©2008 Professor Rui Yao All Rights Reserved

CHAPTER

3

CHAPTER

3The Interest Factor

in Financing

3-2

Chapter ObjectivesChapter Objectives

• Future value of a lump sum

• Present value of a lump sum

• Future value of an annuity

• Present value of an annuity

• Price and yield relationships

• Internal rate of return / yield to maturity

3-3

Future Value of a Lump SumFuture Value of a Lump Sum

• FV = PV (1+i)n

FV = future values; PV = present value i = interest rate, discount rate, rate of return

• The principle of compounding, or interest on interest:

if we know1. An initial deposit - PV2. An interest rate - i3. Time period - n

We can compute the values at some specified future time period.

Q: What happens with simple interests?

3-4

Future Value of aFuture Value of a Lump SumLump Sum: : An ExampleAn Example

• Example: assume Astute investor invests $1,000 today which pays 10 percent, compounded annually. What is the expected future value of that deposit in five years?

• Solution= $1,610.51

3-5

Present Value of a Future SumPresent Value of a Future Sum

• The discounting process is the opposite of compounding

PV = FV / (1+i)n

• Example: assume Astute investor has an opportunity that provides $1,610.51 at the end of five years. If Ms. Investor requires a 10 percent annual return, how much can astute pay today for this future sum?

• Solution= $1,000

3-6

AnnuitiesAnnuities

• Ordinary Annuity Payment due at the end of the period e.g., mortgage payment

• Annuity Due Payment due at the beginning of the

period e.g., a monthly rental payment

3-7

Future Value of an AnnuityFuture Value of an Annuity

• FVA = PMT (1+i )n-1+PMT (1+i )n-2 …+ PMT = PMT [1/i ( (1+i )n-1)]

• Example: assume Astute investor invests $1,000 at the end of each year in an investment which pays 10 percent, compounded annually. What is the expected future value of that investment in five years?

• Solution = $6,105.10

• Q: What happens if i=0%? • Q: What if n goes to infinity?

3-8

Sinking Fund PaymentSinking Fund Payment

• Example: assume Astute investor wants to accumulate $6,105.10 in five years. Assume Ms. Investor can earn 10 percent, compounded annually. How much must be invested each year to obtain the goal?

• Solution= $1,000.00

3-9

Present Value of an AnnuityPresent Value of an Annuity

• PVA = PMT /(1+i)1 + PMT /(1+i)2…+ PMT /(1+i)n

= PMT [1/i (1-1/(1+i)n)]

Special cases:

Q: What happens if i = 0 % ?

Q: What happens if n goes to infinity?

Example: What is the PV of 8-period annuity with pmt of $1,000, and discount rate of 10%

3-10

Investment Yields / Internal Rate of Investment Yields / Internal Rate of ReturnReturn

• The discount rate that sets the present value of future investment cash returns equal to the initial investment costs today

• Example: What is the investment yield if you will receive $400 monthly payment for the next 20 years for an initial investment of $51,593?

3-11

Present Value of an AnnuityPresent Value of an Annuity

• What if compounding frequency is not annual?

Adjust i and n to reflect compounding frequency

Q: What happens if m goes to infinity (continuous compounding)?

3-12

• Bond is exchange of CF now (the PV, or price) for a pattern of cash flows later (coupons + par)

• Bond price = PV(coupon payments)

+ PV(par value) Requires determination of

• Expected cash flows (coupons and par)

• “Required” discount rate, or required yield

Bond PricingBond Pricing

3-13

• Combine our PV for annuity and lump sum

• Example Semiannual, 10%, fixed rate 20-year bond with a par of

$1,000. No credit risk, not callable, etc. Required yield is 11%• C = c × F = 0.1/2 × $1,000 = $50• r = 0.11/2 = 0.055• n = 20 × 2 = 40• P = $50/0.055 × [1- 1/(1.055)40] +

$1,000/(1+0.055)40

= $50 × 16.04613 + $1,000/8.51332 = $802.31 + $117.46

= $919.77 Note that P<F, i.e., bond trades at a discount.

Q: what is the yield to maturity if P=$900?

nn

n

n

tt

r

F

rr

Fc

r

F

r

FcP

11

11

111

3-14

• More on Bond Pricing Yield = Internal Rate of Return (IRR)

IRR sets the NPV to zero for a bond investment Solve using financial calculator, Excel function RATE

or IRR Special case of one future cash flow (zero-coupon

bond):

n

tt

t

n

tt

t

y

CFP

y

CFP

1

1

10

1

1

1/1

n

n

nn

P

CFy

y

CFP

3-15

• More on Bond Pricing The required yield or discount rate can be thought of

simply as another way of quoting the price.

Special case of one future cash flow (zero-coupon bond):

1

1/1

n

n

nn

P

CFy

y

CFP

3-16

• Price-yield relationship: Decreasing For non-callable bonds, convex

• Callable bond and Yield to call

• Ex: Using Excel (or other) show this For the previous example, vary bond yield to

maturity from 5% to 15%

3-17

• More on Bond Pricing If required discount rate remains unchanged, a par

bond’s price will remain unchanged, but a discount bond will appreciate and a premium bond will depreciate over time.

• Why?

• Show with a spreadsheet

• Example: What is your return if you buy a 10% semi-annual coupon bond at 11% yield to maturity and hold for one year while the yield to maturity

stays the same goes up to 12% goes down to 10%

3-18

Other conventional yield measures• Current yield = (annual coupon)/(current price)

E.g.: Face is $100, current price is $80, coupon rate is 8%: yc = 0.08 × 100/80 = 8/80 = 0.1 = 10%

Ignores capital gains (losses) and reinvestment income

• Yield to maturity: yield (IRR) if bond is held to maturity

• Q: what is the “current yield” for a stock?

Q: What is the ranking ofa. couple rate; b. current yield; c. yield to maturity

for

a. par bond;

b. discount bond;

c. premium bond

3-19

Useful Excel FunctionsUseful Excel Functions

• FV• PV• Rate• PMT

IPMT PPMT

• NPER

• NPV• IRR

• GOAL SEEK / SOLVER