Preparation
Simple Loans require payment of one amount which equals the loan principal plus the interest.
Fixed-Payment Loans are loans where the loan principal and interest are repaid in several payments, often monthly, in equal dollar amounts over the loan term.
(I.E)- auto loans and home mortgages are frequently of the fixed-payment type.
Coupon Bond- owners of the bond pay a fixed amount until maturity date, face value is repaid
Discount Bond- is a bought a price below its face value, Face Value is repaid at the maturity.
The concept of present value (or present discounted value) is based on the commonsense notion that a dollar of cash flow paid to you one year from now is less valuable to you than a dollar paid to you today. This notion is true because you could invest the dollar in a savings account that earns interest and have more than a dollar in one year.
When we have council bond, which doesn’t have maturity date.
An asset is a piece of property that is a store of value.
Problem 1
You invested your savings in 6 % coupon bond with the face value of 1000 USD and 3 years maturity. Suppose you have kept this bond for one year and now you are considering your bond for selling before maturity. At what price will you sell the bond? Interest rate is 7%.
Solution: 60/1,07 + 1060/ (1,07) 2 = 981,92
Problem 1.
You invested your savings in 8% coupon bond with the face value of 1000 USD and 3 years maturity. Suppose you have kept this bond for one year and now you are considering your bond for selling before maturity. At what price will you sell the bond if interest rate is 7%?
Solution: 80/1,07 + 1080/(1,07) 2 = 1018,08
Problem 1.
You invested your savings in 8% coupon bond with the face value of 1000 USD and 3 years maturity. Suppose you have kept this bond for one year and now you are considering your bond for selling before maturity. At what price will you sell the bond if interest rate is 9%?
Solution: 80/1,09 + 1080/ (1,09) 2 = 982,41
Problem 1.
You invested your savings in 7% coupon bond with the face value of 1000 USD and 3 years maturity. Suppose you have kept this bond for one year and now you are considering your bond for selling before maturity. At what price will you sell the bond if interest rate is 8%?
Solution: 70/1,08 + 1070/(1,08) 2 = 982,16
Problem 2.
You want to invest in a ten-year, 7% coupon bond with a face value of $1000 that is currently selling for $871.65. Compute your rate of return if you sell the bond next year for $880.10.
R= (70+ 880.10-871.65)/871.65 = 9%
Problem 2.
You decided to buy a 5-year bond, 7% with a face value of $ 1000 that is currently selling for $ 850. Compute your rate of return if you are selling this bond next year for $ 865.
R = (70+ 865-850)/850 = 10%
Problem 2.
You want to invest in a ten-year 8% coupon bond with a face value of $ 1000 that is currently selling for 860. Compute you rate of return if you sell the bond next year for $ 900.
R= (80+ 900 – 860)/860 = 13,95%
Problem 2.
You decided to buy a 5-year bond, 8% with a face value of $ 1000 that is currently selling for $ 850. Compute your rate of return if you are selling this bond next year for $ 890.
R = (80+890-850)/850 = 14,12%
Problem 3.
You are working for BTA and you are asked to estimate the interest rate risk of the bond portfolio (in $ terms) corporate bonds with the 8% coupon rate and 4 years maturity. The face value of a bond is 1000$. Suppose the current interest rate is 9%. Calculate the duration.
73,39 | 73,39 | 0,0758 | |
67,33 | 134,66 | 0,1392 | |
61,77 | 185,31 | 0,1915 | |
765,10 | 3060,40 | 3,1629 |
967,59 3,5694
Problem 3.
Calculate the duration on a four – year 7% coupon bond with the face value of 10 000 if the current interest rate is 8%. You have to draw the table to solve the problem.
648,15 | 648,15 |
| |
600,14 | 1200,28 |
| |
555,68 | 1667,04 |
| |
7,864.82 | 31,459.28 |
|
34,974.75 / 9,668.79 = 3,62
Problem 3.
Calculate the duration on a three – year 7% coupon bond with the face value of 1000 if the current interest rate is 8%. You have to draw the table to solve the problem.
DUR = 2,80
Problem 3.
Calculate the duration on a three – year 6% coupon bond with the face value of 1000 if the current interest rate is 7%. You have to draw the table to solve the problem.
56,07 | 56,07 |
| |
52,41 | 104,82 |
| |
865,28 | 2,595.84 |
| |
|
|
|
|
DUR = 2,756.73/ 973,76 = 2,83
Problem 4.
One-year T-bill rates over the next four years are expected to be 3%, 4%, 5% and 5.5%. If four-year T-bonds are yielding 4.5%, what is the liquidity premium on this bond?
4,5% = (3+4+5+5.5)/4 + Ln; Ln = 0,125
Problem 4.
If the interest rates on one- to five-year bonds are currently 4%, 5%, 6%, 7%, 8% and the term premiums for one- to five-year bonds are 0%, 0.25%, 0.35%, 0.40%, 0.50%, predict the interest rate for four-year bond according to the liquidity premium theory.
Interest rate for four-year bond = (4+5+6+7)/4 + 0,40 = 5,9%
Problem 4.
The one-year interest rate over the next ten years will be 3%, 4.5%, 6%, 7.5%, 9%, 10.5%, 13%, 14.5%, 16% and 17.5%. Using the pure expectations theory, what will be the interest rates on the three-year bond, six-year bond?
Interest rate for three-year bond = 3+ 4.5 +6/3 = 4.5%
Interest rate for six-year bond = 3+ 4.5+6+ 7.5+9+10,5/6 = 6.75%
Problem 5
Calculate the yield for consol bond, which offers 8% coupon bond with the face value of $10,000 if the current market price of the bond is $9800?
Yield = 800/9800 = 0,0816 = 8,16%
Problem 5
Calculate the yield for the consol bond with 9% coupon bond and with the face value of $10,000 if the current market price of the bond is $9560?
Yield = 900/9560 = 9,41%
Problem 5
Calculate the yield for the consol bond, which offers 8% coupon rate with the face value of 10,000 and the current market price of the bond is $9800?
Yield = 800/9800 = 8,16%
Problem 5
Calculate the yield for the consol bond with 7% coupon bond and with the face value of $10,000 if the current market price of the bond is $9660?
Yield = 700/9660 = 7,25%
At what price will you be willing to buy 3-year discount bond if it pays $1000 face value at maturity. The current YTM=9%.
Problem 6
Predict an interest rate 2 years from now if rates are 4%, 5%, 6%, 7%, 8%, premiums are
0%, 0.25%, 0.35%, 0.40%, 0.50%.
I t+2 = ((1+0,06-0,0035)^3/(1+0,05-0,0025)^2))-1 = 0,0747 = 7,47%
Problem 7
You own a $1000-par zero-coupon bond that has five years of remaining maturity. You plan on selling the bond in one year, and believe that the required yield next year will have the following probability distribution.
Probability | Required Yield % |
0.1 | 6.60% |
0.2 | 6.75% |
0.4 | 7.00% |
0.2 | 7.20% |
0.1 | 7.45% |
a. What is your expected price when you sell the bond?
b. What is the standard deviation of the bond price?
Solution:
Probability | Required Yield | Price | Prob ´ Price | Prob *(Price – Exp. Price)2 |
0.1 | 6.60% | $774.41 | $77.44 | 12.84776241 |
0.2 | 6.75% | $770.07 | $154.01 | 9.775668131 |
0.4 | 7.00% | $762.90 | $305.16 | 0.013017512 |
0.2 | 7.20% | $757.22 | $151.44 | 6.862609541 |
0.1 | 7.45% | $750.20 |
|
|
|
|
| $763.07 | 46.08937999 |
a)
The expected price is $763.07.
b) The variance is $46.09, or a standard deviation of $6.79.
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