chapter 4. present and future value future value present value applications irr coupon bonds real...
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Chapter 4. Present and Future ValueChapter 4. Present and Future ValueChapter 4. Present and Future ValueChapter 4. Present and Future Value
• Future Value
• Present Value
• Applications IRR Coupon bonds
• Real vs. nominal interest rates
• Future Value
• Present Value
• Applications IRR Coupon bonds
• Real vs. nominal interest rates
Present & Future ValuePresent & Future ValuePresent & Future ValuePresent & 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 value (FV)example: future value (FV)example: future value (FV)example: future value (FV)
• $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 annual interest rate is i:
= $100(1 + i)n
• with FV, we compound cash flow today to the future
• of $100 in n years if annual interest rate is i:
= $100(1 + i)n
• with FV, we compound cash flow today to the future
Rule of 72Rule of 72Rule of 72Rule of 72
• how long for $100 to double to $200?
• approx. 72/i
• at 5%, $100 will double in 72/5 = 14.4 $100(1+i)14.4 = $201.9
• how long for $100 to double to $200?
• approx. 72/i
• at 5%, $100 will double in 72/5 = 14.4 $100(1+i)14.4 = $201.9
present value (PV)present value (PV)present value (PV)present value (PV)
• 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
• With PV, we discount future cash flows Payment we wait for are worth
LESS
• With PV, we discount future cash flows Payment we wait for are worth
LESS
• i = interest rate
• = discount rate
• = yield
• annual basis
• i = interest rate
• = discount rate
• = yield
• annual basis
About iAbout iAbout iAbout i
n
i
PV
PV
PV, FV and iPV, FV and iPV, FV and iPV, FV and i
• given PV, FV, calculate I
example:
• CD
• initial investment $1000
• end of 5 years $1400
• what is i?
• given PV, FV, calculate I
example:
• CD
• initial investment $1000
• end of 5 years $1400
• what is i?
• is it 40%?
• is 40%/5 = 8%?
• No….
• i solves
• is it 40%?
• is 40%/5 = 8%?
• No….
• i solves
5)1(
1400$1000$
i
i = 6.96%
ApplicationsApplicationsApplicationsApplications
• Internal rate of return (IRR)
• Coupon Bond• Internal rate of return (IRR)
• Coupon Bond
Application 1: IRRApplication 1: IRRApplication 1: IRRApplication 1: IRR
• Interest rate Where PV of cash flows = cost
• Used to evaluate investments Compare IRR to cost of capital
• Interest rate Where PV of cash flows = cost
• Used to evaluate investments Compare IRR to cost of capital
Example Example Example Example
• Computer course $1800 cost Bonus over the next 5 years of
$500/yr.
• We want to know i where
PV bonus = $1800
• Computer course $1800 cost Bonus over the next 5 years of
$500/yr.
• We want to know i where
PV bonus = $1800
Solve the following:Solve the following:Solve the following:Solve the following:
Solve for i?
• Trial & error
• Spreadsheet
• Online calc.
Solve for i?
• Trial & error
• Spreadsheet
• Online calc.
Answer?
• 12.05%
Answer?
• 12.05%
Example Example Example Example
• Bonus: 700, 600, 500, 400, 300
• Solve• Bonus: 700, 600, 500, 400, 300
• Solve
5432 1
300$
1
400$
1
500$
1
600$
1
700$1800
iiiii
i = 14.16%
Example Example Example Example
• Bonus: 300, 400, 500, 600, 700
• Solve• Bonus: 300, 400, 500, 600, 700
• Solve
5432 1
700$
1
600$
1
500$
1
400$
1
300$1800
iiiii
i = 10.44%
Example: annuity vs. lump sumExample: annuity vs. lump sumExample: annuity vs. lump sumExample: annuity vs. lump sum
• choice: $10,000 today $4,000/yr. for 3 years
• which one?
• implied discount rate?
• choice: $10,000 today $4,000/yr. for 3 years
• which one?
• implied discount rate?
32 1
000,4$
1
000,4$
1
000,4$000,10
iii
i = 9.7%
• purchase price, P
• promised of a series of payments until maturity face value at maturity, F
(principal, par value) coupon payments (6 months)
• purchase price, P
• promised of a series of payments until maturity face value at maturity, F
(principal, par value) coupon payments (6 months)
Application 2: Coupon BondApplication 2: Coupon BondApplication 2: Coupon BondApplication 2: Coupon Bond
• size of coupon payment annual coupon rate face value 6 mo. pmt. = (coupon rate x F)/2
• size of coupon payment annual coupon rate face value 6 mo. pmt. = (coupon rate x F)/2
what determines the price?what determines the price?what determines the price?what determines the price?
• size, timing & certainty of promised payments
• assume certainty
• size, timing & certainty of promised payments
• assume certainty
P = PV of payments
• i where P = PV(pmts.) is known as the yield to maturity (YTM)• i where P = PV(pmts.) is known as
the yield to maturity (YTM)
example: coupon bondexample: coupon bondexample: coupon bondexample: 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.
what are the payments?what are the payments?what are the payments?what are the payments?
• 6 mos. $300
• 1 year $300
• 1.5 yrs. $300 …..
• 2 yrs. $300 + $10,000
• a total of 4 semi-annual pmts.
• 6 mos. $300
• 1 year $300
• 1.5 yrs. $300 …..
• 2 yrs. $300 + $10,000
• a total of 4 semi-annual pmts.
• 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
• 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
• 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
Maturity & bond price volatilityMaturity & bond price volatilityMaturity & bond price volatilityMaturity & 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
Real vs. Nominal Interest RatesReal vs. Nominal Interest RatesReal vs. Nominal Interest RatesReal vs. Nominal Interest Rates
• thusfar we have calculated nominal interest rates ignores effects of rising
inflation inflation affects purchasing
power of future payments
• thusfar we have calculated nominal interest rates ignores effects of rising
inflation inflation affects purchasing
power of future payments
exampleexampleexampleexample
• $100,000 mortgage
• 6% fixed, 30 years
• $600 monthly pmt.
• at 2% annual inflation, by 2037 $600 would buy about half as
much as it does today $600/(1.02)30 = $331
• $100,000 mortgage
• 6% fixed, 30 years
• $600 monthly pmt.
• at 2% annual inflation, by 2037 $600 would buy about half as
much as it does today $600/(1.02)30 = $331
• so interest charged by a lender reflects the loss due to inflation over the life of the loan
• so interest charged by a lender reflects the loss due to inflation over the life of the loan
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
[exactly: (1+i) = (1+ir)(1+ πe )]
nominal interest rate = i
expected inflation rate = πe
approximately:
i = ir + πe
• The Fisher equation
or ir = i – πe
[exactly: (1+i) = (1+ir)(1+ 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
inflation and iinflation and iinflation and iinflation and i
• if inflation is high…
• lenders demand higher nominal rate, especially for long term loans
• long-term i depends A LOT on inflation expectations
• if inflation is high…
• lenders demand higher nominal rate, especially for long term loans
• long-term i depends A LOT on inflation expectations