Ch 4. 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) PVIF (k,t)

As we can see, if we know the

We can calculate the PV easily.

This is called PVIF (k,t) , present value interest factor

(4) How to use PVIF table

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 investment opportunities with the maturities and rates, How do you want to evaluate and compare them?

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:

)1ln(

)/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(

.)

II. FV and PV with multiple cash flows

1) FV with multiple cash flows: Two methods

(1) Rolling over FV year by year(2) FV=FV1+FV2+FV3….

Ex) Deposit $100 every year for 3 yrs. And 10% interest rate. FV?

2) PV with multiple cash flows: Two method

(1) Rolling back year by year

(2) PV=PV1+PV2+…..

Ex) You are supposed to need $1000 in one year and $2000 in the second year. If you can earn 9% on your money, how much you have to put up today?

2. Annuities and Perpetuities

1) Def of Annuity:

Constant cash flows for a fixed period of time

Ex) car loan

Ex) Assets with promised to pay $500 at the end of the each of the next three years. What is the price of the asset now?

Answer:

2) Formula for Annuity Present Value

43.1243)1.1/(500)1.1/(5001.1/500 32 PV

k

PVfactorC

k

kCPV

t

1])1/(1[1

kk

CPVTherefore

kCkPV

Ck

CkPVPV

k

C

k

C

k

CCkPV

k

C

k

C

k

CPV

t

t

t

t

t

])1(

11[

]1)1(

1[)(

)1()1(

)1(.....

)1()1()1(

)1(.....

)1()1(

12

2

Ex) You stop by a car dealer shop and find a really good car. The sticker price of the car is $15000. But you don’t have money now. So, want installment payment over 4 yrs. Over conversation, the dealer suggests $632 per month for 48 month at 1% per month.

How much is going to be your PV of total installments?

2-1) How to use the Annuity table in calculation?

- Calculating PV

2-2) Finding C

Ex) You stop by a car dealer shop and find a really good car. The sticker price of the car is $15000. But you don’t have money now. So, if you want installment payment over 4 yrs, how much you have to pay monthly? (Here interest rate is 12%)

2-3) Finding rate

Ex) an insurance company offers to pay you $1000 per year for 10 years if you pay $6710 up front. What rate is used in this annuity?

3) Def of perpetuities:

An annuity in which the cash flow continues forever

4) Formula for PV of perpetuities

PV=C/k

Ex) Preferred stock – promised fixed dividend every period forever.

A company want to sell preferred stock at $100 per share. How much of dividend it has to pay. Currently the similar preferred stock is sold at $40 with $1 dividend.

i) Calculate r:

R= 1/40 = 0.025

ii) Calculate C:

100 = C/0.025. Then, C=2.5

5) FV for Annuities

kkCFV t /]1)1[(

k

kCFV

kCkFV

kCCkFVFV

kCkCkCkCkFV

kCkCkCCFV

t

t

t

t

t

]1)1[(

])1(1[)(

)1()1(

)1(.....)1()1()1()1(

)1(......)1()1(32

12

Ex) $2000 annuity for 30 years and k= 0.08. What is the annuity future value?

6) Annuities due

Def: annuity for which the cash flows occur at the beginning of the period

Annuity due value =

ordinary annuity value * (1+k)

• 7) Uneven Cash Flows;• Summing PV and FV of each cash flows• Using the cash flow patterns to apply

formula

• Ex) If you are supposed to need $100 (1st), $200 (2nd) and 300 (3rd) at the end of each year and your account provides 7% of interest per year, how much do you need to deposit now?

• Ex) If you are supposed to deposit $100 (1st), $200 (2nd) and 300 (3rd) at the end of each year and your account provides 7% of interest per year, how much your total deposit would be at the end of 3rd year?

• You are supposed to need $100 (1st), $200 (2nd) and 300 (3rd) at the end of each year, the end of 3rd year. You have $360 now. What interest rate (return) do you need to cover your needs?

3. Rate

Q1. 10% compounded semi-annually is the same as 10% per year in compounding?

No! here, 10% is stated or quoted rate and actually, 10.25% (=(1+0.05)*(1+0.05)-1) is the effective annual rate.

To compare to other rates, we need to convert quoted rates into the effective rates

EAR is also called EFF %.

Ex)

Bank A: 15% compounding daily

Bank B: 15.5% compounding quarterly

Bank C: 16% annually

1rate/m)] (Quoted[1

Rate) Annual iveEAR(Effectm

3-1) APRs (Annual Percentage Rate)

Def: interest rate charged per period (periodic rate) multiplied by the number of periods per year

APR =EAR?

No!!!!

So, APR is a quoted rate and need to be converted to the EAR

Ex)

One credit card company selling a card by tele-marketing. The company said the card will benefit its cardholders with semi-annual 15%APRs, compared to the other credit card with 16% EAR.

Do you agree or not?

• 6. Fraction time period

• Suppose you deposited $100 in a bank that pays a normal rate of 10%, compounded, based on a 365 –day year. How much would you have after 9 months?

• Periodic rate = 0.1/365 per day

• FV = 100*(1+0.1/365)^(365*9/12) = 107.79

7.Loan types and loan Amortization

• Three types of loans:1) Pure discount loan:Receive money today and repay a single lump sum

in future 2) Interest only loan:Pay interest each period and repay the entire

principal at some point in the future3) Amortized loan: Repay parts of the loan amount over time

Ex) Amortized loans

Year Beginning Amount Payment Interest Repayment of Principal Remaining balance1 1000 374.11 60 314.11 685.892 686 374.11 41 333 3533 353 374.11 21 353 0

• 8. growing annuity

• E.g) suppose a 65-year old is contemplating retirement, expects to live for another 20 years, has a $1 million nest egg, expect the investment to earn a nominal annual rate of 6%, expect inflation to average 3% per year, and wants to withdraw a constant real amount annually over the 20 years so as to maintain a constant standard of living. If the first withdraw is to be made today, What is the amount of that initial withdrawal?

• (1) step1: real rate calculation• Real rate =rr = [(1+rnom)/(1+Inflation)]-1• = [1.06/1.03]-1 =2.9126214%• (2) step 2: using the real rate, calculate Annuity

due (Payment) – mode: beginning.• = 66673.87. Then it grows by 3% (inflation rate)

every year.• What happen if we want to calculate annuity

(payment) at the end of the first year. 66673.87*(1+0.03)= 68674.09

• E.g) You need to accumulate $100,000 in 10 years. You plan to make a deposit ina bank now, at Time 0 and then make 9 more deposits at beginning of each of the following 9 years. The bank pays 6% interest, you expect inflation to be 2% per year and you plan to increase your annual deposits at the inflation rate. How much you have to deposit initially?

• Step 1: calculate real rate = 1.06/1.02 -1 = 0.0392157

• Step 2: real value of 100000 is 100000/(1+0.02)^10 = 82034.83

• Step 3: beginning mode, • N=10, I/YR=3.9215686, PV=0 and FV=

82034.83, PMT =-6598.87.• It means the t=0, deposit is 6598.87, at t=1, it is

6598.87*(1+0.02)