Future Value,Future Value,

Present Value and Present Value and

Interest RatesInterest Rates

Future Value $100 + $100(0.05) = $105

PV + Interest = FV

PV + PV*i = FV

PV = Present Value

FV = Future Value

i = interest rate (as a percentage)

Future Value in one year:

FV = PV*(1+i)

Future Value in two years

$100+$100(0.05)+$100(0.05) + $5(0.05) = $100+$100(0.05)+$100(0.05) + $5(0.05) = $105 + $105(.05) = $110.25 $105 + $105(.05) = $110.25

=Present Value of the Initial Investment(100) =Present Value of the Initial Investment(100) + Interest on the initial $100 in the 1st Year + Interest on the initial $100 in the 1st Year

+ Interest on the initial $100 in the 2nd Year+ Interest on the initial $100 in the 2nd Year + Interest on the $5 interest from the 1+ Interest on the $5 interest from the 1stst Year Year

in the 2in the 2ndnd Year Year = Value in 1= Value in 1stst + Interest on Value in 1 + Interest on Value in 1stst Year Year

In general, future value n years from now compounded at interest rate i

FVn = PV*(1+i)n

Computing Future Value at 5% Annual Interest

Present ValuePresent Value

Present Value (PV) is the value today (in the present) of a payment that is promised n years in the future.

OR

Present Value (PV) is the amount that must be invested today at annual interest rate i in order to have a specific amount n years in the future

Present Value of an amount received in one year:

Solving the Future Value EquationFV = PV*(1+i)

Present Value of $FV received n years in the future:

Present ValuePresent Value

Example

Present Value of $100 received in 2 ½ years and an interest rate of 8%.

PV = $100 / (1.08)2.5 = $82.50

Note:

FV =$82.50 * (1.08)2.5 = $100

Important Properties of Present Value

Present Value is higher:Present Value is higher:

1.1. The higher the future payment (The higher the future payment (FV))

2.2. The shorter the time period until The shorter the time period until payment. (payment. (n))

3.3. The lower the interest rate at which The lower the interest rate at which future receipts are discounted (future receipts are discounted (i))

Internal Rate of Return

The Internal Rate of Return on an investment is the interest rate that equates the present value of its future cash flows with its cost.

Internal Rate of Return

A machine with a price of $1,000,000 that generates $150,000/year for 10 years.

Solving for i, i=.0814 or 8.14%

When future inflows are discounted at a rate of 8.14%, the present value of those inflows equals $1million

Bond PricingBond Pricing

The price of a bond now is the The price of a bond now is the Present ValuePresent Value of its promised of its promised payments in the future.payments in the future.Payment stops at the maturity date (n years)– The last payment is for the face value (F)

{or par value or principal} of the bondIn addition to repayment of principal, Coupon Bonds make annual payments (C) based on an interest rate, the coupon rate (ic),

C=ic*F

Bond PricingBond PricingA discount bond just has a $100 (face A discount bond just has a $100 (face value, F) principle in n years. The present value, F) principle in n years. The present value or price of a discount bond, (Pvalue or price of a discount bond, (PBPBP):):

Bond Pricing: Coupon Bond

If a bond has n coupon payments (C), where C= ic * F, the Present Value (PCP) of the coupon payments is:

Bond Pricing: Coupon Bond[Like n+1 of discount bonds]

Present Value of Coupon Bond (PCB) =

Present value of Yearly Coupon Payments (PCP)

+

Present Value of the Principal Payment (PBP)

Bond PricingBond Pricing

The price of a bond and the interest rate are inversely related

the higher the interest rate, the lower the bond prices.

Real and Nominal Interest Rates

Nominal Interest Rate (i)

Interest Rates expressed in current dollar terms.

Real Interest Rate (r)

Nominal Interest Rate adjusted for (expected) inflation, πe.

Fisher Equation:

i = r + πe

or

r = i - πe