Math 418B Worksheet 3 Callable Bonds
Name: __________________________ Show critical work and then circle the answer you get. In everyday business and financial usage there are three different “yield” associated with a bond:
1. Nominal yield is simply the annualized coupon rate on the bond. If a $100 par value bond has coupons totaling $9 per year, then the nominal yield on the bond is 9% per annum.
2. Current yield is the ratio of the annualized coupon to the original price of the bond.
3. Yield to maturity is the actual annualized yield rate, or internal rate of return defined as “The yield rate is that rate of interest at which the present value of returns from the investment is equal to the present value of contributions into the investment.”
The most problems in Math 418 and FM exam have the yield rates as defined in 3. Yield to maturity.
Callable Bond A callable bond is one for which there is a range of possible redemption dates. The redemption is chosen by the bond issuer. For a specific yield rate, the price of bond will depend on the time until maturity. For a specific bond price, the yield rate on the bond will depend on the time until maturity.
Example A: Pricing callable bonds (yield rate lower than coupon rate)
1. A 10% bond with semi-annual coupons and with face amount 1,000,000 is issued with the condition that
redemption can take place on any coupon date between 12 and 15 years from the issue date. Find the price paid by in investor wishing a minimum yield of i(2)=0.08.
Solution: Using BAII Plus calculator to fill out the following table N
(term, no. of coupons)
I/Y (Yield to maturity)
PV (Bond price)
PMT (Coupon payment)
FV (Redemp. amt)
24 0.04 1152469.63 50,000 1,000,000
25 0.04 1156220.80 50,000 1,000,000
26 0.04 1159827.69 50,000 1,000,000
27 0.04 1163295.86 50,000 1,000,000
28 0.04 1166630.63 50,000 1,000,000
29 0.04 1169837.15 50,000 1,000,000
30 0.04 1172920.33 50,000 1,000,000
Ans. This callable bond has price ranging from 1,152,470 to 1,172,920 depending on the time until it’s called for redemption. It is, thus, most prudent for the investor to offer a price of no more than 1,152,470 to guarantee a minimum yield of i(2)=0.08.
2. For the same callable bond above, suppose the investor pays the minimum of all prices for the range of
redemption dates. Find the yield rate if the issuer chooses a redemption date corresponding to the maximum price.
3. Suppose the investor pays 1,150,000 for this callable bond and holds the bond until it’s called (12-15 years from issue). Find the minimum yield that the investor will obtain.
Answer ______________________8.03%
Example B: Pricing callable bonds (yield rate higher than coupon rate)
1. A 10% bond with semi-annual coupons and with face amount 1,000,000 is issued with the condition that
redemption can take place on any coupon date between 12 and 15 years from the issue date. Find the price paid by in investor wishing a minimum yield of i(2)=0.12.
Solution: Using BAII Plus calculator to fill out the following table N
(term, no. of coupons)
I/Y (Yield to maturity)
PV (Bond price)
PMT (Coupon payment)
FV (Redemp. amt)
24 0.06 874.496.42 50,000 1,000,000
25 0.06 872166.44 50,000 1,000,000
26 0.06 869968.34 50,000 1,000,000
27 0.06 867894.66 50,000 1,000,000
28 0.06 865938.36 50,000 1,000,000
29 0.06 864092.79 50,000 1,000,000
30 0.06 862351.69 50,000 1,000,000
Ans. This callable bond has price ranging from 862,352 to 874,496 depending on the time until it’s called for redemption. It is, thus, most prudent for the investor to offer a price of no more than 862,352 to guarantee a minimum yield of i(2)=0.12.
2. For the same callable bond above, suppose the investor pays the minimum of all prices for the range of
redemption dates. Find the yield rate if the issuer chooses a redemption date corresponding to the maximum price.
Answer ______________________12.215%
3. Suppose the investor pays 850,000 for this callable bond and holds the bond until it’s called (12-15 years from issue). Find the minimum yield that the investor will obtain.
Example C: Pricing callable bonds (variable redemption amounts)
A 15-year 8% bond with face amount 100 is callable on a coupon date in the 10th to 15th years. In the 10th year the bond is callable at par, in the 11th or 12th years at redemption amount 115, or in the 13th, 14th or 15th years at redemption amount 135.
1. What price should an investor pay in order to ensure a minimum nominal annual yield to maturity of 12%? (Note: 6-month yield rate is higher than the modified coupon rate for any redemption amount, that is, bought at a discount)
Solution:
Time N
(term, no. of coupons)
I/Y (Yield to maturity)
PV (Bond price)
PMT (Coupon payment)
FV (Redemp. amt)
9.5 19 6(%) 77.68 4 100
10 20 6(%) 77.06 4 100
10.5 21 6(%) 80.88 4 115
11 22 6(%) 80.08 4 115
11.5 23 6(%) 79.32 4 115
12 24 6(%) 78.60 4 115
12.5 25 6(%) 82.59 4 135
13 26 6(%) 81.69 4 135
13.5 27 6(%) 80.84 4 135
14 28 6(%) 80.03 4 135
14.5 29 6(%) 79.28 4 135
15 30 6(%) 78.56 4 135
Remark: The principal of pricing a callable bond bought at a discount by using the latest redemption date may fail when the redemption amounts are not level. Explain.
Because you would expect the smallest PV to be at latest date when bought at discount but since FV are increasing the smallest PV is at the latest time at smallest FV.
Answer: ____________________________77.06
2. Find the investor’s minimum yield if the purchase price is 80.
Remark: If the bond is bought at a discount to the redemption value, it is to the investor’s disadvantage to have the redemption at the latest date. Explain.
When sold at discount there is a capital gain on purchase day. The sooner the bondholder receives the gain the greater the yield, so if redeemed on last date the yield is smaller. As a result, for a fixed yield, the bond value is smaller if redeemed later.
Example D: Pricing callable bonds (variable redemption amounts)
A 15-year 8% bond with face amount 100 is callable on a coupon date in the 10th to 15th years. In the 10th year the bond is callable at par, in the 11th or 12th years at redemption amount 115, or in the 13th, 14th or 15th years at redemption amount 135.
1. What price should an investor pay in order to ensure a minimum nominal annual yield to maturity of 6%? (Note: 6-month yield rate is lower than modified coupon rate in 10th-12th years—at a premium; higher than modified coupon rate in 13th-15th years—at a discount)
Solution:
Time N
(term, no. of coupons)
I/Y (Yield to maturity)
PV (Bond price)
PMT (Coupon payment)
FV (Redemp. amt)
9.5 19 3(%) 114.32 4 100
10 20 3(%) 114.87 4 100
10.5 21 3(%) 123.48 4 115
11 22 3(%) 123.77 4 115
11.5 23 3(%) 124.04 4 115
12 24 3(%) 124.31 4 115
12.5 25 3(%) 134.13 4 135
13 26 3(%) 134.11 4 135
13.5 27 3(%) 134.08 4 135
14 28 3(%) 134.06 4 135
14.5 29 3(%) 134.04 4 135
15 30 3(%) 134.02 4 135
Remark: What’s the principal of pricing a callable bond bought at a premium? Does the minimum price occur at the latest redemption date? Explain.
It’s the soonest because it’s a premium bond so the coupon rate is greater than yield, so the PV or bond value increases because the expected more coupons are adding values to the bond.
Answer: ____________________________114.32
2. Find the investor’s minimum yield if the purchase price is 120.
Remark: If the bond is bought at a premium to the redemption value, the minimum yield to maturity occurs at the earliest redemption date. Explain.
There will be a capital loss when the bond is redeemed so the earliest the loss occurs, the lower the return received by the bondholder.
Callable
Bonds
Price Formula: Part 1 F changes to C
Price Formula: Part 2 F(r-j) ̅ changes to
C(r-j) ̅
# of Coupons from issue to redemption:
n
Bond Price at issue
Bought at a
Premium
Fixed Positive (since r>j)
More positive More coupons Call later, more valuable Bought at a
Discount
Fixed Negative (since r<j)
More negative More coupons Call sooner, more valuable
Hold to
Maturity
Price Formula: Part 1 F
Price Formula: Part 2 F(r-j) ̅̅̅̅̅̅
# of Coupons until maturity, n-t
Book Value/Outstanding
Balance (at time t)
Bought at a
Premium
Fixed Positive (since r>j) Decreasing Bought at a
Discount
Fixed Negative (since r<j) Increasing
