Chapter 4. Understanding Interest Chapter 4. Understanding Interest RatesRatesChapter 4. Understanding Interest Chapter 4. Understanding Interest RatesRates
• Present Value
• Yield to Maturity
• Other Yields
• Other Measurement Issues
• Present Value
• Yield to Maturity
• Other Yields
• Other Measurement Issues
I. Measuring Interest RatesI. Measuring Interest RatesI. Measuring Interest RatesI. Measuring Interest Rates
A. Credit Market Instruments
• simple loan borrower pays back loan and
interest in one lump sum
A. Credit Market Instruments
• simple loan borrower pays back loan and
interest in one lump sum
• fixed-payment loan loan is repaid with equal (monthly)
payments each payment is combination of
principal and interest
• fixed-payment loan loan is repaid with equal (monthly)
payments each payment is combination of
principal and interest
• coupon bond purchase price (P) interest payments (6 months) face value at maturity (F) size of interest payments
-- coupon rate
-- face value
• coupon bond purchase price (P) interest payments (6 months) face value at maturity (F) size of interest payments
-- coupon rate
-- face value
• discount bond zero coupon bond purchased price less than face
value
-- F > P face value at maturity no interest payments
• discount bond zero coupon bond purchased price less than face
value
-- F > P face value at maturity no interest payments
B. Present & Future ValueB. Present & Future ValueB. Present & Future ValueB. Present & Future Value
• time value of money
• $100 today vs. $100 in 1 year not indifferent! money earns interest over time, and we prefer consuming today
• time value of money
• $100 today vs. $100 in 1 year not indifferent! money earns interest over time, and we prefer consuming today
example: future valueexample: future valueexample: future valueexample: future value
• $100 today
• interest rate 5% annually
• at end of 1 year:
100 + (100 x .05)
= 100(1.05) = $105
• at end of 2 years:
100 + (1.05)2 = $110.25
• $100 today
• interest rate 5% annually
• at end of 1 year:
100 + (100 x .05)
= 100(1.05) = $105
• at end of 2 years:
100 + (1.05)2 = $110.25
future valuefuture valuefuture valuefuture value
• of $100 in n years if interest rate is i:
= $100(1 + i)n • of $100 in n years if interest rate is i:
= $100(1 + i)n
present valuepresent valuepresent valuepresent value
• work backwards
• if get $100 in n years,
what is that worth today?
• work backwards
• if get $100 in n years,
what is that worth today?
PV = $100
(1+ i)n
exampleexampleexampleexample
• receive $100 in 3 years
• i = 5%
• what is PV?
• receive $100 in 3 years
• i = 5%
• what is PV?
PV = $100
(1+ .05)3
= $86.36
n
i
PV
PV
C. Yield to Maturity (YTM)C. Yield to Maturity (YTM)C. Yield to Maturity (YTM)C. Yield to Maturity (YTM)
• a measure of interest rate
• interest rate where• a measure of interest rate
• interest rate where
P = PV of cash flows
example 1: simple loanexample 1: simple loanexample 1: simple loanexample 1: simple loan
• loan = $1500, 1 year, 6%
• future payment
= $1500(1+.06) = $1590
• yield to maturity, i
• loan = $1500, 1 year, 6%
• future payment
= $1500(1+.06) = $1590
• yield to maturity, i
$1500 = $1590
(1+ i)i = 6%
example 2: fixed pmt. loanexample 2: fixed pmt. loanexample 2: fixed pmt. loanexample 2: fixed pmt. loan
• $15,000 car loan, 5 years
• monthly pmt. = $300
• so $15,000 is price today
• cash flow is 60 pmts. of $300
• $15,000 car loan, 5 years
• monthly pmt. = $300
• so $15,000 is price today
• cash flow is 60 pmts. of $300
• YTM solves• YTM solves
• i/12 is monthly discount rate
• i is yield to maturity• i/12 is monthly discount rate
• i is yield to maturity
• how to solve for i? trial-and-error bond table* financial calculator spreadsheet
• how to solve for i? trial-and-error bond table* financial calculator spreadsheet
• payment between $297.02 & $300.57
• YTM is between 7% and 7.5%
(7.42%)
• payment between $297.02 & $300.57
• YTM is between 7% and 7.5%
(7.42%)
example 3: coupon bondexample 3: coupon bondexample 3: coupon bondexample 3: coupon bond
• 2 year Tnote, F = $10,000
• coupon rate 6%
• price of $9750
• what are interest payments?
(.06)($10,000)(.5) = $300 every 6 mos.
• 2 year Tnote, F = $10,000
• coupon rate 6%
• price of $9750
• what are interest payments?
(.06)($10,000)(.5) = $300 every 6 mos.
• YTM solves the equation• YTM solves the equation
• i/2 is 6-month discount rate
• i is yield to maturity• i/2 is 6-month discount rate
• i is yield to maturity
• price between $9816 & $9726
• YTM is between 7% and 7.5%
(7.37%)
• price between $9816 & $9726
• YTM is between 7% and 7.5%
(7.37%)
P, F and YTMP, F and YTMP, F and YTMP, F and YTM
• P = F then YTM = coupon rate
• P < F then YTM > coupon rate bond sells at a discount
• P > F then YTM < coupon rate bond sells at a premium
• P = F then YTM = coupon rate
• P < F then YTM > coupon rate bond sells at a discount
• P > F then YTM < coupon rate bond sells at a premium
• P and YTM move in opposite directions
• interest rates and value of debt securities move in opposite directions if rates rise, bond prices fall if rates fall, bond prices rise
• P and YTM move in opposite directions
• interest rates and value of debt securities move in opposite directions if rates rise, bond prices fall if rates fall, bond prices rise
example 4: discount bondexample 4: discount bondexample 4: discount bondexample 4: discount bond
• 90 day Tbill,
• P = $9850, F = $10,000
• YTM solves
• 90 day Tbill,
• P = $9850, F = $10,000
• YTM solves
36590
1
000109850
i
,$$
9850
00010
365
901
$
,$i
19850
00010
365
90
$
,$i
9850
985000010
365
90
$
,$i
90
365
9850
985000010
$
,$i = 6.18%
• in general,• in general,
D. Current YieldD. Current YieldD. Current YieldD. Current Yield
• approximation of YTM for coupon bonds• approximation of YTM for coupon
bonds
ic =annual coupon payment
bond price
• better approximation when maturity is longer P is close to F
• better approximation when maturity is longer P is close to F
example 5example 5example 5example 5
• 2 year Tnotes, F = $10,000
• P = $9750, coupon rate = 6%
• current yield
• 2 year Tnotes, F = $10,000
• P = $9750, coupon rate = 6%
• current yield
ic =600
9750= 6.15%
• current yield = 6.15%
• true YTM = 7.37%
• lousy approximation only 2 years to maturity selling 25% below F
• current yield = 6.15%
• true YTM = 7.37%
• lousy approximation only 2 years to maturity selling 25% below F
E. Discount YieldE. Discount YieldE. Discount YieldE. Discount Yield
• yield on a discount basis
• approximation of YTM• yield on a discount basis
• approximation of YTM
idb = F - P
Fx
360d
• compare w/ YTM• compare w/ YTM
idb = F - P
Fx
360d
iytm = F - P
Px
365d
iytm > idb
example 6:example 6:example 6:example 6:
• 90-day Tbill, price $9850• 90-day Tbill, price $9850
idb = 10,000 - 9850
9850x
36090
= 6%
true YTM is 6.18%
II. Other measurement issuesII. Other measurement issuesII. Other measurement issuesII. Other measurement issues
A. Interest rates vs. return
• YTM assumes bond is held until maturity
• if not, resale price is important
A. Interest rates vs. return
• YTM assumes bond is held until maturity
• if not, resale price is important
B. Maturity & bond price volatilityB. Maturity & bond price volatilityB. Maturity & bond price volatilityB. Maturity & bond price volatility
• YTM rises from 6 to 8% bond prices fall but 10-year bond price falls the
most
• Prices are more volatile for longer maturities long-term bonds have greater
interest rate risk
• YTM rises from 6 to 8% bond prices fall but 10-year bond price falls the
most
• Prices are more volatile for longer maturities long-term bonds have greater
interest rate risk
• Why? long-term bonds “lock in” a
coupon rate for a longer time if interest rates rise
-- stuck with a below-market coupon rate
if interest rates fall
-- receiving an above-market coupon rate
• Why? long-term bonds “lock in” a
coupon rate for a longer time if interest rates rise
-- stuck with a below-market coupon rate
if interest rates fall
-- receiving an above-market coupon rate
C. Real vs. Nominal Interest RatesC. Real vs. Nominal Interest RatesC. Real vs. Nominal Interest RatesC. Real vs. Nominal Interest Rates
• thusfar we have calculated nominal interest rates ignores effects of rising inflation
• thusfar we have calculated nominal interest rates ignores effects of rising inflation
real interest rate, ireal interest rate, irrreal interest rate, ireal interest rate, irr
nominal interest rate = i
expected inflation rate = πe
approximately:
i = ir + πe
• The Fisher equation
or ir = i - πe
nominal interest rate = i
expected inflation rate = πe
approximately:
i = ir + πe
• The Fisher equation
or ir = i - πe
• real interest rates measure true cost of borrowing
• why? as inflation rises, real value of loan
payments falls, so real cost of borrowing falls
• real interest rates measure true cost of borrowing
• why? as inflation rises, real value of loan
payments falls, so real cost of borrowing falls
