EC1008: Chapter 4 questions and answers
If the interest rate is 15%, what is the present value of a security that pays you $1,100 next year, $1,250 the year after, and $1,347 the year after that?
$1,100/(1 + 0.15) + $1,250/(1 + 0.15)^2 + $1,347/(1 + 0.15)^3 = $2,787.38
Write down the formula that is used to calculate the yield to maturity on a twenty-year 12% coupon bond with a $1,000 face value that sells for $2,500.
$2,500 = $120/(1 + i) + 120/(1 + i)^2 + . . . + 120/(1 + i)^20 + $1,000/(1 + i)^20. Solving for i gives the yield to maturity.
Consider a coupon bond that has a $900 par value and a coupon rate of 6%. The bond is currently selling for $860.15 and has two years to maturity. What is the bond's yield to maturity (YTM)?
$860.15 = $54/(1 + i) + $54/(1 + i)^2 + $900/(1 + i)^2. Solving for i gives a yield to maturity of 0.085, or 8.5%.
What is the yield to maturity on a $10,000-face-value discount bond maturing in one year that sells for $9,523.81?
($10,000 - $9,523.81) / $ 9,523.81 = $476.19 / $9,523.81 = 0.05 = 5%
What is the yield to maturity (YTM) on a simple loan for $1,500 that requires a repayment of $15,000 in five years' time?
58.5%, derived as follows: The present value of the $15,000 payment five years from now is $15,000 /(1 + i)^5, which equals the $1,500 loan. Thus 1500 = 15000 /(1 + i)^5. Solving for i = 0.585 = 58.5%
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.)
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
A lottery claims its grand prize is $15 million, payable over 5 years at $3,000,000 per year. If the first payment is made immediately, what is this grand prize really worth? Use an interest rate of 7%
In present value terms, the lottery prize is worth $3,000,000 + $3,000,000/(1.07) + $3,000,000/(1.07)^2 + $3,000,000/(1.07)^3 + $3,000,000/(1.07)^4 = $13,161,634
A financial adviser has just given you the following advice: "Long-term bonds are a great investment because their interest rate is over 20%." Is the financial adviser necessarily right?
No. If interest rates rise sharply in the future, long-term bonds may suffer such a sharp fall in price that their return might be quite low, possibly even negative.
If the interest rate is 5% in the first year, and 10% in the second year, what is the present value of a security that pays you $100 next year, and $200 the year after?
PV = 100/(1+0.05) + 200/(1+0.05)(1+0.10) = $268
If the interest rate is 5%, what is the present value of a security that pays you £100 next year, -£50 the year after, and £35 the third year?
PV = 100/(1+0.05) - 50/(1+0.05)^2 + 35/(1+0.05)^3 = $80.12
If the interest rate is -0.01, what is the present value of a security that pays you £100 next year? How does the present values change if the interest rate is 0.01, instead?
PV = 100/(1-0.01) = 101.01 vs. PV = 100/(1+0.01) = 99.01
Calculate the present value of a $1,300 discount bond with seven years to maturity if the yield to maturity is 8%
PV = FV /(1 + i)^n , where FV = 1300, i = 0.08, n = 7. PV = 1300/(1+0.08)^7 . Thus, PV = 758.54.
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?
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%) and 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.)
A $1,100-face-value bond has a 5% coupon rate, its current price is $1,040, and it is expected to increase to $1070 next year. Calculate the current yield, the expected rate of capital gains, and the expected rate of return.
The coupon payment C = $55, thus the current yield is $55/$1040 = 0.053, or 5.3%. The expected rate of capital gain, g = ($1070 - $1040)/$1040 = 30/1040 = 0.028, or 2.9%. The expected rate of return, R = iC + g = 5.3% + 2.9% = 8.2%.
Which $10,000 bond has the higher yield to maturity, a twenty-year bond selling for $8,000 with a current yield of 20% or a one-year bond selling for $8,000 with a current yield of 10%?
The current yield is a good approximation for the yield to maturity of long-term bond, but not for a short-term bond. The formula for the current yield is i=C/P. Re-working the formula we can find the coupon C= P x i= 8000 x 0.1= 800 Using the Coupon-bond formula, we can then derive the yield to maturity for the one- year coupon-bond: 8000 = (800 + 10000)/(1+i) 8000 + 8000i = 10800 i= 2800/8000 = 0.35 or 35% The yield to maturity on the bond given above is greater than the YTM of a similar $10,000 20-year bond with a current yield of 20% selling for $8,000.
When is the current yield a good approximation of the yield to maturity?
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.
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?
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.
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?
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.
What is the price of a perpetuity that has a coupon of $70 per year and a yield to maturity of 1.5%? If the yield to maturity doubles, what will happen to the perpetuity's price?
The price would be $70/0.015 = $4667. If the yield to maturity doubles to 3%, the price would fall to half its previous value, to $2333 = $70/0.03.
Assume you just deposited $1,250 into a bank account. The current real interest rate is 1%, and inflation is expected to be 5% over the next year. What nominal rate would you require from the bank over the next year? How much money will you have at the end of one year? If you are saving to buy a fancy bicycle that currently sells for $1,300, will you have enough money to buy it?
The required nominal rate would be: i = rr +π e = 1% + 5% = 6%. At this rate, you would expect to have $1,250 x 1.06, or $1,325 at the end of the year. Can you afford the bicycle? It is uncertain. This depends on whether the price of the bicycle increases with inflation.
Property taxes in a particular district are 2% of the purchase price of a home every year. If you just purchased a $150,000 home, what is the present value of all the future property tax payments? Assume that the house remains worth $150,000 forever, property tax rates never change, and a 4% interest rate is used for discounting.
The taxes on the $150,000 home are $150,000 × 0.02 = $3,000 per year. The PV of all future payments = $3,000/0.04 = $75,000 (a perpetuity).
Would $175, to be received in exactly one year, be worth more to you today when the interest rate is 15% or when it is 20%?
Today, it would be worth 175 / (1 + 0.15) = $152 (rounded to the nearest whole number) when the interest rate is 15%, rather than 175 / (1 + 0.20) = $146 when the interest rate is 20%. Thus, $175 received in one year would be worth less to you today if the interest rate rose to 20%.
True or False: With a discount bond, the return on the bond is equal to the rate of capital gain.
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.
Do bondholders fare better when the yield to maturity increases or when it decreases? Why?
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.
What relationships do you observe between years to maturity, yield to maturity, and the current price?
When the yield to maturity is greater than the coupon rate, the bond's current price is below its face value. For a given maturity, the bond's current price falls as the yield to maturity rises. For a given yield to maturity, a bond's value rises as its maturity increases. When the yield to maturity is equal to the coupon rate, a bond's current price equals its face value regardless of the number of years of maturity.
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?
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. 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.
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?
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.
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?
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.