Sistema finanziario - Financial System

INT INT INT + M

M = face value

Coupon bonds = interest INT is higher than 0, C > 0 pay interest on a stated coupon



rate, and the interest (that is, coupon) payments per year, INT, are generally constant over

the life of the bond.

Zero Coupon bonds = interest INT is equal to 0 ; C = 0 do not pay coupon interest.



Coupon rate C

To calculate that we have to use the ( ) (the interest rate that must be used to

calculate the periodic cash flow). M

face par value

The or of the bond, , is a lump sum payment received by the

bondholder at maturity.

- Face value is generally set at $1,000 in the U.S. bond market.

- When new bonds are issued, the coupon rate on the new bonds is typically set at the

current required rate of return.

Calculate the Present Value of a Bond (annual coupon)

Start by considering the riskiness related to that bond.

First part of the equation = annuity

M = lump sum

semiannual

With : rb/2 and 2T



Ex. Coupon bonds

Maturity = 3 years (semi-annual)

C = 8% (annual coupon rate) semi-annual



r = 10% 5% semi-annual



M = 1000

Fair price of discounted cash flows = 40/(1 + 5%)^1 = 38 total = 776.064



If the market price is 970, I won’t value that bond (it is overvalued by the market).

Premium, Discount, Par Bond

Description of a and

Discount bond less

= when the market price is (<) than the face value of the bond,

coupon rate less required rate of return

when the on a bond is than the on the bond,

fair present value less face value

the is than the of the bond.

coupon rate less yield to maturity

When the on a bond is than the on the bond, the

current market price less face value

is than the of the bond.

Premium coupon rate

= current market price > face value; when the on a bond is

greater required rate of return fair present value greater

than the on the bond, the is

face value

than the of the bond (sold at premium).

coupon rate greater yield to maturity

When the on a bond is than the on the bond, the

current market price greater face value

is than the of the bond.

Par bond = current market price = face value of the bond,

coupon rate equal required rate of return

when the on a bond is to the on the bond, the

fair present value equal face value

is to the of the bond.

coupon rate equal yield to maturity

When the on a bond is to the on the bond, the

current market price equal face value

is to the of the bond.

2% 5% 8%

1 20 50 80

2 20 50 80

3 20 50 80

4 20 50 80

5 20 50 80

6 20 50 80

7 20 50 80

8 20 50 80

10 1020 1050 1080

Face value = 1000

r = 5%

20/(1 + 5%)^1 - 1 equal

5% coupon rate exactly to the

 par bond

interest rate always have a

 less

2% coupon rate to the interest rate

 

discount bond more premium bond

8% coupon rate to the interest rate

 

Bond Valuation Formula used to calculate Yield to Maturity

Yield to Maturity = expected interest rate on a bond supposing that the

bondholder buys it at the current market price the return that we receive on a bond.



We assume that all the payments are invested periodically all at the same rate.

It is the return, or yield, the bondholder will earn on the bond if he or she buys it at

its current market price � , receives all coupon and principal payments as promised, and

� Yield to Maturity

holds the bond until maturity is the (ytm).

The yield-to-maturity calculation implicitly assumes that all coupon payments periodically

received by the bondholder can be reinvested at the same rate — that is, reinvested at the

calculated yield to maturity.

Ex.

Coupon rate 5%

Price = 100

Face value (M) = 100

t P = 100

1 = 5%*100 = 5

2 5

3 5

4 5

5 100 + 5 = 105

Find an IR such that we will obtain a market price equal to 100.

Because the price is equal to the face value, we are dealing with a par bond (the sum of all

the discounted cash flows will give me 100) the interest rate is exactly 5%



We suppose that all the cash flows are reinvested at the same IR once we have received 5$



we are going to invest them at the same IR 5%, and I repeat it for all the periods:

= 5*(1 + 5%)^4

t P = 100 c = 5%

1 = 5%*100 = 5 = 5*(1+5%)^4

= = 6.08



5*(1+5%)^4 =

6.08

2 5 = 5*(1+5%)^3

= 5.69

3 5 5.51

4 5 5.25

5 100 + 5 = 105 105

Total = 127.63

We have Invested 100 to receive 127.63 in 5 years

The IR is ((127.63/100)^1/5) -1= 5%

equal

In order to receive a return to the yield (YTM) we have to reinvest all the cash

lower

flows (monetary units) paid by the bond, otherwise we will get a return than the yield

to maturity.

higher higher expected return

The the interest rate, the the .

YTM = Expected rate of return (E(r)).

 Equity Valuation

preferred common stock)

Valuation process for an equity instrument (such as or involves

finding the present value of an infinite series of cash flows on the equity discounted at an

appropriate interest rate.

Cash flows from holding equity come from dividends paid out by the firm over the life of the



stock.

Shares have no maturity, we are supposed to receive the dividends over the life of the

financial instrument (an infinite number of dividends) we can use different variables:



(The higher the DP, the higher IR)

Div

- t

= Dividend paid to stockholders at the end of the year

t

P

- t

= Price of a firm’s common stock at the end of the year

t

P

- = Current price of a firm’s common stock

0

r

- = Interest rate used to discount cash flows on an investment in a stock

s

Present value methodology applies time value of money to evaluate a stock’s cash flows

over its life as follows: (Div

The price of a stock is equal to the present value of its future dividends ), whose

t

values are uncertain.

We use 3 models to estimate the flow of dividends over the life of the stock:

Zero growth in dividends over (infinite) life of stock

1. No growth in the dividends

 

we will always receive the same;

Constant growth rate in dividends over (infinite) life of stock

2. There is a



monetary growth tomorrow I will receive higher and higher dividends;



Nonconstant growth in dividends over (infinite) life of stock

3. Abnormal growth



in dividends.

Zero growth in dividends all the dividends are equal:



It means that dividends are expected to remain at a constant level forever.

We have to discount all the dividends (is the easiest model):

Constant growth in dividends

It means that dividends equal the second one plus a constant g (dividends are expected

g

to grow at a constant rate, , each year into the future):

The rate of return on the stocks, if they were purchased at price P , may be

0

determined as follows:

Ex. g = 2.5% (growth in dividends)

r = 9%

D2 = 10€

P = D0 * (1 + g) / (rs - g) = 10/6.5% = 153.85€ < 150€ (the stock is undervalued by the

market)

Dividend growth assumed by the market participants?

150 = 10/(9% - g)

If dividends increase their price P increases, if the riskiness decreases we will observe an

increase in the market price

Supernormal (or Nonconstant) growth in dividends

Firms often experience periods of Supernormal or Nonconstant dividend growth, after

which dividend growth settles at some constant rate. To find the present value of a

stock experiencing supernormal or nonconstant dividend growth, a 3-step process is used:

1. Find the present value of the dividends during the period of supernormal (nonconstant)

growth.

2. Find the price of the stock at the end of the supernormal (nonconstant) growth period

using the constant growth in dividends model. Then, discount this price to a present value.

3. Add the 2 components of the stock price together.

Same Ex.

You are evaluating Financial system Inc (FS) Stock. The stock is expected to experience a

supernormal growth in dividends of 20% over the next 3 years. Following this period dividends

are expected to grow at a constant rate of 3%. The stock paid a dividend of 0.50 last year. The

required rate of return on the stock is 15%.

Calculate fair price of stock.

Hint: you have to use a three-step process

Step 1: Find the present value of dividends during supernormal growth period

Step 2: Find the present value of dividends after supernormal growth period

Step 3: Add the two components of the stock price together

For 3 years dividends will grow at 20% then they will grow at a constant rate of 3%, r =



15%

t 0.5 = 0.5/(1+15%)^1 = 0.52

1 0.6

2 (0.5*(1+20%)^2) = 0.72

3 0.864

4 = 0.864*(1+3%)^4 = 0.88 Total = 1.63426

Find present value (sum first 3 values).

Second step:

Basically, firstly you calculate the pink value (present value of

stock at beginning of first period, after year 3), then compute

present value at end of supernormal growth period.

Third step: Sum values found in first and second step = 1.63 +

4.87 = 6.5

Impact of the Interest Rate Changes on Security Values

IR and securities value are inversely related: as interest rates increase, present values of

bonds (and bond prices) decrease at a decreasing rate.

Impact of Maturity and Coupon Rates

on Security Values

Price sensitivity to changes in IR = measured by

duration

using the .

A bond’s price sensitivity is measured by the

percentage change in its present value for a

given change in interest rates.

The shorter the time remaining to

 maturity, the closer a bond’s price is to its face value.

The further a bond is from maturity, the more sensitive the price of the bond as

 interest rates change.

The relationship between bond price sensitivity and maturity is not linear.

 As the time remaining to maturity on a bond increases, price sensitivity

 decreasing

increases but at a rate.

The following relationships hold when evaluating either required rates of return and the

resulting fair present value of the bond or expected rates of return and the current market

price of the bond.

The higher the bond’s coupon rate, the higher its present value at any given

 interest rate.

The higher the bond’s coupon rate, the smaller the price changes on the bond for a

 given change in interest rates. pull to parity

I few consider a zero-coupon bond over time its value tends to 100 (the



value of the financial instrument tends to the par value): the closer the bond is to

maturity, the closer the Interest rate is to 0. Pull to parity

and price-return

Example of Market

Efficiency

Unicredit we expect the



securities to increase their value positive response in the market