Chapter 4 ECN 241

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?

$1,100/(1 + 0.10) + $1,210/(1 + 0.10)2 + $1,331/(1 + 0.10)3 = $3,000.

Write down the formula that is used to calculate the yield to maturity on a twenty-year 10% coupon bond with a $1,000 face value that sells for $2,000.

$2,000 = $100/(1 + i) + $100/(1 + i)2 + . . . + $100/(1 + i)20 + $1,000/(1 + i)20. Solving for i gives the yield to maturity.

What is the yield to maturity on a simple loan for $1 million that requires a repayment of $2 million in five years' time?

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 i = 5 2 −1= 0.149 =14.9%.

What is the yield to maturity on a $1,000-face-value discount bond maturing in one year that sells for $800?

25% = ($1,000 - $800)/$800 = $200/$800 = 0.25.

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.

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 cur- rent yield of 5%?

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.

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 yield to maturity doubles, what will happen to the perpetuity's price? $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%

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%.

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.

Would a dollar tomorrow be worth more to you today when the interest rate is 20% or when it is 10%?

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.

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.

Calculate the present value of a $1,000 discount bond with five years to maturity if the yield to maturity is 6%

PV = FV/(1 + i)n, where FV = 1000, i = 0.06, n = 5. Thus, PV = 747.26

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%). 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.)

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 $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?

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.

Assume you just deposited $1,000 into a bank account. The current real interest rate is 2%, and inflation is expected to be 6% 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 $1,050, will you have enough money to buy it?

The required nominal rate would be: 2%+ 6% =8%. i = rr +π e At this rate, you would expect to have $1,000 X 1.08, or $1,080 at the end of the year. Can you afford the bicycle? In theory, the price of the bicycle will increase with the rate of inflation. So, one year later, the bicycle will cost $1,050 X 1.06, or $1,113. You will be short by $33. If the bicycle does not increase in price with inflation, then you will have enough to purchase it.

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 taxes on the $250,000 home are $250,000 Å~ 0.04 = $10,000 per year. The PV of all future payments = $10,000/0.06 = $166,666.67 (a perpetuity).

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.

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. 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.

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.

Consider a bond with a 4% annual coupon and a face value of $1,000. Complete the following table. What relationships do you observe between years to maturity, yield to maturity, and the current price?

Years to Maturity Yield to Maturity Current Price 2 2% 2 4% 3 4% 5 2% 5 6% Years to Maturity Yield to Maturity Current Price 2 2% 1038.83 2 4% 1000.00 3 4% 1000.00 5 2% 1094.27 5 6% 915.75 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

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.