Money and Banking Chapter 4
You would rather be holding long-term bonds because their price would increase more than the price of the short-term bonds, giving them a higher return. Longer-term bonds are more susceptible to higher price fluctuations than shorter-term bonds, and hence have greater interest-rate risk.
1. If interest rates decline, which would you rather be holding, long-term bonds or short-term bonds? Why? Which type of bond has the greater interest-rate risk?
People are more likely to buy houses because the real interest rate when purchasing a house has fallen from 3% (= 5-2%) to 1% (= 10-9%). The real cost of financing the house is thus lower, even though nominal mortgage rates have risen. (If the tax deductibility of interest payments is allowed for, then it becomes even more likely that people will buy houses.)
1. If mortgage rates rise from 5% to 10%, but the expected rate of increase in housing prices rises from 2% to 9%, are people more or less likely to buy houses?
$1,100/(1 + 0.10) + $1,210/(1 + 0.10)2 + $1,331/(1 + 0.10)3 = $3,000.
1. If the interest rate is 10%, what is the present value of a security that pays you $1,100 next year, $1,210 the year after, and $1,331 the year after that?
The economists are right. They reason that nominal interest rates were below expected rates of inflation in the late 1970s, making real interest rates negative. The expected inflation rate, however, fell much faster than nominal interest rates in the mid-1980s, so nominal interest rates were above the expected inflation rate and real rates became positive.
1. Interest rates were lower in the mid-1980s than in the late 1970s, yet many economists have commented that real interest rates were actually much higher in the mid-1980s than in the late 1970s. Does this make sense? Do you think that these economists are right?
While it would appear to them that their wealth is declining as nominal interest rates fall, as long as expected inflation falls at the same rate as nominal interest rates, their real return on savings accounts will be unaffected. However, in practice, expected inflation as reflected by the cost of living for seniors and retired persons often is much higher than standard measures of inflation, thus low nominal rates can adversely affect the wealth of senior citizens and retired persons.
1. Retired persons often have much of their wealth placed in savings accounts and other interest-bearing investments and complain whenever interest rates are low. Do they have a valid complaint?
The yield to maturity of these bonds solves the following equation: 5,000/(1+i) = 5,012. After some algebra, the yield to maturity happens to be around - 0.24%. This is not a typical situation. In normal times, banks will not choose to pay more than the face value of a discount bond, since that implies negative yields to maturity. This example illustrates situations as the ones described in the Global Box in this chapter.
1. Suppose that a commercial bank wants to buy Treasury bills. These instruments pay $5,000 in one year and are currently selling for $5,012. What is the yield to maturity of these bonds? Is this a typical situation? Why?
The rate of capital gain is the part of the rate of return formula that incorporates future changes in the price of the bond. The other part of the formula, the current yield, is composed of the coupon payment (completely determined by the bond´s face value and coupon rate) and the price you paid for the bond today. The rate of capital gain incorporates the future price of the bond and is therefore the part of the formula that reflects the consequences of future price changes.
1. Suppose today you buy a coupon bond that you plan to sell one year later. Which part of the rate of return formulation incorporates future changes into the bond?Note: Check Equations 7 and 8 in this chapter.
In present value terms, the lottery prize is worth $2,000,000 + $2,000,000/(1.06) + $2,000,000/(1.06)2 + $2,000,000/(1.06)3 + $2,000,000/(1.06)4, or $8,930,211.
1. A lottery claims its grand prize is $10 million, payable over 5 years at $2,000,000 per year. If the first payment is made immediately, what is this grand prize really worth? Use an interest rate of 6%.
PV = FV/(1 + i)n, where FV = 1000, i = 0.06, n = 5. Thus, PV = 747.26.
1. Calculate the present value of a $1,000 discount bond with five years to maturity if the yield to maturity is 6%
When the yield to maturity increases, this represents a decrease in the price of the bond. If the bondholder were to sell the bond at a lower price, the capital gains would be smaller (capital losses larger), and therefore, the bondholder would be worse off.
1. Do bondholders fare better when the yield to maturity increases or when it decreases? Why?
When comparing amounts of money that are disbursed at different dates, one has to take into consideration the concept of present value of money. To calculate the present value of the $5,500 promised one year from today one needs to know the annual interest rate. In this case, for an interest rate larger than 10%, one would prefer to accept the $5,000 today (since one can deposit that amount and receive more than $5,500 one year from today).
1. Explain which information you would need to take into consideration when deciding to receive $5,000 today or $5,500 one year from today.
If the interest rate were 12%, the present discounted value of the payments on the government loan are necessarily less than the $1,000 loan amount because they do not start for two years. Thus, the yield to maturity must be lower than 12% in order for the present discounted value of these payments to add up to $1,000.
1. To help pay for college, you have just taken out a $1,000 government loan that makes you pay $126 per year for 25 years. However, you don't have to start making these payments until you graduate from college two years from now. Why is the yield to maturity necessarily less than 12%? (This is the yield to maturity on a normal $1,000 fixed-payment loan on which you pay $126 per year for 25 years.)
True. The return on a bond is the current yield iC plus the rate of capital gain, g. A discount bond, by definition, has no coupon payments, thus the current yield is always zero (the coupon payment of zero divided by current price) for a discount bond.
1. True or False: With a discount bond, the return on the bond is equal to the rate of capital gain.
The current yield will be a good approximation to the yield to maturity whenever the bond price is very close to par or when the maturity of the bond is over about ten years. This is because cash flows farther in the future have such small present discounted values that the value of a long-term coupon bond is close to a perpetuity with the same coupon rate.
1. When is the current yield a good approximation of the yield to maturity?
The near-term costs to maintaining a given size loan are much smaller for a perpetuity than for a similar fixed payment loan, discount, or coupon bond. For instance, assuming a 5% interest rate over 10 years, on a $1000 loan, a perpetuity costs $50 a year (or $500 in payments over 10 years). For a fixed payment loan, this would be $129.50 per year (or $1295 in payments over the same 10-year period). For a discount loan, this loan would require a lump sum payment of $1628.89 in 10 years. For a coupon bond, assuming the same $50 coupon payment as the perpetuity implies a $1000 face value. Thus, for the coupon bond, the total payments at the end of 10 years will be $1500.
1. Why would a government choose to issue a perpetuity, which requires payments forever, instead of a terminal loan, such as a fixed-payment loan, discount bond, or coupon bond?
It would be worth 1/(1 + 0.20) = $0.83 when the interest rate is 20%, rather than 1/(1 + 0.10) = $0.91 when the interest rate is 10%. Thus, a dollar tomorrow is worth less with a higher interest rate today
1. Would a dollar tomorrow be worth more to you today when the interest rate is 20% or when it is 10%?
14.9%, derived as follows: The present value of the $2 million payment five years from now is $2/(1 + i)5 million, which equals the $1 million loan. Thus 1 = 2/(1 + i)5. Solving for i, (1 + i)5 = 2, so that
18. What is the yield to maturity on a simple loan for $1 million that requires a repayment of $2 million in five years' time?
When yield to maturity is above the coupon rate, the bond's current price is below its face value. The opposite holds true when yield to maturity is below the coupon rate. For a given maturity, the bond's current price falls as yield to maturity rises. For a given yield to maturity, a bond's value rises as its maturity increases. When yield to maturity equals the coupon rate, a bond's current price equals its face value regardless of years to maturity.
19. Consider a coupon bond that has a $1,000 par value and a coupon rate of 10%. The bond is currently selling for $1,044.89 and has two years to maturity. What is the bond's yield to maturity?
If the one-year bond did not have a coupon payment, its yield to maturity would be ($1,000-$800)/ $800 = $200/$800 = 0.25, or 25%. Because it does have a coupon payment, its yield to maturity must be greater than 25%. However, because the current yield is a good approximation of the yield to maturity for a twenty-year bond, we know that the yield to maturity on this bond is approximately 15%. Therefore, the one-year bond has a higher yield to maturity.
19. Which $1,000 bond has the higher yield to maturity, a twenty-year bond selling for $800 with a current yield of 15% or a one-year bond selling for $800 with a current yield of 5%?
The taxes on the $250,000 home are $250,000 ´ 0.04 = $10,000 per year. The PV of all future payments = $10,000/0.06 = $166,666.67 (a perpetuity).
22. A $1000-face-value bond has a 10% coupon rate, its current price is $960, and its price is expected to increase to $980 next year. Calculate the current yield, the expected rate of capital gain, and the expected rate of return.
The price would be $50/0.025 = $2000. If the yield to maturity doubles to 5%, the price would fall to half its previous value, to $1000 = $50/0.05.
22. Property taxes in a particular district are 4% of the purchase price of a home every year. If you just purchased a $250,000 home, what is the present value of all the future property tax payments? Assume that the house remains worth $250,000 forever, property tax rates never change, and a 6% interest rate is used for discounting.
The bank will charge you a nominal interest rate equal to 1% + 3% = 4%. However, if actual inflation turns out to be lower than expected, then you will be worse off than originally planned, since the real cost of borrowing (measured by the real interest rate) turned out to be 4-0.5% = 3.5%.
22. Suppose that you want to take out a loan and that your local bank wants to charge you an annual real interest rate equal to 3%. Assuming that the annualized expected rate of inflation over the life of the bond is 1%, determine the nominal interest rate that the bank will charge you. What happens if, over the life of the loan, actual inflation is 0.5%?
$1044.89 = $100/(1 + i) + $100/(1 + i)2 + $1,000/(1 + i)2. Solving for i gives a yield to maturity of 0.075, or 7.5%.
22. What is the price of a perpetuity that has a coupon of $50 per year and a yield to maturity of 2.5%? If the yield to maturity doubles, what will happen to the perpetuity's price?
Present Value
Let i=.10 1 year: 100*(1+0.10)=110 2 years: 110*(1+0.10)=121 3 years: 121*(1=0.10)=133 Or 100*(1+i)^n with n being years
PV=CF/(1+i)^n
Simple Present Value PV=today's(present) value CF=future cash flow (payment) i= the interest rate
Fixed-Payment Loan
The same cash flow payment every period throughout the life of the loan LV = loan value FP = fixed yearly payment n = number of years until maturity LV=(FP/1+i)+(FP/(1+i^2))
Whenever the current price P is greater than face value F of a discount bond, the yield to maturity will be negative. It is possible for a coupon bond to have a negative nominal interest rate, as long as the coupon payment and face value are low relative to the current price. As an example, with a one-year coupon bond, the yield to maturity is given as i = (C + F - P)/P; in this case whenever C + F < P, i will be negative. It is impossible for a perpetuity to have a negative nominal interest rate, since this would require either the coupon payment or the price to be negative.
Under what conditions will a discount bond have a negative nominal interest rate? Is it possible for a coupon bond or a perpetuity to have a negative nominal interest rate?
Coupon Bond
Using the same strategy used for the fixed-payment loan: P = price of coupon bond C = yearly coupon payment F = face value of the bond n = years to maturity date P=C/1+i
Ex post real interest rate
adjusted for actual changes in the price level
Real interest rate
adjusted for changes in price level so it more accurately reflects the cost of borrowing
Ex ante real interest rate
adjusted for expected changes in the price level
Discount Bond
i=F-P/P F = Face value of the discount bond P = Current price of the discount bond
Nominal interest rate
makes no allowance for inflation
Yield to maturity
the interest rate that equates the present value of cash flow payments received from a debt instrument with its value today
Four Types of Credit Market Instruments
•Simple Loan •Fixed Payment Loan •Coupon Bond Discount Bond
Consol or perpetuity
•a bond with no maturity date that does not repay principal but pays fixed coupon payments forever P = C / ic Pc = price of the consol C = yearly interest payment Ic = yield to maturity of the consol can rewrite above equation as this: ic = C/Pc
Present value
•a dollar paid to you one year from now is less valuable than a dollar paid to you today.
