Institutions Ch 3 (Done)

The term interest rates can have many different meanings depending on the time frame used for analysis and the type of security being analyzed 

Coupon rate

interest rate on a bond instrument used to calculate the annual cash flow the bond issuer promises to pay the bond holder.

Required rate of return

interest rate an investor should receive on a security given its risk. Required rate of return is used to calculate the fair present value on a security.

Expected rate of return

interest rate an investor expects to receive on a security if he or she buys the security at its current market price, receives all expected payments, and sells the security at the end of his or her investment horizon.

Realized rate of return

actual interest rate earned on an investment in a financial security. Realized rate of return is a historical (ex post) measure of the interest rate.

required rate of return (r)

The interest rate used to find the fair present value of a financial security is called the ___________

The interest rate used to find the fair present value of a financial security is called the required rate of return (r)

• Function of the various risks associated with a security and is thus the interest rate the investor should receive on the security given its risk (default risk, liquidity risk, etc.) 

• Ex ante measure of the interest rate on a security

  

Expected Rate of Return

E(r), on a financial security is the interest rate a market participant expects to earn by buying the security at its current market price (P), receiving all projected cash flow payments (CFs) on the security, and selling the security at the end of the participant's investment horizon

Market efficiency

The speed with which financial security prices adjust to unexpected news, to maintain equality with the fair present value of the security, is referred to as ___________.

The speed with which financial security prices adjust to unexpected news, to maintain equality with the fair present value of the security, is referred to as market efficiency

• If financial markets are efficient, which tends to be the case most of the time, the current market price of a security tends to equal its fair price present value 

• When an event occurs that unexpectedly changes interest rates or a characteristic of a financial security, the current market price of a security can temporarily diverge from its fair present value

The realized rate of return, ( r )

on a financial security is the interest rate actually earned on an investment in a financial security

The realized rate of return, ( r ), on a financial security is the interest rate actually earned on an investment in a financial security

• Ex post measure of the interest rate on the security 

• If the realized rate of return ( r ) is greater (less) than the required rate of return (r), the market participant earned more (less) than was needed to be compensated for the ex ante or expected risk of investing in the security

Bond valuation employs time value of money concepts

• Fair value of a bond reflects the present value of all cash flows promised or projected to be received on that bond discounted at the required rate of return (r_b) 

• Expected rate of return, E(r_b), is the interest rate that equates the current market price of the bond with the present value of all promised cash flows received over the life of the bond 

• Realized rate of return (r_b) on a bond is the actual return earned on a bond investment that has already taken place

Promised cash flows on bonds come from two sources:

1. Interest or coupon payments paid over the life of the bond

2. Lump sum payment (face or par value) when a bond matures

Coupon bonds

pay interest based on a stated coupon rate, and the interest (i.e., coupon) payments per year, INT, are generally constant over the life of the bond

Zero-coupon bonds

do not pay coupon interest

• In other words, INT = 0

The face or par value of the bond, M, is a lump sum payment received by the bond holder 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

Calculating Present Value of a Bond

Description of a Premium, Discount, and Par Bond

Premium bond

when the coupon rate on a bond is greater than the required rate of return on the bond, the fair present value is greater than the face value of the bond.

When the coupon rate on a bond is greater than the yield to maturity on the bond, the current market price is greater than the face value of the bond.

Discount bond

when the coupon rate on a bond is less than the required rate of return on the bond, the fair present value is less than the face value of the bond.

When the coupon rate on a bond is less than the yield to maturity on the bond, the current market price is less than the face value of the bond.

Par bond

when the coupon rate on a bond is equal to the required rate of return on the bond, the fair present value is equal to the face value of the bond.

When the coupon rate on a bond is equal to the yield to maturity on the bond, the current market price is equal to the face value of the bond.

yield to maturity (ytm)

The return, or yield, the bond holder 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 holds the bond until maturity is the _______________

YTM may be solved for as follows:

Equity Valuation

Valuation process for an equity instrument (such as preferred or common stock) 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

These variables are often used in equity valuation:

Present value methodology applies time value of money to evaluate a stock’s cash flows over its life as follows:

The price of a stock is equal to the present value of its future dividends (Div_t), whose values are uncertain

Common assumptions made regarding expected pattern of uncertain flow of dividends over life of the stock 

1. Zero growth in dividends over (infinite) life of stock 

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

3. Nonconstant growth in dividends over (infinite) life of stock 

Zero growth in dividends means that dividends are expected to remain at a constant level forever 

In generalized form:

Constant growth in dividends means that dividends on a stock are expected to grow at a constant rate, g, each year into the future

Rate of return on the stock, if it were purchased at a price P₀, may be determined as follows:

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 three-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 two components of the stock price together. 

Impact of Interest Rate Changes on Security Values

As interest rates increase, present values of bonds (and bond prices) decrease at a decreasing rate

price sensitivity

A bond’s ______________ is measured by the percentage change in its present value for a given change in interest rates

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 

• Relationship between bond price sensitivity and maturity is not linear 

  • As the time remaining to maturity on a bond increases, price sensitivity increases but at a decreasing 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

Duration

_________ is the weighted-average time to maturity on an investment using the relative present values of the cash flows as weights

Duration is the weighted-average time to maturity on an investment using the relative present values of the cash flows as weights

• Produces an accurate measure of the price sensitivity of a bond to interest rate changes for relatively small changes in interest rates

  • Less accurate measure of price sensitivity the larger the change in interest rates

elasticity

In addition to being a measure of the average life of an asset or liability, duration also has economic meaning as the sensitivity, or _______, of that asset or liability’s value to small interest rate changes (either required rate of return or yield to maturity)

Duration, for any fixed-income security that pays interest annually, may be calculated using the following formula:

• Recall that most bonds pay interest semiannually rather than annually 

• For bonds that pay interest semiannually, the duration equation becomes:

Duration: Example

Duration: Example (text)

Suppose that you have a bond that offers a coupon rate of 10 percent paid semiannually (or 5 percent paid every 6 months). The face value of the bond is $1,000, it matures in four years, its current rate of return (r) is 8 percent, and its current price is $1,067.34. See Table 3-8 for the calculation of its duration. As the calculation indicates, the dura- tion, or weighted-average time to maturity, on this bond is 3.42 years. In other words, on a time value of money basis, the initial investment of $1,067.34 is recovered after 3.42 years.

Table 3-9 shows that if the annual coupon rate is lowered to 6 percent, the duration of the bond rises to 3.60 years. Since 6 percent annual coupon payments are smaller than 10 percent coupon payments, it takes longer to recover the initial investment with the 6 percent coupon bond. In Table 3-10, duration is calculated for the original 10 percent coupon bond, assuming that its rate of return (r) increases from 8 percent to 10 percent. Now duration falls from 3.42 years (in Table 3-8) to 3.39 years. The higher the rate of return on the bond, the more the investor earns on reinvested coupons and the shorter the time needed to recover his or her initial investment. Finally, as the maturity on a bond decreases, in this case to 3 years in Table 3-11, duration falls to 2.67 years (i.e., the shorter the maturity on the bond, the more quickly the initial investment is recovered).

When understanding duration, you need to focus on this part: Suppose that you have a bond that offers a coupon rate of 10 percent paid semiannually (or 5 percent paid every 6 months). The face value of the bond is $1,000, it matures in four years, its current rate of return (r) is 8 percent, and its current price is $1,067.34.

As the calculation indicates, the duration, or weighted-average time to maturity, on this bond is 3.42 years. In other words, on a time value of money basis, the initial investment of $1,067.34 is recovered after 3.42 years.

(Note: How long it takes to get your money back)

Table 3-9 shows that if the annual coupon rate is lowered to 6 percent, the duration of the bond rises to 3.60 years.

Since 6 percent annual coupon payments are smaller than 10 percent coupon payments, it takes longer to recover the initial investment with the 6 percent coupon bond.

shorter time needed to recover his or her initial investment

The higher the rate of return on the bond, the more the investor earns on reinvested coupons, and the ____________________________.

The current price that an investor is willing to pay for a zero-coupon bond, assuming semiannual compounding of interest, is equal to the present value of the single, fixed (face value) payment on the bond that is received on maturity (here, $1,000): 

Duration and coupon interest

The higher the coupon or promised interest payment on the bond, the shorter its duration

• The larger the coupon or promised interest payment, the more quickly investors receive cash flows on a bond and the higher are the present value weights of those cash flows in the duration calculation

Duration and rate of return

Duration decreases as the rate of return on the bond increases

Duration and maturity

Duration increases with maturity, but at a decreasing rate

Economic Meaning of Duration

In addition to being a measure of the average life of a bond, duration is also a direct measure of its price sensitivity to changes in interest rates, or elasticity

For small changes in interest rates, bond prices move in an inversely proportional manner based on the size of D

Annual compounding of interest:

Semiannual compounding of interest:

Modified duration (MD):

Duration accurately measures the price sensitivity of financial securities only for small changes in interest rates of the order of one or a few basis points

A basis point is equal to one-hundredth of 1 percent

Duration misestimates the change in the value of a security following a large change (either positive or negative) in interest rates

• As a result of convexity, the capital loss effect of large rate increases tends to be smaller than the capital gain effect of large rate decreases 

• Convexity is the degree of curvature of the price-interest rate curve around some interest rate level

Convexity

_______ is the degree of curvature of the price-interest rate curve around some interest rate level

Convexity is desirable.

The greater the convexity of a security or portfolio of securities, the more insurance or interest rate protection an investor or FI manager has against rate increases and the greater the potential gains after interest rate falls.

Convexity increases the error in duration as an investment criterion.

The larger the interest rate changes and the more convex a fixed-income security or portfolio, the greater the error in using just duration (and duration matching) to immunize exposure to interest rate shocks.

All fixed-income securities are convex.

That is, as interest rates change, bond prices change at a nonconstant rate.