a Discount Basis

Yield to MaturityYield to Maturity

Interest rate that equates PV of cash flow Interest rate that equates PV of cash flow payments received with its value todaypayments received with its value today

Most accurate measure of interest ratesMost accurate measure of interest rates1. Simple Loans1. Simple Loans2. Fixed-Payment Loans2. Fixed-Payment Loans3. Coupon Bonds3. Coupon Bonds4. Perpetuity4. Perpetuity5. Discount Bonds5. Discount Bonds

Yield to MaturityYield to Maturity

Simple LoansSimple LoansQ. Pete borrows $100 from Bob and Bob Q. Pete borrows $100 from Bob and Bob

wants back $110 from him next year. wants back $110 from him next year. YTM?YTM?

A. PV = FV / ( 1 + i ) ^ tA. PV = FV / ( 1 + i ) ^ t100 = 110 / (1+ i ) ^ 1100 = 110 / (1+ i ) ^ 1 r = 0.1 = 10%r = 0.1 = 10% In a simple loan, i = YTMIn a simple loan, i = YTM

Yield to MaturityYield to Maturity

Fixed Payment LoanFixed Payment LoanQ. $100,000 loan, payments of $9439.29 Q. $100,000 loan, payments of $9439.29

for next 20 years. YTM?for next 20 years. YTM?LV = FP / (1 + i) + FP / (1 + i)^2 + ... + LV = FP / (1 + i) + FP / (1 + i)^2 + ... +

FP / (1 + i) ^ nFP / (1 + i) ^ n LV = Loan ValueLV = Loan Value FP = Fixed Yearly PaymentFP = Fixed Yearly Payment n = number of years until maturityn = number of years until maturity

A. 0.7 = 7%A. 0.7 = 7%

Yield to MaturityYield to Maturity

Coupon BondCoupon BondPP = C ((1-(1/(1+i)^n)/i) + FV/ (1+i)^nPP = C ((1-(1/(1+i)^n)/i) + FV/ (1+i)^n PP = Value of BondPP = Value of Bond C = Coupon PaymentC = Coupon Payment i = YTMi = YTM N = number of periodsN = number of periods FV = Face value of bondFV = Face value of bond

YTMYTM

Price of 8% coupon bond is $1122, FV of Price of 8% coupon bond is $1122, FV of $1000, 12 years to maturity, paid annually, $1000, 12 years to maturity, paid annually, what is YTM?what is YTM?

PP = C ((1-(1/(1+i)^n)/i) + FV/ (1+i)^nPP = C ((1-(1/(1+i)^n)/i) + FV/ (1+i)^n PP = Value of BondPP = Value of Bond C = Coupon PaymentC = Coupon Payment i = YTMi = YTM N = number of periodsN = number of periods FV = Face value of bondFV = Face value of bond

A. 0.65 = 6.5%A. 0.65 = 6.5%

Yield to MaturityYield to Maturity

PerpetuityPerpetuity Coupon bond with no maturity date (goes on Coupon bond with no maturity date (goes on

forever)forever) P = C / iP = C / i P = Price of perpetuityP = Price of perpetuity C = Yearly PaymentC = Yearly Payment i = YTMi = YTM

Q. Bond with price of $2000 and pays $100 Q. Bond with price of $2000 and pays $100 annually forever, what is YTM?annually forever, what is YTM?

Answer: 100 / 2000 = 0.05 = 5%Answer: 100 / 2000 = 0.05 = 5%

Yield to MaturityYield to Maturity

Discount BondDiscount Bond i = (F – P) / Pi = (F – P) / P F = Face ValueF = Face Value P = Current PriceP = Current Price

Ex. Treasury Bills that pays a FV of $1000 Ex. Treasury Bills that pays a FV of $1000 in 1 year. Purchase Price is $900. YTM?in 1 year. Purchase Price is $900. YTM?

A. i = (1000 – 900) / 900 = 0.111 = 11.1%A. i = (1000 – 900) / 900 = 0.111 = 11.1%

Yield to MaturityYield to Maturity

Yield to Maturity for bonds is the actual Yield to Maturity for bonds is the actual interest rateinterest rate

Bond prices and interest rates are Bond prices and interest rates are negatively relatednegatively related

Current YieldCurrent Yield

Yield of a bond at a point in timeYield of a bond at a point in timeDoes not reflect total returnDoes not reflect total returnUsed as approximation to describe interest Used as approximation to describe interest

rates for long term bondsrates for long term bondsCY = C / PCY = C / P C = Coupon paymentC = Coupon payment P = Current priceP = Current price

Current YieldCurrent Yield

Q. What is the current yield of a $1000, Q. What is the current yield of a $1000, 8% coupon bond maturing in 3 years and 8% coupon bond maturing in 3 years and selling for a price of $1200?selling for a price of $1200?

A. CY = C / P A. CY = C / P = 80 / 1200 = 80 / 1200 = 0.0666 = 6.66%= 0.0666 = 6.66%

Yield on a Discount BasisYield on a Discount Basis

Most accurate measure of interest rates Most accurate measure of interest rates for short term bondsfor short term bonds

Yield of bills with a maturity of less than Yield of bills with a maturity of less than one yearone year

i = (F – P) / P * 365 / Days to Maturityi = (F – P) / P * 365 / Days to Maturity i = Yield on a discount basisi = Yield on a discount basis F = Face ValueF = Face Value P = Purchase PriceP = Purchase Price

Yield on a Discount BasisYield on a Discount Basis

Q. 91 day treasury bill selling for $988 Q. 91 day treasury bill selling for $988 with FV of $1000with FV of $1000

A. A. i = (F – P) / P * 365 / Days to Maturityi = (F – P) / P * 365 / Days to Maturity i = (1000 – 988) / 988 * 365 / 91 = i = (1000 – 988) / 988 * 365 / 91 =

0.0487 = 4.87%0.0487 = 4.87%

Central Banks and Interest Central Banks and Interest RatesRates

Very important in determining interest Very important in determining interest ratesrates

Central banks determine lending rates and Central banks determine lending rates and heavily influence market interest ratesheavily influence market interest rates

Lending rate is the rate that the central Lending rate is the rate that the central bank will lend to other banks, thereby bank will lend to other banks, thereby insuring that all lending rates (inter-bank or insuring that all lending rates (inter-bank or otherwise) are above this rateotherwise) are above this rate

Central BanksCentral BanksCan be government controlled, or Can be government controlled, or

independent of government (Bank of independent of government (Bank of Canada, Federal Reserve)Canada, Federal Reserve)

Determine countries monetary policyDetermine countries monetary policySole issuer of the countries' currencySole issuer of the countries' currencySupervises banking industrySupervises banking industry ““Lender of last resort” and governments Lender of last resort” and governments

bankerbankerManages gold reservesManages gold reservesSets interest rateSets interest rate

International Interest RatesInternational Interest Rates

Canada - 4.5%Canada - 4.5%United States - 4.75%United States - 4.75%United Kingdom - 5.75%United Kingdom - 5.75%Europe - 4.0%Europe - 4.0%Japan - 0.5%Japan - 0.5%Australia - 6.5%Australia - 6.5%China - 7.02% China - 7.02% India - 7.75%India - 7.75%

CanadaCanada

Determined by the Bank of CanadaDetermined by the Bank of Canada

United StatesUnited States

Determined by the Federal ReserveDetermined by the Federal Reserve

United KingdomUnited KingdomDetermined by the Bank of EnglandDetermined by the Bank of England

JapanJapan Determined by Bank of JapanDetermined by Bank of Japan

Japan Interest Rates

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JapanJapan

Island nation with no natural resources Island nation with no natural resources and very high population, and yet has the and very high population, and yet has the second largest economy in the world with second largest economy in the world with the highest life expectancy, and very the highest life expectancy, and very technologically advancedtechnologically advanced

For past 15 years has been in prolonged For past 15 years has been in prolonged stagnation due to many factorsstagnation due to many factors

JapanJapan

As a result, the interest rate has been very As a result, the interest rate has been very low (0.1% for past 5 years) in order to low (0.1% for past 5 years) in order to stimulate the economystimulate the economy

Extremely low compared to Canada Extremely low compared to Canada (4.5%)(4.5%)

Low interest rate has resulted in negative Low interest rate has resulted in negative inflationinflation

Interest Rates vs ReturnsInterest Rates vs Returns

Rate of return on a bond not necessarily the yield to Rate of return on a bond not necessarily the yield to maturity of the bondmaturity of the bond

Rate of return is dependant on behaviour of market Rate of return is dependant on behaviour of market interest rates over the holding period of the bondinterest rates over the holding period of the bond

If a bond is held for its entire life, then yield to maturity is If a bond is held for its entire life, then yield to maturity is equal to the return of the bondequal to the return of the bond

RET = (C / Pt) + (Pt+1 – Pt) / PtRET = (C / Pt) + (Pt+1 – Pt) / Pt RET = ReturnRET = Return Pt = Initial Price of BondPt = Initial Price of Bond Pt+1 = Selling Price of BondPt+1 = Selling Price of Bond C = Coupon PaymentC = Coupon Payment

Interest Rates vs ReturnsInterest Rates vs Returns

Q. What is the rate of return on a bond Q. What is the rate of return on a bond bought for $988 and sold one year later for bought for $988 and sold one year later for $1200? The coupon rate is 5%, and face $1200? The coupon rate is 5%, and face value is $1000.value is $1000.

A. (C / Pt) + (Pt+1 – Pt) / PtA. (C / Pt) + (Pt+1 – Pt) / Pt

= ( 50 / 988) + (1200 – 988) / 988= ( 50 / 988) + (1200 – 988) / 988

= 0.0506 + 0.2146= 0.0506 + 0.2146

= 0.2652 = 26.5%= 0.2652 = 26.5%

Volatility of BondsVolatility of Bonds

Longer term = larger risk/more volatilityLonger term = larger risk/more volatilityKnown as interest rate riskKnown as interest rate risk

Real vs Nominal RatesReal vs Nominal Rates

So far, i = nominal interest rateSo far, i = nominal interest rateReal interest rate is a more accurate rate Real interest rate is a more accurate rate

that is adjusted for inflationthat is adjusted for inflation i = real interest rate + expected inflation i = real interest rate + expected inflation OR OR Real interest rate = i – expected inflationReal interest rate = i – expected inflation i = nominal ratei = nominal rateActual inflation is not known until laterActual inflation is not known until later

Real vs Nominal RatesReal vs Nominal Rates

When real interest rate is low, there are When real interest rate is low, there are greater incentives to borrow and fewer to greater incentives to borrow and fewer to lendlend

Measured by indexed bondsMeasured by indexed bonds Interest and principal payments are adjusted Interest and principal payments are adjusted

for price levels by indexing to the Consumer for price levels by indexing to the Consumer Price Index (CPI)Price Index (CPI)

Real vs Nominal RatesReal vs Nominal Rates

Real rate is lower than nominal rate Real rate is lower than nominal rate because of inflationbecause of inflation

Learning ObjectivesLearning Objectives

1. Detail terms present value and interest rate1. Detail terms present value and interest rate 2. Discern measurements of interest rates: YTM, 2. Discern measurements of interest rates: YTM,

Current Yield, Yield on Discount basisCurrent Yield, Yield on Discount basis 3. Illustrate inverse relationship of bond prices 3. Illustrate inverse relationship of bond prices

and interest ratesand interest rates 4. Explain difference between nominal and real 4. Explain difference between nominal and real

interest ratesinterest rates 5. Explain difference between interest rates and 5. Explain difference between interest rates and

rates of returnrates of return

The EndThe End

Questions?