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Review questions

1.Define the concepts of compounding and discounting. Use future values and present values to explain how these concepts are related.

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Compounding is a method used to find out the future value of a present sum-‑ that is, what is the future value of money lent (or borrowed) today.

Unlike compounding, which is forward looking, discounting is in effect backward looking. Discounting is the method used to figure out what the present value of money is to be received (or paid) in the future.

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2.Use the concept of present value to explain why a trip to Hawaii next year would mean more to most people than the same trip in the year 2015.

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A trip to Hawaii would mean more to most people next year than in the year 2015. The present value of a trip next year is much higher than the present value of a trip in year 2015.

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3.Under what conditions will a bond sell at a premium above par? At a discount from par?

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If the interest rate increases, a bond will sell at a discount from par. If the interest rate increases, the present value of the future stream of income from the bond falls and therefore its price falls.

If the interest rate decreases, a bond will sell at a premium above par. If the interest rate decreases, the present value of the future stream of income from the bond rises and therefore its price rises.

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4.During the Great Depression of the 1930s, nominal interest rates were close to zero. Explain how real interest rates could be very high even though nominal interest rates were very low. (Hint: Prices fell during parts of the Great Depression.)

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The nominal interest rate is composed of the real interest rate plus an inflation premium. The real interest rate is therefore the nominal interest rate minus the inflation premium.

During the Great Depression, if prices were going down instead of up, the inflation premium was actually negative.

In this case, to find the real rate, we subtract a negative inflation rate from a low nominal rate. Thus, the more negative the inflation rate is, the higher the real interest rate will be for any nominal interest rate.

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5.Assume that after you graduate, you get a job as the chief financial officer of a small company. Explain why being able to forecast the direction of interest rate changes may be critical for your success in that position. Likewise, why are investment bankers concerned about future changes in the interest rate?

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As the chief financial officer of a small company, the interest rate is very important.

In all probability, the company holds some long term financial instruments. The fluctuations of interest rates have a direct effect on the value of these long term investments.

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For example, if interest rates increase, the value of long term bonds decreases and vice versa.

If a company holds long term financial instruments, an increase in interest rates will cause the value of these instruments to fall.

The chief financial officer will be much more successful if he/she can anticipate or forecast changes in interest rates and sell the long term financial instruments before interest rates do increase and the value of the instruments fall.

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Likewise, if the chief financial officer expects rates to fall, he/she should move available funds into long term instruments before rates do fall.

If successful in this strategy, long-term capital gains will be increased and capital losses minimized.

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Investment bankers hold stockpiles( 储备） of long term financial instruments such as bonds.

Any increase or decrease of interest rates can drastically change the values of bonds and expose the investment banker to potential capital gains or losses.

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6.What factors affect the demand for loanable funds? The supply of loanable funds?

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The demand for loananble funds comes from household, business, government and foreign net borrowers.

As spending plans of these units increase, the demand for loanable funds increases and vice versa.

Changes in the demand for loanable funds are positively or directly related to changes in GDP.

When GDP increases, the demand for loanable funds increases and vice versa.

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The supply of loanable funds comes from household, business, government and foreign net lenders.

In addition, the supply of loanable funds is affected by the actions of the Fed (monetary policy) in speeding up or slowing down the provision of reserves to the banking system. When monetary policy is expansionary, the supply of loanable funds increases. When monetary policy is contractionary, the supply of loanable funds decreases.

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7.In general, discuss the movement of interest rates, the money supply, and prices over the business cycle.

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In expansions, GDP increases causing interest rates generally to rise because of increased demand, and inflation to pick up speed. Likewise, to prevent overheating of the economy, the Fed is generally reducing the money supply and further increasing interest rates.

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In recessions, GDP is falling, the interest rate is falling due to reduced demand and expansionary Fed policy may be causing the money supply to increase and interest rates to further fall. At the same time, upward price pressures are reduced.

Thus, but not always, interest rates and inflation fluctuate pro-cyclically.

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8.A young couple is borrowing $100,000 to buy their first home. An older couple is living off the interest income from the $100,000 in financial assets they own. How does the interest rate affect each couple? If the interest rate increases, could that change the behavior of either couple? How and why?

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If the young couple borrowed through a fixed loan, higher interest rates would affect their payment and perhaps increase so much that they would be unable to borrow the $100,000. If the young couple borrowed through a variable interest rate loan, then the higher the interest rate, the worse off they will be.

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If the older couple invested their $100,000 in long term financial instruments, then the higher the interest rate would reduce the value of those instruments. If the $100,000 was invested in shorter term instruments, higher interest rates would lead to higher income for the older couple. In either case, spending would be affected for both parties.

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If the interest rate increased, the young couple may have to postpone the purchase of a house either because they can no longer qualify for the higher payment or were reluctant to take on the higher payment.

If the older couple had long term financial instruments, falls in the values of those instruments could cause them to reduce their spending. If they held short term instruments, increases in interest income would cause increases in spending.

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If interest rates become volatile, the young couple might decide to look for a fixed interest rate loan and the older couple might decide to leave the bond market. In both cases, they (or their loans and assets) will not be so tied to interest rate fluctuations.

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Analytical Questions

9.What is the present value of each of the following income streams?

a.$100 to be received at the end of each of the next three years

b.$100 to be received at the end of each of the next three years plus an additional payment of $1,000 at the end of the third year

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The present value of $100 at the end of each of the next three years depends on the current interest rate and can be found by the following formula:

$100/(1+i)1 + $100/(1+i)2 + $100/(1+i)3

For an interest rate of 10 percent, the present

value is found by substituting .10 for i in the above formula:

$90.91 + $82.64 + $75.19 = $248.74

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The present value of $100 at the end of each of the next three years plus an additional payment of $1,000 at the end of the third year depends on the current interest rate and can be found by the following formula:

$100/(1+i)1 + $100/(1+i)2 + $100/(1+i)3 + $1,000/(1+i)3

For an interest rate of 10 percent, the present value is

found by substituting .10 for i would be equal to $90.91 + $82.64 + $75.19 + $751.88 = $1,000.62

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10.What is the price of a bond that pays the income stream in question 9 (b)?

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The price of a bond that pays the income stream in question 9 is the present value of the stream of income. If the interest rate is 10 percent, the present value is $1,000.62.

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11.Assume that a bond with five years to maturity, a par value of $1,000, and a $60 annual coupon payment costs $1,100 today. What is the coupon rate? What is the current yield?

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The coupon rate is the coupon payment ($60) divided by the par value ($1,000). In this case, it is therefore 6 percent ($60/$1,000).

The current yield is the coupon payment ($60) divided by the price of the bond today ($1,100). In this case, it is therefore 5.45 percent ($60/$1,100).

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12.The nominal interest rate is 12 percent, and anticipated inflation is 8 percent. What is the real interest rate?

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The real interest rate is the nominal rate minus the anticipated inflation. In this case it is 4 percent (12 percent – 8 percent = 4 percent).

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13.Graph the demand and supply for loanable funds. If there is an increase in income, ceteris paribus, show what happens to the interest rate, the demand for loanable funds, and the quantity supplied of loanable funds. If the Fed orchestrates a decrease in the money supply growth rate, ceteris paribus, show what happens to the interest rate, the supply of loanable funds, and the quantity demanded of loanable funds.

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The supply of and demand for funds

loanable funds

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If there is an increase in income, the demand curve for the loanable funds shifts from D to D1. At the new equilibrium point (where S and D1 intersect), interest rate, demand for loanable funds, and quantity supplied of loanable funds increase.

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The supply of and demand for funds

loanable funds

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If the Fed orchestrates a decrease in the money supply growth rate, the supply curve will shift from S to S1. At the new equilibrium point (where D and S1 intersect), the interest rate increases. At the new equilibrium point, the supply of loanable funds and the quantity demanded of loanable funds decrease.

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14.As an enrolling freshman, would you have been willing to pay $18,000 for four years’ tuition rather than $5,000 per year for four years? (Assume you would be able to do so and that you have no fear of flunking out of college before you graduate.)

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Whether I would be willing to pay $18,000 tuition now or $5,000 per year for four years depends on the current interest rate.

The present value of $18,000 is $18,000. The present value of $5,000 per year at the beginning of each of the 4 years is equal to $5,000 + $5,000/(1+i)1 + $5,000/(1+i)2 + $5,000/(1+i)3.

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If the interest rate is 6 percent, this is $5,000 + $5,000/(1.06)1 + $5,000/(1.06)2 + $5,000(1.06)3 = $5,000 + $4,716.98 + $4,464.29 + $4,201.68 = $18,382.95. In this case, it would be better to pay the $18,000 up front because it has a lower present value.

If the interest rate is 10 percent, then it would be better to make 4 payments of $5,000 because they have a present value lower than $18,000. It is $5,000 + $5,000/(1.10)1 + $5,000/(1.10)2 + $5,000/(1.10)3 = $5,000 + $4,545.45 + $4,132.29 + $3,759.40 = $17,437.14.

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15.You win a million dollar lottery to be paid out in 20 annual installments of $50,000 over the next 20 years. Assuming an interest rate of 6 percent, how large a payment would you accept today for this future stream of income?

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To answer this question, you must find the present value of the 20 future annual payments of $50,000 each over the next 20 years.

We use the present value formula, $50,000/(1+i)1 + $50,000/(1+i)2 +…….+ $50,000/(1+i)20 =

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16.Jake is given $10,000 in a CD that matures in 10 years. Assuming interest payments are reinvested during the life of the CD, how much will the CD be worth at maturity if the interest rate is 5 percent? If the interest rate is 10 percent?

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To answer the questions, you need to find the future value 10 years from now of the $10,000 invested in the CD today.

If the interest rate is 5 percent, the CD will be worth $16,300 ($10,000 x (1.05)10). If the interest rate is 10 percent, the CD will be worth $25,900 ($10,000 x 1.110).

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17.Henry and Sheree just had a baby. How much will they have to invest today for the baby to have $100,000 for college in 18 years if the interest rate is 5 percent? If the interest rate is 10 percent?

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To answer the question, you need to find the present value of $100,000 in 18 years.

If the interest rate is 5 percent, the present value is $41,493.70 ($100,000/1.0518).

If the interest rate is 10 percent, the present value is $17,985.61 ($100,000/1.118).

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18.Use graphical analysis to show that if Y and M both increase, the interest rate may increase, decrease, or stay the same. In each case, what happens to the equilibrium quantity demanded and supplied?

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If Y and M increase, both the demand and the supply curves will shift to the right. Graph I shows that at the new equilibrium point (where the D1 and S1 curves intersect), the interest rate is lower than at the original equilibrium point (where the D and S curves intersect).

Graph II shows that at the new equilibrium point (where the D1 and S1 curves intersect), the interest rate is higher than at the original equilibrium point (where the D and S curves intersect).

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As both graphs illustrate, the change in interest rate will depend on the magnitude of the shifts of the supply and demand curves.

In both graphs, the quantity demanded and quantity supplied change in accordance to the magnitude of the shifts of the supply and demand curves.

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19.Using Exhibit 6-6, determine in what years real interest rates were at their highest and lowest levels.

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From Exhibit 6-6, it appears that real interest rates were highest in the early to mid-1980s (roughly 1982 to 1986) and lowest in the mid-to-late 1970s. During the latter period, real interest rates were sometimes negative.

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20.What is the price of a consol with a coupon payment of $200 per year if the interest rate is 10 percent? What is the interest rate on a consol if the coupon payment is $400 and the price of the consol is $8,000? (Appendix 6-A)

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The price of a consol with a coupon payment of $200 per year and an interest rate of 10 percent is $2,000 ($200/.10 = $2,000).

If the price of a consol is $8,000 and the coupon payment is $400, then the interest rate is 5 percent ($8,000 = $400/.05).

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21.I purchase a consol with a coupon payment of $100 when the interest rate is 10 percent. When I sell the consol, the interest rate has risen to 20 percent. What is the amount of my capital gain or loss? (Appendix 6-A)

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The price of a consol with a coupon payment of $100 and a 10 percent interest rate is $1,000 ($100/.10 = $1,000).

The price of a consol with a coupon payment of $100 and a 20 percent interest rate is $500 ($100/.20 = $500).

Therefore, if I buy the consol when the interest rate is 10 percent for $1,000 and sell the consol when the interest rate is 20 percent for $500, then I make a capital loss of $500.