- CFA Exams
- 2025 Level II
- Topic 6. Fixed Income
- Learning Module 26. The Term Structure and Interest Rate Dynamics
- Subject 1. The Forward Rate Model

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##### Subject 1. The Forward Rate Model PDF Download

A [1+ r(T*+T)]
This is called the

f(10,2) = 11.14%[1+r(T)]
You can think of the spot rate for a given maturity as a sort of average of the forward rates for all maturities up to and including the spot rate's maturity. If the spot rate is increasing, so spot rate n is greater than spot rate n - 1, then forward rate must be above spot curve. If the spot curve is downward sloping, the forward curve will lie below the spot curve.

**spot rate**, r(T), is the interest rate today for a specified period of time T.Consider two zero coupon bonds. Bond A is a 1-year bond and bond B is a 2-year bond. Both have face value of $1,000. Assume annual compounding. The 1-year interest rate, r(1), is 8%. The 2-year interest rate, r(2) is 10%.

The present value for bond A and B:

- PV
_{A}=$1,000/1.08^{1}= $925.93 - PV
_{B}=$1,000/1.10^{2}= $826.45

P(T) = 1/(1+r(T) )

^{T}is called the**discount factor**. In our example, P(1) = 0.92593, and P(2) = 0.82645A

**forward rate**, f(T*, T), is the interest rate beginning some time in the future, T*, for a period of time T.Let's say an investor wants to invest $1,000 for 2 years. He is faced with 2 choices:

- Directly invest in the 2-year bond. The value of his investment after 2 years will be $1,000 x (1 + 0.1)
^{2}= $1,210 - Invest in the 1-year bond, and again invest the proceeds after 1 year in a 1-year bond.

The returns from the 2 choices should be the same. The value of the second choice after 2 years should be 1,000 x (1 + 0.08) x (1 + F(1,1)) = 1,210 so F(1,1) = 12.04%

F(1,1) is the 1-year forward rate starting 1 year from now.

Given spot rates for maturities of T* and T* + T, we can compute the forward rate starting at T* for T:

^{(T*+T)}=[1+r(T*)]

^{T*}[1+f(T*,T)]

^{T}

**forward rate model**.

*Example*Ed is considering a bond purchase but cannot decide if he wants to hold the bond for 10 years or 12 years. The spot rate is 5% for 10 years, and 6% for 12 years. What is the forward rate starting 10 years from now, for 2 years?

(1 + r(10 + 2)]

^{10 + 2}= [1 + r(10)]^{10}[1 + f(10, 2)]^{2}f(10,2) = 11.14%

Similarly, we can calculate spot rates from forward rates. The spot rate is the geometric mean of the forward rates.

^{T}= [1+r(1)] [1+f(1,1)] [1+f(2,1)]...[1+f(T-1,1)]

The

**yield-to-maturity**measures an investor's return from the bond correctly only if these assumptions are true:

- The bond is held to maturity. The issuer does not default: All coupon and principal payments are made in full when due.
- The coupon reinvestment rate is the same as the YTM.
- A flat yield curve.

For example, consider a 10%, 2-year bond selling for $1,036.30 (selling at premium). The cash flows for this bond are (1) 4 payments every six months of $50, and (2) a payment 2 years from now of $1,000.

$1036.30 = $50/(1 + YTM/2)

^{1}+ $50/(1 + YTM/2)^{2}+ $50/(1 + YTM/2)^{3}+ $1050/(1 + YTM/2)^{4}. By trial and error or using a financial calculator, YTM is found to be 8%.YTM can be viewed as a weighted average of the spot rates applying to the bond's cash flows.

YTM measures an investor's return from the bond correctly only if these assumptions are true. However, in reality the YTM of a bond may not be realized because:

- Future interest rates may be less than the YTM. This is known as
**reinvestment risk**. - The bond cannot be held to maturity, and may have to be sold for less than the purchase price because the interest rate required by the market is higher than the YTM. That is, there is
**interest rate risk**.

The yield computed at the time of purchase (i.e., YTM) considers all three sources of return, including, coupon payments, capital gain/loss, and reinvestment income. The amount of actual reinvestment income affects the return the investor can realize from the bond. It depends on the level of reinvestment rate. If the reinvestment rate is less than YTM, the realized return will be less than the expected level of return from the bond.

The

**realized return**is the actual return on the bond during the time an investor holds the bond.###
**User Contributed Comments**
2

User |
Comment |
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youyoumm |
Can anyone show me how to use Texas calculater to compute the sample of YTM step by step? Many thanks!!! |

yekky |
Sure youyoumm. Take the first example for the YTM where it mentions a 10%, 2 year bond, selling at 1036.30. Also, I have gone into some simple detail to explain this in case anyone else is having difficulty. Very simply, always think of these bond style calculations as this: You are always given at least 3 of the inputs, if not 4 of them and you have to find one of them. Its like basic algebra where you have to solve for x. Here the YTM or I/Y on your calculator is the variable we need to find, or our x. Also, make sure you realize that the data in this question is given to you on an annual basis and you need to find it semi annually (like for most Level 1 questions). So take note of all the data available, to input into your Texas BAii plus calculator: N = 4 (annually its 2 years, so thats 2 coupon payments. But we want the semi-annual data so thats twice every year OR 4 coupon payments. Try to think of N as the number of coupon payments rather than time) I/Y = x (we need to SOLVE for this) PV = -1036.30 as given to us which is the premium price we paid for the bond. NOTE: Make sure you put the negative sign in front of the PV value or else your calculator will give you an error. We do this simply because the only time we are GIVING CASH AWAY is when we buy the bond right? We bought it at 1036.30 so its a negative sign to show that we have spent that money to buy the bond.) PMT = 50(PMT = payment we receive from the coupons which is again on a semi-annual basis. So the question tells us the bond is a 10% bond meaning it pays 10% interest to us annually. But we want the semi-annual data so instead of 10% yearly, we are receiving it every 6 months so ist 10/2 = 5%. All you do now is find out what 5% of 1000 is so that you know how much interest payment we receive every 6 months = $50.) FV = is the principal payment we receive at the end of the bonds maturity. So its obviously going to be the $1000 as we always regard the principal or par amount to be $1000. So now that you know how each input is found, you have all your values to input. You can do this simply by starting from the N and working your way all the way to the FV or however you like to do it. Note: Remember that on the Texas Baii plus, you enter the values with the number first. So for example if you N = 4, you press 4 first and then N and thats how you plug in all the other data as well. So type in the data like this: 4 = N (skip I/Y as thats what we need to compute at the end) -1036.30 = PV 50 = PMT 1000 = FV then press the yellow 2nd button and then press CPT (compute) and then press the I/Y button. = it should give you an answer of 4. Remember you just solved everything on a SEMI-ANNUAL BASIS. So if the SEMI value is 4, the WHOLE value is your 4 multipled by 2 = 8% ANNUALLY. I hope you actually made it to this point because such calculations are crucial for the exam. All the best. |

I used your notes and passed ... highly recommended!