Bond YTM Question during Interview

I got this question during a finance interview with a top prop shop. I dont think I answered correctly. Can someone help me figure out the correct answer?

What is the 1-year holding period return of a 30 year US Treasury if it is currently selling at par ($100) with a 7% coupon and the Yield to Maturity a year from now is 11%?

(Assume entire 7% coupon is paid at the end of year rather than semianually)

 

Alright, so lets do it this way:

1) Find the Price of the Bond at t=1 We know Price and Yield have an inverse relationship, therefore intuitively the price should be lower at t=1 since yield increased to 11% (since its selling at par, current YTM is 7%).

Solving for price you get 65.40... if you want an estimate you can use Current Yield

CY= C/P 11%=7/P P= 63.6

Remember for discount bonds CY is greater than YTM and for premium bonds CY is less than YTM.

2) Now we can solve for HPY

HPY= (7+65.40-100)/100 = -27.6%

I believe this is how it should be done.

Jack: They’re all former investment bankers who were laid off from that economic crisis that Nancy Pelosi caused. They have zero real world skills, but God they work hard. -30 Rock
 

You could use a financial calculator or excel ( =PV(11%,29,7,100) ):

11% YTM = Yield 29 = Periods Remaining 7 = Coupon 100 = Face Value

=65.40

Like I said, you could use Current Yield to estimate it, since the number of periods is large, it should be relatively close.

Jack: They’re all former investment bankers who were laid off from that economic crisis that Nancy Pelosi caused. They have zero real world skills, but God they work hard. -30 Rock
 

Revsly has a pretty good answer, although in reality the US Treasury is paying you semi-annual interest.

FV= 100 N= 58 (instead of 29 years, you have 58 six month time periods) PMT = 3.5 (you receive half your coupon every 6 months) I = 5.5 (divide the annual YTM in half to account for 6 month time period)

PV = 65.26

HPY= (7+65.26-100)/100 = -27.74%

 

The formula I know is

Annuity of a bond = perpetuity(1-perpetuity(discounted to the last period) + principle(discounted to last period)

Bond PV = 7/0.11(1-1/0.11(1.11)^58)) + 100/1.11^58 = $62.51 or 62.511%

I could be wrong but don't think you need to divide the yield to maturity to 5.5% if you have doubled the discounting period.

 
righton:
From a theoretical standpoint, if you have to choose between the YTM and the current yield, always use YTM since it accounts for the gain or loss that will occur when the par value is repaid. The current yield is just an approximation of the YTM.

^^^^ Right on Righton.

Jack: They’re all former investment bankers who were laid off from that economic crisis that Nancy Pelosi caused. They have zero real world skills, but God they work hard. -30 Rock
 

Project the cash flows with the dates and use XIRR. Technically getting refi'd out would be a yield to call, rather than a YTM, not sure why they would call it YTM.

There have been many great comebacks throughout history. Jesus was dead but then came back as an all-powerful God-Zombie.
 

You are right,, this probably seems easier than relying on a formula.

Kenny_Powers_CFA:
Project the cash flows with the dates and use XIRR. Technically getting refi'd out would be a yield to call, rather than a YTM, not sure why they would call it YTM.
 
Best Response

For a semi-annual coupon, the iterative process such as Newton-Raphson method helps find the YTM

You would have to solve for the interest rate in the bond price formula, it will also require knowing the 1st order derivative of the bond price

f(i) = 1000 + -900 * (1+i)^10 + 25 [(1+i)^10 - 1]/i

f'(i) = 10 * -900 * (1+i)^9 + 25 * (10 i (1 + i)^9 - (1 + i)^10 + 1) / (i^2)

i = 0.1 f(i) = -935.9326 f'(i) = -19311.0161 i1 = 0.1 - -935.9326/-19311.0161 = 0.051533746738395 Error Bound = 0.051533746738395 - 0.1 = 0.048466 > 0.000001

i1 = 0.051533746738395 f(i1) = -170.853 f'(i1) = -12666.9086 i2 = 0.051533746738395 - -170.853/-12666.9086 = 0.03804560833436 Error Bound = 0.03804560833436 - 0.051533746738395 = 0.013488 > 0.000001

i2 = 0.03804560833436 f(i2) = -9.9487 f'(i2) = -11217.2599 i3 = 0.03804560833436 - -9.9487/-11217.2599 = 0.037158700659625 Error Bound = 0.037158700659625 - 0.03804560833436 = 0.000887 > 0.000001

i3 = 0.037158700659625 f(i3) = -0.04 f'(i3) = -11127.2132 i4 = 0.037158700659625 - -0.04/-11127.2132 = 0.037155107831769 Error Bound = 0.037155107831769 - 0.037158700659625 = 4.0E-6 > 0.000001

i4 = 0.037155107831769 f(i4) = -0 f'(i4) = -11126.8497 i5 = 0.037155107831769 - -0/-11126.8497 = 0.037155107773083 Error Bound = 0.037155107773083 - 0.037155107831769 = 0

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