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CHAPTER FIVE
BONDS, PRICES, BOND YIELDS, AND INTEREST RATE RISK
There is strong relationship between interest rate changes and yield of financial instruments, a relationship known as interest rate risk. To understand this, we need to understand how these instruments are priced. Pricing of financial instruments is based on the concept of time value of money.
THE TIME VALUE OF MONEY
This is based on the concept of positive time preference for consumption. People prefer current consumption than future consumption. This indicates that money have different value in different time periods.
The time value of money can be simply stated as a given amount of money today is worth more than the same amount received at some future date. That is money will have a value at the present, different than its value in the future.
To compare between money in different time periods, we use the concept of future value (compounding) and present value (discounting).
FUTURE VALUE
If a person has a certain amount of money today (present value), how much will it be worth in the future (future value) at a given rate of interest?
The formula for calculating the future value (compounding) is:
FV = PV (1+i)ⁿ
Where:
FV = future value of an amount of money n periods in the future.
PV = present value (the value of money today)
i = interest rate
n = number of compounding periods
Example:
Suppose you have BD 2000 in saving account paying 8% annually, how much would you get after 2 years?
Solution: Using the formula: FV = PV (1+i)ⁿ, we have
FV = 2000 (1 + .08)² = 2000 (1.08) ² = BD 2332.8
If the bank decides to pay interest semiannually (2 times a year) then:
- The number of compounding periods = 2 Х 2 = 4
- Interest rate = 8/2 = 4%
- So, FV = 2000 (1.04) 4 = BD 2339.7
If interest is to be paid quarterly (4 times a year), then
- The number of compounding periods = 2 Х4 = 8
- Interest rate = 8/4 = 2%
- So, Fv= 2000(1.02) 8 = 2343.3
If the interest is paid daily:
- The number of compounding periods = 2 Х 365 = 370
- Interest rate = 8/365 = 0.0219%
- So, FV = 2000(1.00219) 370 =
Note that FV increases as compounding period increases.
PRESENT VALUE:
Present value (PV) is the opposite of Future value. If you will get an amount of money in the future, how much does worth at the present?
The formula of calculating the present value (discounting) is:
1
PV = FV [------], or
(1+i) n
FV
= ------
(1+i) n
Example: If you are to be given BD 2332.8 after 2 years, and interest rate is 8%, how much does it worth at the present?
1 2332.8
PV = 2332.8 (------) = ------= BD 2000
(1.08) ² 1.1664
BOND PRICING
A bond is a contractual obligation of a borrower to make a periodic cash interest payments to a lender for a fixed period of years, and to pay upon maturity, the principal or the original sum of the borrowed money.
- The periodic interest payments, is known ascoupon payments.
- The number of years over which the bond contract extends is known asthe term-to-maturity.
- The principal or the original sum paid to the lender (the original amount borrowed) upon maturity is known as the par value (or face value).
So, a bond is a series of cash payments (coupon payment and principal) for a fixed number of years.
The coupon payment is typically expressed in percentage of the principle amount and is known as the coupon rate. That is coupon rate is coupon payment expressed as percentage of the principle amount. It is computed by the formula:
C
CR = ------X 100
F
Where:
CR= coupon rate
C = annual coupon payment
F = par or face value of the bond
Needless to say, the equation contains three unknowns, and knowing any two, the missing one could be calculated. In general the coupon rate is fixed first, then coupon payments are calculated. Coupon rates are usually fixed or set at or near market rate (yield) of similar bonds available in the market.
THE BONDS PRICING FORMULA:
Because a bond is a borrower’s contractual promise to make a series of future cash payments, the pricing of a bond is an application of the present value formula. Thus, the price of a bond is the present value of the future cash flow (coupon payments and principle amount) discounted by the interest rate that people place on the time value of money. In other words, a bond is a series of cash payments (coupon payments and principle) of a fixed amount for a specified number of years. Thus, each future cash payment must be individually discounted from the date received back to the present.
Thus the formula for price of a fixed-coupon-rate bond with n periods to maturity is:
C 1 C 2 C n+ Fn
PB = ------+ ------+ ------+ ------
(1+i) 1 (1+i) ² (1+i) n
Where:
PB = price of the bond or present value of the stream of cash payments.
C t = coupon payment in period t.
Fn = par value or face value (principle amount) to be paid at maturity.
i = interest rate (discount rate) or yield to maturity.
n = number of periods to maturity.
Example:
What is the price of corporate bond maturing in 5 years, with 5% coupon rate, BD1000 par value (face value) if interest rate (yield) on similar bonds is 7%?
Solution: First, we have to calculate coupon payments using the formula:
C
C.R. = ------X 100, which by substitution gives: [C/1000]*100= 5%
F
Then solving for C, we get: C = 1000 X 5% = 50. Then using the bond price formula, the price of this bond is calculated as follows:
50 50 50 50 1050
PB = ------+ ------+ ------+ ------+ ------= $918
(1.07) 1 (1.07) 2 (1.07) 3 (1.07) 4 (1.07) 5
Note that the final cash payment consists of the final coupon payment ($50) and the face value of the bond ($1000).
If interest is paid more than once a year, the PB equation is adjusted as:
C1/m C2/m C mn/m+ Fmn
PB = ------+ ------+ ------+ ------
(1+i/m)1 (1+i/m) ² (1+i/m) mn
Where:
m is the number of times interest is paid (compounded) each year.
mn is the total number of interest payments over the life of the bond.
Therefore, in our example if interest is paid semiannually (2 times a year), then: m = 2, i = 7%/2 = 3.5%, mn = 5 X 2 = 10 and C/m =50/2=25, and the price of the bond is given as:
25 25 1025
PB = ------+ ------+ ------+ ------+ ------=
(1.035) 1 (1.035) 2 (1.035) 10
How changes in interest rate affect the price of the bond?
In our example above (5 years, 5% coupon rate, and BD1000 face value bond):
When interest rate was 7%, PB was calculated as BD918
If interest rate increases to 10%, then:
50 50 1050
PB = ------+ ------+ ------+ ------= BD810
(1.1) 1 (1.1) 2 (1.1) 5
If the interest rate decreases to 3%, then:
50 50 1050
PB = ------+ ------+ + ------= BD1092
(1.03) 1 (1.03) 2 (1.03) 5
If interest rate is 5% (equal to coupon rate), then:
50 50 1050
PB = ------+ ------+ + ------= BD1000
(1.05) 1 (1.05) 2 (1.05) 5
From this we note that:
There is inverse relationship between interest rate and bond price. That is, as interest rate increases PB decreases and vice versa.
So, depending on the interest rate (i), the price of the bond could be: (the bond can be sold at):
- Below its face value, and the bond is known discount bond (i.e. bond is selling at discount)(This happens when i > CR; 10%>5%).
- Above its face value, and it is called premium bond (i.e. bond sells at premium) (This happens when i < CR; 3%<5%).
- Equal to face value, and bond is known as par bond (i.e. bond sells at par) (This happens when i = CR; 5%=5%).
ZERO COUPOND BONDS
These are securities that have no coupon payment but promise a single payment at maturity. The interest paid to the holder is the difference between the price paid for the securities and the amount received upon maturity (or price received when sold).
A major attraction of zero coupon bonds for investors is that there is no coupon reinvestment risk because there are no coupon payments to reinvest. (That is zero coupon bonds eliminate reinvestment risk).
The price of zero coupon bonds is given by this formula:
Fmn
PB zero coupon bond = ------
(1 +i/m)mn
Where:
PB = price of the zero coupon bond
Fn = amount of cash payments at maturity.
i = interest rate (yield) for n periods
n = number of years until the payment is due
m = number of times interest is compounded each year
Thus, the price of a ten-year zero coupon bond with a BD1000 face value and a 12% (6% semiannual rate) market rate of interest is:
Solution:
mn = 10X2 = 20i = 12%/2 = 6%
$1000
FB = ------= $ 311.80
(1.06) 20
BOND YIELD MEASURES
The cash flows that an investor might obtain from a bond come from three main sources:
1-Coupon payments.
2-Interest income from reinvesting coupon payments.
3-Any capital gains or losses resulting from changes in market interest rates.
Thus while the coupon rate on a bond reflects only the annual cash flows promised by borrowers to the lender, the actual rate of return (bond yield) that the lender may earn depend on several key risks:
- Credit risk or default risk (the chance that the borrowers fails to make coupon or principal payments).
- Reinvestment risk (market interest rate may change, causing the lender to have to reinvest coupon payments at interest rate different than the interest rate at the time the bond was purchased).
- Price risk (interest rate changes cause the market value of a bond to rise or fall, resulting in capital gains or losses to the investor).
The ideal yield measure should capture all three potential sources of risk. So we differentiate between three yield measures: yield to maturity, realized yield, and expected yield.
1-YIELD TO MATURITY
This is calculated by using the bond-pricing formula when thebond price is known. It is the yield (i), which equates the discounted cash flows to the price of the bond. That is the yield to maturity is found by solving the following equation (bond price formula) for the interest rate (i) when all unknowns, except i, are given:
C1 C2 C3 Cn+ Fn
PB = ------+ ------+ ------+ ------+ ------
(1+i) 1 (1+i) 2 (1+i) 3 (1+i) n
Example: Calculate the yield to maturity of five years, 5 percent coupon, BD1000 face value bond, priced at BD918?
First, we have to calculate coupon payments = c =$1000 X 5% = 50
Second: substitute in the above equation and solve for i, such that:
50 50 50 50 1050
BD918 = ------+ ------+ ------+ ------+ ------
(1+i) 1 (1+i) 2 (1+i) 3 (1+i) 4 (1+i) 5
Unfortunately, the yield to maturity (i) cannot be determined algebraically but must be found by trail and error, or by using financial calculator.
Note that yield to maturity is also known as promised yield, since it is the yield that is promised to the bondholder on the assumption that:
-The bond will be held to maturity.
-All coupon and principle payments will be made as promised (no default risk).
-The coupon payments will be reinvested at the bond’s promised yield for the remaining terms to maturity (no change in interest rate).
2-REALIZED YIELD
This is the return earned on a bond given the cash flows actually received by the investor and assuming that the coupon payments are reinvested at the promised yield. The actual cash flows could be different than promised cash flows for number of reasons, including:
- If bond is not held to maturity (it will be sold at current market price rather than the promised face value)
- If payments are not paid as promised (there is default).
- If interest rate changes, so reinvestment takes place at different rate
This means that we should calculate a yield measure which shows the actual yield of the bond to the investor. This yield is measured by using the actual cash flows obtained from the bond.
That is, realized yield is the rate that equates the original price of the bond (par/face value, or price paid by the investor) to the discounted actual cash flows from the bond.
It is calculated in 2 steps.
1-Calculate the current market price of the bond.
2-Find i, which equates the original price paid by the investor (par value or otherwise) to the present value of the actual cash flows.
Example: In our example (5%, 5 years, BD1000 face value bond)
If the interest rate increases to 10% (from 7%) and the holder of the bond decides to sell the bond after 2 years (holding for 2 years instead of 5 years maturity period), what is the realized yield?
Solution:
1-Calculate the current market price of the bond, and since the holder will hold for 2 years, we calculate the current market price for the remaining 3 years of the life of the bond, as:
50 50 1050
PB = ------+ ------+ ------= BD876
(1.1) 1 (1.1) 2 (1.1) 3
2-Find i such that:
50 50 + 876
BD1000 = ------+ ------
(1+i) 1 (1+i) 2
Again the realized yield (i) cannot be determined algebraically but must be found by trail and error, or by using financial calculator. Note that realized rate helps to evaluate return of the bond ex-post (after the end of the holding period or investment horizon).
3-EXPECTED YIELD
Investors and financial institutions that plan to sell their bonds before maturity would like to know the potential impact of interest rate changes on the returns of their bond investments ex-ante (before the fact).
So, if a bond is to be sold before maturity you need to know its expected yield based on information about the money supply, inflation rates, economic activity, and the past behavior of interest rate.
Expected yield is calculated in 3 steps:
1-Predict or forecast the nterest rate in the future.
2-Calculate the expected price of the bond using the predicted interest rate.
3-Calculate expected yield (i), which equates the original price paid by the investor (price or par value) to the discounted expected cash flows from the bond.
Example:
In our example, you bought the bond (BD1000, 5%, 5 years) and intend to sell it after three years. If after three years interest rate is forecasted as 3%, what is the expected yield?
Solution:
1-Future predicted interest rate is 3%, as forecasted.
2-Calculate the current market price of the bond for the remaining 2 years of the life of the bond, using the predicted interest rate in 1 above:
50 1050
PB = ------+ ------= $ 1038
(1+0.03)1 (1+0.03)2
3-Calculate i such that:
50 50 50 + 1038
$1000 = ------+ ------+ ------
(1+i)1 (1+i)2 (1+i)3
Again, expected yield (i) cannot be determined algebraically but must be found by trail and error, or by using financial calculator.
Yield and Bond Price:
How changes in yield affect bond prices?
For bonds and any other financial claims, a strict relationship exists between price and yield (or interest rate). Bond price and yield vary inversely. Specifically, as a bond market price rises, its yield declines; or as the price declines, its yield increases (as i, PB). So the higher the price an investor must pay for a bond, the lower is the realized rate or rate of return on the investment.
This inverse relationship exists because the coupon rate (and coupon payments) on a bond is fixed at the time the bond is issued. This indicates that price of the bond changes as market interest rate changes. But how we can measures the response of bond price changes to changes in interest rate? This is measured by bond price volatility (or price variability).
BOND PRICE VOLATILITY:
This is the percentage change in bond price for a given change in interest rate (yield). It is calculated using the following equation:
Pt – Pt-1
% ∆ PB = ------X 100
Pt-1
Where:
% ∆ PB = the percentage change in price (price volatility).
Pt = the new price in period t.
Pt-1 = the bond’s price one period earlier, (t-1).
Example: In our example (BD1000, 5%, 5 years)
When interest rate = 7%, the PB = BD918
When interest rate increases to 10%, PB = BD810
So, as interest rate increased from 7% to 10%, the price of the bond fell from BD918 to BD810 with a decline is 918 – 810 = BD108, and price volatility is:
810 - 918
% ∆ PB = ------X 100 = - 11.76%
918
If interest rate decreases to 5% (from 7%), then PB = BD1000
So price volatility is:
1000 - 918
% ∆ PB = ------X 100 = 8.93%
918
Thus, a bond’s price volatility shows how sensitive a bond’s price is to changes in yield or market interest rate.
Price volatility depends on term to maturity and coupon rate.
BOND PRICE VOLATILITY AND MATURITY:
There is a strong relationship between price volatility and maturity of the bond. Take a simple example:
The price of a $1000, 5% coupon bond, maturity in one year is:
1050
- When i = 5% PB = ------= $1000
(1.05) 1
1050
- When i = 6% PB = ------= $991
(1.06) 1
991- 1000
Therefore, price volatility = % ∆ PB = ------X 100 = - 0.9%
1000
Now, price of the same bond, but matures in three years is:
-When i = 5% PB = BD1000
50 50 1050
-When i = 6% PB = ------+ ------+ ------= $973
(1.06) 1 (1.06) 2 (1.06) 3
973 - 1000
Therefore, price volatility =% ∆ PB = ------X 100 = - 2.7%
1000
So, bonds with longer maturity periods (n) have higher price variability (volatility) than bonds with shorter maturity. That is as n increases, volatility increases. (See table 5.3 in the text for more detailed calculations).
BOND PRICE VOLATILITY AND COUPON RATE:
Another important factor that affects the price volatility of a bond is the bonds’ coupon rate. Take a simple example again:
The price of a BD1000, 5% coupon bond, maturity in three years is:
-When i = 5% PB = $1000
-When i = 6% PB = $973
973 - 1000
Therefore, price volatility = % ∆ PB = ------X 100 = - 2.7%
1000
Now, for the same bond assume that coupon rate is 1% (instead of 5%), then the bond price is (note coupon payment now is C= 1000*1% =10):
10 10 1010
- When i = 5% PB = ------+ ------+ ------= $981
(1.05) 1 (1.05) 2 (1.05) 3
10 10 1010
- When i = 6% PB = ------+ ------+ ------= $866
(1.06) 1 (1.06) 2 (1.06) 3
866 - 981
Therefore, price volatility =% ∆ PB = ------X 100 = - 11.7%
981
So, the lower the coupon rate, the higher the price volatility. That is holding maturity constant, bonds with lower coupon rates have higher price volatility than bonds with higher coupon rates.
To summarize, we note three important properties of the relationship between bond prices and yield:
1-Bond price are inversely related to bond yield.
2-The price volatility of a long-term bond is greater than that of a short-term bond, holding the coupon rate constant.
3-The price volatility of a lower coupon bond is greater than that of a higher coupon bond, holding maturity constant.
So, the lower the coupon rate, the higher the price volatility. Or, price volatility varies inversely (or indirectly) with coupon rate, while it varies directly with maturity.
INTEREST RATE RISK AND DURATION
In this section we formally present the concept of interest rate risk and show how investors and financial institutions attempt to manage it using a risk measure called duration.
Interest rate risk is the risk related to changes in interest rate that cause a bond’s realized yield to differ from the promised yield. Two factors affects interest rate risk: (1) price risk and (2) reinvestment risk.
- Price risk: Interest rate changes cause the market value of a bond to rise or fall, resulting in capital gains or losses to the investor, (recall the inverse relationship between i and PB).
- Reinvestment risk: market interest rate may change, causing the lender to have to reinvest coupon payments at interest rate different from the interest rate at the time the bond was purchased. That is it refers to the change in bond yield resulting from reinvesting coupon payments at different rate.
Price risk versus reinvestment risk:
It is very important to recognize that price risk and reinvestment risk partially offset one another. When interest rate decline, bond price increases, resulting in a capital gain (good news), but the gain is partially offset by lower coupon reinvestment income (bad news). On the other hand, when interest rate rise, the bond suffers a capital loss (bad news), but the loss is partially offset by higher coupon reinvestment income (good news). So, price risk and reinvestment risk potentially offset each other. But how can we measure interest risk?