Ch 5. Time Value of Money

Goal: • to learn time value of money and discounted cash

flows• To understand a tool to value the expected future

value in terms of present value.

• Cash flow: Cash in (inflow) or out (outflow) over times.

• Why we need this tool?

- Mainly for financial decisions:

a) Project valuation

b) Security valuation – stock and bond

I. Time Value of Money: Single Time.

1. Future Value and Compounding

• Future value:

The amount of money an investment will grow to over some period of time.

Ex) Investing $200 today and after 2 yrs, the investment will become $400. The $400 is the Future value.

2) FV calculation

1) A single period:

FV = Investment * (1+k)

Ex) Invest $100 in the saving accounts with the 10% interest per year.

FV =100*(1+0.1)=110

Future value is $110

2) More than one period

Ex) Invest $100 in the saving account with the 10% interest rate for 2 yrs

FV1 = 100*(1+0.1)=110

FV2 = 110*(1+0.1)=121.

tk)(1 Investment FV

20.1)(1100 2 FV

Here, we reinvest the first interest to get the future value. This is the compounding. That is, compounding the interest means earning interest on interest.

The simple interest means no reinvestment on the interest.

Ex) invest $100 with 10% with simple interestFV=100+2*0.1*100=120

3) Decomposing FV and Impact of compounding

• FV = investment + simple interest + compound interest

• The impact of compounding is small over the short period

2. Present value and discounting

- Def: the current value of future cash flows discounted at the appropriate discount rate. In other word, converting FV to PV with discount rate

- Why we need PV?

We use the PV in evaluating projects or securities with different maturities and FV

1) How to calculate PV

• Starting from the FV concept

)(

)1(

valuepresentVP

kinvestmentFV t

(1) Single period case

PV =FV/(1+r)

Ex) You need $400 to buy text books next year and you can earn 7% on your money

How much you have to put up today?

PV =400/(1+0.07)=373.83

(2) Multi-period

Ex) Need $1000 to buy a text book after 2 yrs and you can earn 7% on your money

nkFVPV )1/(

PV 1000 / (1+ 0.07)2

3. Why we need the FV and PV concept?

If you have to pick up one out of three saving accounts with the same maturity but different rates, How do you want to evaluate and compare the accounts?

A) $1000, 8% and 3yrs

B) $2000, 6% and 3yrs

C) $1500, 7% and 3yrs

• If you have to pick up one out of three saving accounts with the maturities and rates, How do you want to evaluate and compare the accounts?

A) $3000, 8% and 1yrs

B) $4000, 6% and 2yrs

C) $5000, 7% and 3yrs

4. Determining the discount rate

• How to find k (rate)?

(1) Use Future value table

(2) Approximation

( / )FV PV t

1

1

5. Finding the number of periods

• Approximation:

)ln(

)/ln(

k

PVFVt

6. More about Multiple Periods

• Until now, we mainly deal with cases with yearly maturities. That is 1 yr, 2 yrs, or 3 yrs

• What happen if we have to deal with semiannual, quarterly or monthly.

• Do we have to use the same FV-PV equation

• Yes! But need some revisions for more compounding.

R: annual ratet: yearsm: revision for different time frame

ex) Yearly: m=1 Semiannual : m=2 Quarterly: m=4

Monthly: m=12 Continuous compounding:

mt

m

kPVFV )1(

...)7183.2( eePVFV t

• Ex) Initial investment is $100 and semi-annually compounding for next 2 yrs. And current interest rate is 7%. What is the future value of $100 after 2 yrs?

FV 100 10 07

22 2(

.)