GF500 Unit 3 Interest Rate Problem Set Example

Six Rate Problems, Each With a Line on What It Means: A Unit 3 Interest Rate Problem Set

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Purdue University Global

GF500: Financial Institutions and Markets

Unit 3 Interest Rate Problem Set

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[Date]

Prices, rates and the borrower in each problem are illustrative; the books and article cited are real.

Problem 1: Pricing a Zero-Coupon Bond

Given. A zero-coupon bond pays $1,000 in three years. The market yield is 4.5 percent a year, compounded annually.

Formula. Price equals face value divided by one plus the yield, raised to the number of years.

Substitution. $1,000 divided by 1.045 cubed, which is 1.141166.

Result. $876.30.

Meaning. A lender willing to wait three years gives up $876.30 today to receive $1,000; the $123.70 difference is the whole of the return, since there are no coupons.

Problem 2: Pricing a Coupon Note

Given. A five-year note pays a 5 percent annual coupon, $50 a year, on $1,000 face value. The market yield is 6 percent.

Formula. Price equals the present value of the five coupons plus the present value of the face value.

Substitution. The coupons are worth $50 times the five-year annuity factor at 6 percent, 4.21236, or $210.62. Discounting the $1,000 face value five years at 6 percent gives $747.26.

Result. $957.88.

Meaning. The note sells below face value because its coupon trails the market rate; a buyer is compensated for the low coupon by paying less now and receiving full face value at maturity.

Problem 3: Solving for Yield to Maturity

Given. A four-year bond with a 6 percent annual coupon trades at $1,043.30.

Method. We need the one annual rate at which four coupons and the final $1,000, each discounted, add up to $1,043.30. There is no closed formula, so trial rates are used.

Trials. At 4.5 percent the price would be $1,053.81, too high, so the yield must be higher. At 5.0 percent it would be $1,035.46, too low, so the yield lies between. Narrowing the interval gives a yield of about 4.78 percent.

Result. About 4.78 percent.

Meaning. A buyer at $1,043.30 earns less than the 6 percent coupon suggests, because part of each coupon is effectively returning the premium paid over face value.

Problem 4: Holding Period Return When Yields Fall

Given. An investor buys a ten-year bond with a 4 percent annual coupon at par, $1,000. One year later, market yields on nine-year bonds have fallen to 3.5 percent, and the investor sells after collecting the first coupon.

Formula. Holding period return equals the coupon plus the change in price, divided by the purchase price.

Substitution. The selling price is the value of nine remaining coupons of $40 and $1,000 at maturity, discounted at 3.5 percent: $1,038.04. Return equals $40 plus $38.04, divided by $1,000.

Result. 7.80 percent.

Meaning. Falling yields handed the investor a capital gain nearly as large as the coupon, which shows why bondholders gain when rates fall and lose when they rise.

Problem 5: Deriving a Forward Rate

Given. The one-year spot rate is 4.80 percent and the two-year spot rate is 4.40 percent.

Formula. One plus the two-year rate, squared, equals one plus the one-year rate times one plus the one-year rate one year forward.

Substitution. 1.044 squared is 1.089936. Dividing by 1.048 gives 1.040015.

Result. About 4.00 percent.

Meaning. The market is pricing a one-year rate of about 4 percent a year from now, lower than today's 4.80 percent, which suggests investors expect rates to fall, though part of the difference may reflect a term premium.

Problem 6: Real and Nominal Rates

Given. A one-year certificate pays 5.2 percent. Expected inflation over the year is 3.0 percent.

Formula. By the Fisher relation, one plus the nominal rate equals one plus the real rate times one plus expected inflation (Fisher, 1930).

Substitution. 1.052 divided by 1.030 is 1.021359.

Result. A real rate of about 2.14 percent. The common shortcut, nominal minus inflation, gives 2.2 percent, close but slightly high.

Meaning. A saver's purchasing power grows by about 2.1 percent, not 5.2, which is the figure that matters when deciding whether to save or spend.

Every problem in the set rests on one idea, discounting future cash at a market rate, and the differences between them lie only in which quantity is unknown: the price, the yield, the future rate or the real return.

Verifying Each Result

A second route confirms every answer. In Problem 1, compounding $876.30 at 4.5 percent for three years returns $1,000. In Problem 2, a note priced below par must have a yield above its coupon, and 6 percent exceeds 5. In Problem 3, discounting the payments at 4.78 percent reproduces the price to within a few cents. In Problem 4, the gain is consistent with a bond whose duration is a little under eight years, since a half-point fall in yield should raise its price by roughly 4 percent. In Problem 5, investing for two years at 4.40 percent gives the same ending value as one year at 4.80 and a second at about 4.00. These checks catch most keying errors before an answer is submitted.

Notes on Conventions

All rates are annual and compounded once a year, prices are rounded to the cent and rates to two decimal places. Market prices in practice also reflect accrued interest and semiannual coupons for most US bonds; the problems use annual payments to keep the arithmetic visible (Mishkin & Eakins, 2018). Forward rates computed this way blend expectations and risk premiums, a point the yield curve unit takes up (Brealey et al., 2020).

References

Brealey, R. A., Myers, S. C., & Allen, F. (2020). Principles of corporate finance (13th ed.). McGraw-Hill Education.

Fisher, I. (1930). The theory of interest. Macmillan.

Mishkin, F. S., & Eakins, S. G. (2018). Financial markets and institutions (9th ed.). Pearson.