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Chapter 4. Present and Future Value. Future Value Present Value Applications IRR Coupon bonds Real vs. nominal interest rates. 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.

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Chapter 4. Present and Future Value

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### Chapter 4. Present and Future Value

• Future Value

• Present Value

• Applications

• IRR

• Coupon bonds

• Real vs. nominal interest rates

### 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

### 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

### future 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

### Rule 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

### present value (PV)

• work backwards

• if get \$100 in n years,

what is that worth today?

\$100

PV

=

(1+ i)n

### example

• receive \$100 in 3 years

• i = 5%

• what is PV?

\$100

PV

=

=

\$86.36

(1+ .05)3

• With PV, we discount future cash flows

• Payment we wait for are worth LESS

• i = interest rate

• = discount rate

• = yield

• annual basis

n

PV

PV

i

### PV, FV and 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

i = 6.96%

### Applications

• Internal rate of return (IRR)

• Coupon Bond

### Application 1: IRR

• Interest rate

• Where PV of cash flows = cost

• Used to evaluate investments

• Compare IRR to cost of capital

### Example

• Computer course

• \$1800 cost

• Bonus over the next 5 years of \$500/yr.

• We want to know i where

PV bonus = \$1800

Solve for i?

Trial & error

Online calc.

12.05%

### Example

• Bonus: 700, 600, 500, 400, 300

• Solve

i = 14.16%

### Example

• Bonus: 300, 400, 500, 600, 700

• Solve

i = 10.44%

### Example: annuity vs. lump sum

• choice:

• \$10,000 today

• \$4,000/yr. for 3 years

• which one?

• implied discount rate?

i = 9.7%

### Application 2: Coupon Bond

• purchase price, P

• promised of a series of payments until maturity

• face value at maturity, F

(principal, par value)

• coupon payments (6 months)

• size of coupon payment

• annual coupon rate

• face value

• 6 mo. pmt. = (coupon rate x F)/2

### what determines the price?

• 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)

### example: 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.

### 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.

• YTM solves the equation

• i/2 is 6-month discount rate

• i is yield to maturity

• how to solve for i?

• trial-and-error

• bond table*

• financial calculator

• price between \$9816 & \$9726

• YTM is between 7% and 7.5%

(7.37%)

### P, 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 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 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

• 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 Rates

• thusfar we have calculated nominal interest rates

• ignores effects of rising inflation

• inflation affects purchasing power of future payments

### example

• \$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

### real interest rate, ir

nominal interest rate = i

expected inflation rate = πe

approximately:

i = ir + πe

• The Fisher equation

orir = 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

### inflation 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