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