Chapter 4. Present and Future Value Future Value Present Value Applications IRR Coupon bonds Real...
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Transcript of Chapter 4. Present and Future Value Future Value Present Value Applications IRR Coupon bonds Real...
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