**26. **To find the capital gains yield and the current yield, we need to find the price of the bond. The current price of Bond P and the price of Bond P in one year is:

P: *P*0 = $84(PVIFA7%,8) + $1,000(PVIF7%,8) = $1,083.60

*P*1 = $84(PVIFA7%,7) + $1,000(PVIF7%,7) = $1,075.45

Current yield = $84/$1,083.60

Current yield = .0775, or 7.75% The capital gains yield is:

Capital gains yield = (New price – Original price)/Original price

Capital gains yield = ($1,075.45 – 1,083.60)/$1,083.60

Capital gains yield = –.0075, or –.75%

The current price of Bond D and the price of Bond D in one year is: D: *P*0 = $56(PVIFA7%,8) + $1,000(PVIF7%,8) = $916.40

*P*1 = $56(PVIFA7%,7) + $1,000(PVIF7%,7) = $924.55

Current yield = $56/$916.40

Current yield = .0611, or 6.11%

Capital gains yield = ($924.55 – 916.40)/$916.40

Capital gains yield = .0089, or .89%

All else held constant, premium bonds pay a high current income while having price depreciation as maturity nears; discount bonds pay a lower current income but have price appreciation as maturity nears. For either bond, the total return is still 7 percent, but this return is distributed differently between current income and capital gains.

**27. ***a*. The rate of return you expect to earn if you purchase a bond and hold it until maturity is the YTM. The bond price equation for this bond is:

*P*0 = $865 = $55(PVIFA*R%*,21) + $1,000(PVIF *R%*,21)

Using a spreadsheet, financial calculator, or trial and error we find:

*R *= YTM = 6.72%

*b*. To find our HPY, we need to find the price of the bond in two years. The price of the bond in two years, at the new interest rate, will be:

*P*2 = $55(PVIFA5.72%,19) + $1,000(PVIF5.72%,19)

*P*2 = $975.15

To calculate the HPY, we need to find the interest rate that equates the price we paid for the bond with the cash flows we received. The cash flows we received were $55 each year for two years, and the price of the bond when we sold it. The equation to find our HPY is:

*P*0 = $865 = $55(PVIFA*R%*,2) + $975.15(PVIF*R%*,2) Solving for *R*, we get:

*R *= HPY = 12.36%

The realized HPY is greater than the expected YTM when the bond was bought because interest rates dropped by 1 percent; bond prices rise when yields fall.

**28. **The price of any bond (or financial instrument) is the PV of the future cash flows. Even though Bond M makes different coupon payments, to find the price of the bond, we just find the PV of the cash flows. The PV of the cash flows for Bond M is:

*P*M = $800(PVIFA2.95%,16)(PVIF2.95%,12) + $1,000(PVIFA2.95%,12)(PVIF2.95%,28) +

$20,000(PVIF2.95%,40)

*P*M = $17,791.23

Notice that for the coupon payments of $800, we found the PVA for the coupon payments, and then discounted the lump sum back to today.

Bond N is a zero coupon bond with a $20,000 par value; therefore, the price of the bond is the

PV of the par, or:

*P*N = $20,000(PVIF2.95%,40)

*P*N = $6,251.38

**29. **To calculate this, we need to set up an equation with the callable bond equal to a weighted average of the noncallable bonds. We will invest X percent of our money in the first noncallable bond, which means our investment in Bond 3 (the other noncallable bond) will be (1 – X). The equation is:

C2 = C1 X + C3(1 – X)

7.60 = 5.50X + 8.40(1 – X)

7.60 = 5.50X + 8.40 – 8.40X X = .27586

So, we invest about 28 percent of our money in Bond 1, and about 72 percent in Bond 3. This combination of bonds should have the same value as the callable bond, excluding the value of the call. So:

*P*2 = .27586P1 + .72414P3

*P*2 = .27586(106.375) + .72414(108.21875)

*P*2 = 107.71

The call value is the difference between this implied bond value and the actual bond price. So, the call value is:

Call value = 107.71 – 103.50 = 4.210

Assuming a $1,000 par value, the call value is $42.10.

**30. **In general, this is not likely to happen, although it can (and did). The reason that this bond has a negative YTM is that it is a callable U.S. Treasury bond. Market participants know this. Given the high coupon rate of the bond, it is extremely likely to be called, which means the bondholder will not receive all the cash flows promised. A better measure of the return on a callable bond is the yield to call (YTC). The YTC calculation is the basically the same as the YTM calculation, but the number of periods is the number of periods until the call date. If the YTC were calculated on this bond, it would be positive. As we mentioned in the chapter, several European government bonds have also had negative YTMs.

**31. **To find the real annual increase in salary, we need to find the nominal growth rate and the annual inflation rate. So, the nominal growth rate in salary was:

$97,500 = $22,400(1 + *R*)30

*R *= ($97,500/$22,400)1/30 – 1

*R *= .0502, or 5.02%

And the annual inflation rate was:

1,021.39 = 415.23(1 + *h*)30

*h *= (1,021.39/415.23)1/30 – 1