Question

5a FYI bonds have a par value of $1,000. The bonds pay an 8% annual coupon and will mature in 11 years. i) Calculate the price if the yield to maturity on the bonds is 7%, 8% and 9%, respectively. ii) What is the current yield on these bonds if the YTM on the bonds is 7%, 8% and 9%, respectively. Hint, you can only calculate current yield after you have determined the intrinsic value (price) of the bonds. iii) Explain the impact on price if the required rate of return increases Bonds paying an 8% coupon have payments of $80 per year. The future value = the par value, which is returned at bond maturity. N is the years (number of coupon payments) to maturity. 56 Let's look at the 8% coupon bond again. Assume that the required rate of return is now 9% and that rate stays constant until the bond matures. What is the price when there are 10 years to maturity, then at 5 years to maturity and then 1 year to maturity. 6 LRO Inc. bonds on July 1, 2006. The bonds had a coupon rate of 5.5%, with interest paid semiannually. The face value of the bonds is $1,000 and the bonds mature on July 1, 2025. What is the intrinsic value of an LRQ Corporation bond on July 1, 2012 to an investor with a required return of 7%? Hints: The bonds pay interest (coupon payments) every six months. The coupon rate is 5.5%, or $55 per year. Half of this amount is paid every 6 months. Original years to maturity 2025-2006 = 19 years. But we are now at 2012, and there are 13 years to maturity remaining. However, N is the number of payments (of $55/2) and that would be 13 * 2. As with all TVM problems, N, I/Y and PMT must be on the basis. We need the required rate per 6 month period = 7%/2 = 3.5% per 6 months. 7 Bart's Moving Company bonds have a 11% coupon rate. Interest is paid annually. The bonds have a par value of $1,000 and will mature 8 years from now. Compute the value of Bart's Moving Company bonds if investors' required rate of return is 9.5%. 8 Rework the above problem assuming semi-annual coupon payments.

Answer 1

Solution:

Data given: Par value of Bond $ 1,000

   Annual coupon rate 8%

   Maturity period 11 years

Part 5a)

i) To find current price -

if the yield to maturity (YTM) is 7%

The general formula for current price of a bond is explained in Exhibit 1 below:

(1) WHERE COLPON RATE IS SEMI - ANNUAL ( PAYMENT DATES A YEAR) - CURRENT PRICE of BOND P = / & c/2 + ... + 2 + F WHERE coupon RATE IS ANNUAL (SPECIFICALLY MENTIONED) P = C WHERE (1+r) (1+r) 2 (1+rn P = CURRING PRICE c = Coupon PAYMENT (FACE VALUE X Coupon RATE ). r = YIELD TO MATURITY OR DISCOUNT RATE n = YEARS TO MATURITY COUPON = INTEREST RATE, SAY 8% GOC BOND (FOR E.g.)

Further, the present value of an annuity (viz. the annual coupon payment) is explained in Exhibit 2 below and applied to the above formula to enable you to calculate the current price

PRESENT VALUE OF Annuity (FIXED SUM RECEVED IN PERIODIC / AMMUAL INSTALMES) + PVACA ((1+r) - 1] PVA = PRESENT VALUE of ANNUITY OF THE Coupon PAYMENT A = ANNUITY OF COUPUN PAYMENT (OR C AS DEMOTED ABOVE) r&N DENOTED ABOVE So PIN ABOVE BECOMES P = , 7 PVAC + F Critr OR larin ON THE is THAT Icon BE MODIFIED WHERE PROBLEM IS SILE PAYMENT HOMENT DATES PER YEAR DATES PER YEAR / usus (Usually CONVENTIUM BOND PAYMENTS OF Couron AMOUNT ARE SEMI-AMMUNE

Now applying the formula P = C [ {(1 + r)ⁿ - 1} / r(1 + r)ⁿ ] + F / (1 + r)ⁿ

where Face value, F = $ 1000

   Coupon payment, C = $1000 x 8% = $ 80

   Yield to maturity, r = 7% or 0.07

   Maturity period, n = 11

   P = $ 80 x [{(1 + 0.07)¹¹ - 1} / 0.07 x (1 + 0.07)¹¹] + $ 1000 / (1 + 0.07)¹¹

   = $ 80 x [(1.07¹¹ - 1) / 0.07 x 1.07¹¹ ] + $1000 / 1.07¹¹

= $ 80 x 1.1048 / (0.07 x 2.1048) + $1000 / 2.1048

   = $ 80 x 1.1048 / 0.1473 + $ 1000 / 2.1048

   = $ 80 x 7.500 + $ 475.10

   = $ 600 + $ 475.10

   = $ 1,075.10

Similarly if we apply 8 % or 9 % as r in above formula we get the results:

if the yield to maturity (YTM) is 8% the current price is $ 1,000

if the yield to maturity (YTM) is 9% the current price is $ 931.95

ii) Find yield if the YTM is 7%, 8% and 9%

The yield is the coupon payment divided by the current price

So Yield if rate is 7% is 80 / 1075.10 = 0.0744 i.e 7.44%

   … is 8% is 80 / 1000 = 8%

   … is 9% is 80 / 931.95 = 0.0858 i.e. 8.58%   

iii) If the required rate of return or the yield to maturity increases (as we have seen from 7% to 9%), the current price has an inverse relationship i.e. it decreases. Hence, if the required rate on a bond (or intrinsic rate of return) is higher the bond is discounted further to a lower price.

Part 5b) Solution is provided in Exhibit 3 viz the answers are $ 935.81, $ 961.10 and $ 990.83 in the three cases:

را 8oX۱۰۵۹ و جما ديز - 8* ۱۰ 367 د 2 ما +اہا .3\5 = 4 1 دا با 238 0.4 ×۱۰۵۹۱۰ ۱۰۵۹۱ 237 09-0 =4435,61 دنیا 0538 وه _ ما ۹۰۹ماهی ۱-3۱۱ ) ** 1 ددها ( ا tes93 تیک ۱ - ؟ 1053 6.04x1-09 -6 ۹۶ / 5366 8 10واک جدي | م. ۹.ولادة ۔'۰۵۹/x * P80 ' ' $ 450,63 - { س ۵۰۵۹۶۱-۵۶۱۰۵۹ 0.00 ۶۰۵۹ اوي-| |