Assume that you are nearing graduation and have applied for a job with a local b
ID: 2778479 • Letter: A
Question
Assume that you are nearing graduation and have applied for a job with a local bank. The bank’s evaluation process requires you to take an examination that covers several financial analysis techniques. Use the following information for Questions 1 through 2:
What is the present value of the following uneven cash flow stream $50, $100, $75, and $50 at the end of Years 0 through 3? The appropriate interest rate is 10%, compounded annually.
Suppose that on January 1 you deposit $100 in an account that pays a nominal (or quoted) interest rate of 11.33463%, with interest added (compounded) daily. How much will you have in your account on October 1, or 9 months later?
Use the following information for Questions 3 and 4:
A firm issues a 10-year, $1,000 par value bond with a 10% annual coupon and a required rate of return is 10%.
What is the yield to maturity on a 10-year, 9% annual coupon, $1,000 par value bond that sells for $887.00? That sells for $1,134.20? What does a bond selling at a discount or at a premium tell you about the relationship between rd and the bond’s coupon rate?
What are the total return, the current yield, and the capital gains yield for the discount bond in Question #3 at $887.00? At $1,134.20? (Assume the bond is held to maturity and the company does not default on the bond.)
Explanation / Answer
PV = Cash Flow / (1+i)^n
where i is the interest rate and n is the number of the period.
2)
Ending amount = Principal amount * (1+i)^n
where i is the rate of interest and n is the number of period.
THE YIELD TO MATURITY (YTM) IS THE DISCOUNT RATE THAT EQUATES THE PRESENT VALUE OF A BOND'S CASH FLOWS TO ITS PRICE. IN OTHER WORDS, IT IS THE PROMISED RATE OF RETURN ON THE BOND. (NOTE THAT THE EXPECTED RATE OF RETURN IS LESS THAN THE YTM IF SOME PROBABILITY OF DEFAULT EXISTS.) ON A CASH FLOW TIME LINE, WE HAVE THE FOLLOWING SITUATION WHEN THE BOND SELLS FOR $887:
0 1 9 10
3)))))))))3)))))))!!!))))))))3))))))))3
$90 $90 + $ 90
PV1 =)))))- * 1,000
. * *
. k = ? * *
PV10 =)))))))))))))))))))))))))))- *
PVM =)))))))))))))))))))))))))))))))))-
____
SUM = PV = $887
WE WANT TO FIND k IN THIS EQUATION:
Vd = PV = INT/(1+k)1 + … + INT/(1+k)N + M/(1+k)N .
TO GET THE EXACT VALUE OF THE YTM FOR THIS BOND, WE HAVE TO USE EITHER A FINANCIAL CALCULATOR OR A TRIAL-AND-ERROR PROCESS. WITH A FINANCIAL CALCULATOR, WE CAN SOLVE FOR k BY ENTERING THE KNOWN DATA INTO A FINANCIAL CALCULATOR AND THEN PRESSING THE I = k BUTTON. THE YTM IS FOUND TO BE 10.91%.
ALTERNATIVELY, WE COULD USE PRESENT VALUE INTEREST FACTORS:
$887 = $90(PVIFAk,10) + $1,000(PVIFk,10).
GOING TO THE PV TABLES, WE WOULD SUBSTITUTE FACTORS FOR VARIOUS INTEREST RATES, IN A TRIAL-AND-ERROR MANNER, UNTIL WE FOUND THE RATE THAT PRODUCES THE EQUALITY. THIS IS TIRESOME, AND THE PROCEDURE WILL NOT GIVE AN EXACT ANSWER UNLESS THE YTM IS A WHOLE NUMBER.
WE CAN TELL FROM THE BOND'S PRICE, EVEN BEFORE WE BEGIN THE CALCULATIONS, THAT THE YTM MUST BE ABOVE THE 9 PERCENT COUPON RATE. WE KNOW THIS BECAUSE THE BOND IS SELLING AT A DISCOUNT, AND DISCOUNT BONDS ALWAYS HAVE kd > COUPON RATE.
IF THE BOND WERE PRICED AT $1,134.20, THEN IT WOULD BE SELLING AT A PREMIUM. IN THAT CASE, IT MUST HAVE A YTM THAT IS BELOW THE 9 PERCENT COUPON RATE, BECAUSE ALL PREMIUM BONDS MUST HAVE COUPONS THAT EXCEED THE GOING INTEREST RATE. GOING THROUGH THE SAME PROCEDURES AS BEFORE--PLUGGING THE APPROPRIATE VALUES INTO A FINANCIAL CALCULATOR AND THEN PRESSING THE k = I BUTTON, WE FIND THAT AT A PRICE OF $1,134.20, kd = YTM = 7.08%.
PV = Cash Flow / (1+i)^n
where i is the interest rate and n is the number of the period.
2)
Ending amount = Principal amount * (1+i)^n
where i is the rate of interest and n is the number of period.
THE YIELD TO MATURITY (YTM) IS THE DISCOUNT RATE THAT EQUATES THE PRESENT VALUE OF A BOND'S CASH FLOWS TO ITS PRICE. IN OTHER WORDS, IT IS THE PROMISED RATE OF RETURN ON THE BOND. (NOTE THAT THE EXPECTED RATE OF RETURN IS LESS THAN THE YTM IF SOME PROBABILITY OF DEFAULT EXISTS.) ON A CASH FLOW TIME LINE, WE HAVE THE FOLLOWING SITUATION WHEN THE BOND SELLS FOR $887:
0 1 9 10
3)))))))))3)))))))!!!))))))))3))))))))3
$90 $90 + $ 90
PV1 =)))))- * 1,000
. * *
. k = ? * *
PV10 =)))))))))))))))))))))))))))- *
PVM =)))))))))))))))))))))))))))))))))-
____
SUM = PV = $887
WE WANT TO FIND k IN THIS EQUATION:
Vd = PV = INT/(1+k)1 + … + INT/(1+k)N + M/(1+k)N .
TO GET THE EXACT VALUE OF THE YTM FOR THIS BOND, WE HAVE TO USE EITHER A FINANCIAL CALCULATOR OR A TRIAL-AND-ERROR PROCESS. WITH A FINANCIAL CALCULATOR, WE CAN SOLVE FOR k BY ENTERING THE KNOWN DATA INTO A FINANCIAL CALCULATOR AND THEN PRESSING THE I = k BUTTON. THE YTM IS FOUND TO BE 10.91%.
ALTERNATIVELY, WE COULD USE PRESENT VALUE INTEREST FACTORS:
$887 = $90(PVIFAk,10) + $1,000(PVIFk,10).
GOING TO THE PV TABLES, WE WOULD SUBSTITUTE FACTORS FOR VARIOUS INTEREST RATES, IN A TRIAL-AND-ERROR MANNER, UNTIL WE FOUND THE RATE THAT PRODUCES THE EQUALITY. THIS IS TIRESOME, AND THE PROCEDURE WILL NOT GIVE AN EXACT ANSWER UNLESS THE YTM IS A WHOLE NUMBER.
WE CAN TELL FROM THE BOND'S PRICE, EVEN BEFORE WE BEGIN THE CALCULATIONS, THAT THE YTM MUST BE ABOVE THE 9 PERCENT COUPON RATE. WE KNOW THIS BECAUSE THE BOND IS SELLING AT A DISCOUNT, AND DISCOUNT BONDS ALWAYS HAVE kd > COUPON RATE.
IF THE BOND WERE PRICED AT $1,134.20, THEN IT WOULD BE SELLING AT A PREMIUM. IN THAT CASE, IT MUST HAVE A YTM THAT IS BELOW THE 9 PERCENT COUPON RATE, BECAUSE ALL PREMIUM BONDS MUST HAVE COUPONS THAT EXCEED THE GOING INTEREST RATE. GOING THROUGH THE SAME PROCEDURES AS BEFORE--PLUGGING THE APPROPRIATE VALUES INTO A FINANCIAL CALCULATOR AND THEN PRESSING THE k = I BUTTON, WE FIND THAT AT A PRICE OF $1,134.20, kd = YTM = 7.08%.
THE CURRENT YIELD IS DEFINED AS FOLLOWS:
CY = ANNUAL COUPON INTEREST PAYMENT/CURRENT PRICE OF THE BOND
THE CAPITAL GAINS YIELD IS DEFINED AS FOLLOWS:
CGY = EXPECTED CHANGE IN BOND’S PRICE/BEGINNING-OF-YEAR PRICE
THE TOTAL EXPECTED RETURN IS THE SUM OF THE CURRENT YIELD AND THE EXPECTED CAPITAL GAINS YIELD:
EXPECTED EXPECTED EXPECTED CAPITAL
TOTAL RETURN = CURRENT YIELD + GAINS YIELD.
THE TERM YIELD TO MATURITY, OR YTM, IS OFTEN USED IN DISCUSSING BONDS. IT IS SIMPLY THE EXPECTED TOTAL RETURN (ASSUMING NO DEFAULT RISK), SO ^k = EXPECTED TOTAL RETURN = EXPECTED YTM.
RECALL ALSO THAT SECURITIES HAVE REQUIRED RETURNS, k, WHICH DEPEND ON A NUMBER OF FACTORS:
REQUIRED RETURN = k = k* + IP + LP + MRP + DRP.
WE KNOW THAT (1) SECURITY MARKETS ARE NORMALLY IN EQUILIBRIUM, AND (2) THAT FOR EQUILIBRIUM TO EXIST, THE EXPECTED RETURN, ^k = YTM, AS SEEN BY THE MARGINAL INVESTOR, MUST BE EQUAL TO THE REQUIRED RETURN, k. IF THAT EQUALITY DOES NOT HOLD, THEN BUYING AND SELLING WILL OCCUR UNTIL IT DOES HOLD, AND EQUILIBRIUM IS ESTABLISHED. THEREFORE, FOR THE MARGINAL INVESTOR:
^k = YTM = k.
FOR OUR 9 PERCENT COUPON, 10-YEAR BOND SELLING AT A PRICE OF $887 WITH A YTM OF 10.91%, THE CURRENT YIELD IS:
CURRENT YIELD = $90/$887 = 0.1015 = 10.15% .
KNOWING THE CURRENT YIELD AND THE TOTAL RETURN, WE CAN FIND THE CAPITAL GAINS YIELD:
YTM = CURRENT YIELD + CAPITAL GAINS YIELD
CAPITAL GAINS YIELD = YTM - CURRENT YIELD = 10.91% - 10.15% = 0.76%.
THE CAPITAL GAINS YIELD CALCULATION CAN BE CHECKED BY ASKING THIS QUESTION: "WHAT IS THE EXPECTED VALUE OF THE BOND ONE YEAR FROM NOW, ASSUMING THAT INTEREST RATES REMAIN AT CURRENT LEVELS?" THIS IS THE SAME AS ASKING, "WHAT IS THE VALUE OF A 9-YEAR, 9 PERCENT ANNUAL COUPON BOND IF ITS YTM (ITS REQUIRED RATE OF RETURN) IS 10.91 PERCENT?" THE ANSWER, USING THE BOND VALUATION FUNCTION OF A CALCULATOR, IS $893.87. WITH THIS DATA, WE CAN NOW CALCULATE THE BOND'S CAPITAL GAINS YIELD AS FOLLOWS:
CAPITAL GAINS YIELD = (Vd1 - Vd0)/Vd0
= ($893.87 - $887)/$887 = 0.0077 = 0.77%,
WHICH AGREES WITH OUR EARLIER CALCULATION (EXCEPT FOR ROUNDING).
WHEN THE BOND IS SELLING FOR $1,134.20 AND PROVIDING A TOTAL RETURN OF k = YTM = 7.08%, WE HAVE THIS SITUATION:
CURRENT YIELD = $90/$1,134.20 = 7.94%
AND
CAPITAL GAINS YIELD = 7.08% - 7.94% = -0.86%.
THE BOND PROVIDES A CURRENT YIELD THAT EXCEEDS THE TOTAL RETURN, BUT A PURCHASER WOULD INCUR A SMALL CAPITAL LOSS EACH YEAR, AND THIS LOSS WOULD EXACTLY OFFSET THE EXCESS CURRENT YIELD AND FORCE THE TOTAL RETURN TO EQUAL THE REQUIRED RATE.
Year Cash flow PV of cash flow 0 -50 (50.00) 1 100 90.91 2 75 61.98 3 50 37.57 NPV 140.46