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Chapter 4. Present and Future Value

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- Future Value
- Present Value
- Applications
- IRR
- Coupon bonds

- Real vs. nominal interest rates

- time value of money
- $100 today vs. $100 in 1 year
- not indifferent!
- money earns interest over time,
- and we prefer consuming today

- $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

- 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

- 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

- work backwards
- if get $100 in n years,
what is that worth today?

$100

PV

=

(1+ i)n

- 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

- 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%

- Internal rate of return (IRR)
- Coupon Bond

- Interest rate
- Where PV of cash flows = cost

- Used to evaluate investments
- Compare IRR to cost of capital

- 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

Spreadsheet

Online calc.

Answer?

12.05%

- Bonus: 700, 600, 500, 400, 300
- Solve

i = 14.16%

- Bonus: 300, 400, 500, 600, 700
- Solve

i = 10.44%

- choice:
- $10,000 today
- $4,000/yr. for 3 years

- which one?
- implied discount rate?

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)

- face value at maturity, F

- size of coupon payment
- annual coupon rate
- face value
- 6 mo. pmt. = (coupon rate x F)/2

- 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)

- 2 year Tnote, F = $10,000
- coupon rate 6%
- price of $9750
- what are interest payments?
(.06)($10,000)(.5) = $300

- every 6 mos.

- 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
- spreadsheet

- price between $9816 & $9726
- YTM is between 7% and 7.5%
(7.37%)

- 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

- 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

- thusfar we have calculated nominal interest rates
- ignores effects of rising inflation
- inflation affects purchasing power of future payments

- $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

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

- if inflation is high…
- lenders demand higher nominal rate, especially for long term loans
- long-term i depends A LOT on inflation expectations