# Class Notes

**Topics:**Compound interest, Time value of money, Net present value

**Pages:**37 (6260 words)

**Published:**January 13, 2013

Bang Dang Nguyen b.nguyen@jbs.cam.ac.uk http://www.jbs.cam.ac.uk/research/faculty/nguyenb.html

Principles of Finance

Page 1

Course Road Map

I. Present Value and Stock Valuation

II. Project Appraisal and Capital Budgeting III. Risk and Return and Portfolio Selection IV. CAPM and WACC V. Capital Structure and Dividend Policy VI. Options and Real Options

Principles of Finance

Present Value - Page 2

Present Value - Contents

• Valuing Cash Flows – The Time Value of Money – Future Value – Present Value – Value Additivity • Project Evaluation – Net Present Value – The Net Present Value Rule • Shortcuts to Special Cash Flows – Perpetuities - Growing Perpetuities – Annuities - Growing Annuities • Compound Interest Rates – Compound Interest versus Simple Interest – Discrete Compounding – Continuous Compounding – Effective Annual Yield • Adjusting for Inflation Principles of Finance Present Value - Page 3

Valuing Cash Flows

• Most investment decisions involve trade-offs over time. – Within a project - Trade-off between • payoff now, or • investing now and receive payoff later – Across projects - Trade-off between • investment 1 which involves a stream of payoffs, or • investment 2 with different stream of payoffs.

Problem: How do we quantitatively compare cash flows that occur at different times? What is the time value of money? Principles of Finance Present Value - Page 4

The Time Value of Money

Suppose you are asked if interested in: • Investing $1 today to • Receive $0.50 each of the next 2 years.

The answer is not ambiguous: You should certainly NOT do it

The reason is that having one dollar today is worth more than having the same dollar two years in the future.

Principles of Finance

Present Value - Page 5

Time Value of Money

If you have one dollar today, you can invest. If the rate of return is 5% per year, you would receive: • $1 × 1.05 = $1.05 one year from now

If after one year you invest the principal together with the interest for a second year, you then receive: $1.05 × 1.05 = $1.1025 two years from now

[ or

$1 × (1.05)2

]

This certainly better than the proposed $0.5+$0.5.

Principles of Finance

Present Value - Page 6

Future Value

If we continued one more year, we would receive: • $1.1025 × 1.05 = $1.157625 three years from now

[ or

$1 × (1.05)3

]

More generally, the Future Value of a cash flow of C dollars in T years when invested at a rate-of-return r is: FV(C) = $C × (1 + r)T

Principles of Finance

Present Value - Page 7

Present Value

Let us now flip the story: • Question: How much is $1 to be received in 3 years, worth to us today We know it is less than $1 ...

Answer: It is worth today the amount we would have to invest today to receive $1 in 3 years.

Principles of Finance

Present Value - Page 8

Present Value

• We have seen that, if we invest $C today at a rate-of-return r, it’s Future Value in 3 years is FV(C) = $C × (1 + r)3 • Hence, to receive $1 in 3 years, we must deposit today an amount $C such that FV(C) = $1 That is: $C × (1 + r)3 = $1 => $C

$1 (1 r )3

Principles of Finance

Present Value - Page 9

Present Value

• We say the Present Value of $1 to be received in 3 years is

$C

$1 (1 r )3

• If the interest rate is 5%, then the present value of $1 to be received in 3 years from now is $1/(1.05)3 = $0.864. More generally, the Present Value of C dollars to be received in T years, when the interest rate is r, is

PV (C )

where

$C = $C [Discount factor at r , maturity T ] (1 r )T 1 (1 r )T

Present Value - Page 10

[Discount factor at r , maturity T ]

Principles of Finance

Present Value - Example

• Receive either – A. $10M in 5 years, or – B. $15M in 15 years. Which is better if r = 5% ? Calculate the respective present values:

PV A

PVB

$10 = 7.84 (1 0.05)5

$15 = 7.22 (1 0.05)15

we find that opportunity A is...

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