# Interest Rates and Bond Valuation

1 January 2017

As a consumer, the effective annual rate is the more important rate since it represents the rate actually paid or earned.  110. You are considering two annuities, both of which make total annuity payments of \$10,000 over their life. Which would be worth more today: annuity A, which pays \$1,000 at the end of each year for the next 10 years, or annuity B, which pays \$775 at the end of the first year, but the annuity payment grows by \$50 each year, reaching \$1,225 at the end of year 10? Are there any circumstances in which the two would be equal? Explain.

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The second annuity weights its payments more toward the back of the period, rather than the front, making it less valuable unless the discount rate is zero. Some students may get tripped up by the fact that the two annuities have the same total payments. This would clearly demonstrate a lack of understanding of the time value of money. There are three factors that affect the present value of an annuity. Explain what these three factors are and discuss how an increase in each will impact the present value of the annuity. The factors are the interest rate, payment amount, and number of payments.

An increase in the payment and number of payments will increase the present value, while an increase in the interest rate will decrease the present value. There are three factors that affect the future value of an annuity. Explain what these three factors are and discuss how an increase in each will impact the future value of the annuity. The factors are the interest rate, payment amount, and number of payments. An increase in any of these three will increase the future value of the annuity.

Should lending laws be changed to require lenders to report the effective annual rate rather than the annual percentage rate? Explain the reasoning for your answer. It would be more meaningful for consumers to know the effective annual rate rather than the annual percentage rate. The effective annual rate is slightly more difficult to calculate and also more difficult to explain, and may add confusion to the loan process. However, regardless of the costs, it would appear that consumers would benefit from learning what the effective annual rate is as opposed to the annual percentage rate.

Annuity A makes annual payments of \$813. 73 for each of the next 10 years, while annuity B makes annual payments of \$500 per year forever. At what interest rate would you be indifferent between the two? At interest rates above this break-even rate, which annuity would you choose? How about at interest rates below the break-even rate? This requires the students to actually use the present value formulas, setting the present value annuity equal to the present value of a perpetuity and solving for the interest rate that makes the two equivalent.

The first step is recognizing that the indifference point occurs when the two present values are equal. The break-even rate is 10 percent: below that rate, the perpetuity is better, while above that rate, the 10-year annuity is preferred.  A friend who owns a perpetuity that promises to pay \$1,000 at the end of each year, forever, comes to you and offers to sell you all of the payments to be received after the 25th year for a price of \$1,000.

At an interest rate of 10 percent, should you pay the \$1,000 today to receive payment numbers 26 and onwards? What does this suggest to you about the value of perpetual payments? The present value of the perpetuity is \$10,000, and the present value of the first 25 payments is \$9,077. 04, thus you should be willing to pay only \$922. 96 for payments 26 and onwards. This suggests that the value of a perpetuity is derived primarily from the payments received early in its life, and the payments to be received later have little worth today

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