Sep 05

Yield to Maturity (YTM). 2007 FRM

by David Harper, CFA, FRM, CIPM

FRM | Quant |

ytmIntro2

Learning Outcome

  • LO 21.2: Calculate a bond's yield to maturity (YTM) using a calculator with time value functions.

Yield to Maturity (YTM) is a bond's internal rate of return (IRR)

Except for a nuance (see below), the following three metrics are the same:

  • Internal rate of return (IRR)
  • Money-weighted return (MWR)
  • Yield to Maturity (YTM)

These are all measures that reduce a series of cash-flows into a single, internally consistent return. The yield to maturity (YTM) is merely bond-specific language for the IRR (or as Fabozzi writes "YTM is simply a special case of an IRR").

Example

Consider this example:

  • A bond has three years (remaining) to maturity
  • The bond pays a 12% semiannual coupon
  • The bond's face value is $1,000 and its market price is $1,162  (The face value has four synonyms, believe it or not: par value, principal value, redemption value and maturity value. They all refer to the amount the bond issuer agrees to repay on, or before if the bond is called, the maturity date. The reason I prefer to call it 'face value' is simply because its corresponding keystroke on the TI BAII+ is FV.)

The yield to maturity (YMT) is the single internal rate of return that proves the current market price of the bond correct. Specifically, if we discount the future, expected cash flows by a discount rate equal to the YTM, we should get the bond's current price. In this example, we expect six semiannual coupons paying $60 (12% x $1,000 = $120 per year, payable $60 every six months) plus we expect the principal of $1,000 in three years. If that series of cash flows is discounted semi-annually assuming a 6% YTM, the present value of the seven (7) future cash-flows equals about $1,162.

ytmDiagram

Using the TI BAII+ Calculator

The TI BAII+ calculator keystrokes reflect time value of money (TVM) assumptions. That's okay, bond math is all about the time value of money. See below how the time value of money matches the bond lingo; e.g., the face value of the bond is the 'future value' of the cash flows. 

tiCalculator2

 

There are three tips to avoiding trouble when you use the calculator to solve for bond prices/yields/payments/etc:

  • Establish your periodicity; e.g., six months, one year, one month
  • Identify what you are solving for. Given any four above, you can solve for the fifth
  • After you solve, make sure you have converted to the right periodicity to express your answer (typically annual)

The periodicity is the interval. For bonds, the periodicity is often six months because often coupons pay semiannually. However, simple problems sometime use one year. Also, mortgage bonds use one month because the "coupon" (the mortgage payment) is paid monthly. But, if nothing is given, you are best to assume (i) the periodicity of the coupon payments, or lacking that (ii) six months (semiannual) periods

We've established our periodicity (six months). Next we identify what we are solving for; that is I/Y, which is the yield-to-maturity (YTM).

Then we input four known values...

  • N = 6 (3 years x twice per year, since our periodicity is six months)
  • PV = -1,162 (negative because it is a cash outflow: we need to pay it)
  • PMT = $60 (12% x $1,000 face = $120 per year. Therefore, the coupon is $60 every six months)
  • FV = $1,000

...and solve for our unknown value (I/Y):

  • CPT I/Y = 3%

Since we used six months as the periodicity, 3% is the six-month yield to maturity (YTM). However, by convention, we would annualize (double) this to 6%. And we would say the yield-to-maturity is 6%. If you noticed that doubling is inferior to compounding, congratulation! (1.03)(1.03)-1 equals about 6.1% not 6%. Although compounding is technically correct, doubling the six-month yield is the common practice; it is called the bond-equivalent yield.

In Formal Terms

A more formal expression for YTM is given here. Where P(T) is the market price of a bond, c/2 is the semiannual coupon (i.e., c = annual coupon), and F is the face (principal) value, then (y) is the annual (bond-equivalent) yield to maturity:

ytm

The Nuance

I earlier said that YTM = MWR = IRR. This is true except for the following nuance. MWR and IRR often refer to the single historical return that matches historical cash flows. A bond's yield to maturity is forward-looking, so it means "the IRR if the bond is held to maturity" and, here's the nuance, "if the coupons are reinvested at the same YTM."

In our example above, we are implicitly assuming the coupons are reinvested to earn 6% annually. In the intervening three years, rates may shift, and our realized return may vary from our YTM, even if we hold the bond to maturity. Our TYM will be 6% if and only if our coupons are reinvested at 6%.

Comments

  1. BOA 03 Dec 2007

    This shit is not useful at all, so if you show your generosity would younplease clean this site, and put smth useful that I can use for my coursework.

    Thanks for your consideration!

    SIncerely,

    BOA and Sirojiddin

Leave a Comment