# Everything you need to know about fixed income analysis, spot and forward rates, duration and convexity for the CFA Level 1

# FIXED INCOME ANALYSIS & VALUATION

Below we summarize the most important aspects of Fixed Income Analysis, Spot and Forward Yields, Duration, and Convexity for CFA Level 1 exam.

### INTRODUCTION TO DEBT SECURITIES VALUATION

The value of a bond is the present value of its cash flows. If a single yield to maturity is given, you can simply use the following steps on your financial calculator to find the bond value:

N =number of coupons received

I/Y=yield per period

PMT=Value of each coupon payment

FV=Face value of the bond

PV=Price of the bond

If different yields apply to different maturities, each cash flow must discounted at the appropriate rate. This method is called the arbitrage-free valuation. Using the TD Bank 2.375% October 2016 bond rated AAA. We will use the U.S. treasury spot rates plus a spread of 0.71% which is the BofA Merrill Lynch AAA corporate spread to the treasury spot rate.

Unfortunately, we cannot input all these rates on the financial calculator. Discounting the cash flows at their appropriate discount rate yields a bond value of 102.13 which is very close the quoted price of 103 (as per Yahoo! Finance), the small difference could be the result of accrued interest and slightly different yields.

### YIELD MEASURES, SPOT AND FORWARD RATES

**Current Yield**

This is the simplest yield measure, it shows the investor how much he will earn in coupons from invest- ing in the bond today. Taking the TD Bank corporate bond from the previous section, we would earn $2.37 annually from buying this bond at a price of $103. This results in a current yield of 2.3%

**Coupon rate**

This shows the coupon payment as a percentage of the face value. Unless it is a floating rate bond, the coupon rate does not change during the life of the bond.

**Yield to Maturity**

YTM is the rate that makes the present value of the cash flow equal to the price of the bond. It cannot be computed easily since only a process of trial and error would allow us to compute the YTM. There- fore, the financial calculator must be used to find the YTM (I/Y).

The underlying assumption of the YTM is that the coupons are reinvested at the YTM which is not nec- essarily realistic since the reinvestment rate available to an investor could be different. This is the main limitation of YTM measure.

The bond equivalent yield is the semi-annual YTM multiplied by 2. When the bond is callable before maturity, the yield to call is calculated the same way as the YTM except the call date is used as the maturity. Also, the face value used is the call price which is usually larger than the 100 standard bond face value.

**Other yield measures:**

- Yield to worst: worst yield outcome (YTM or yield to call).
- Yield to refunding: YTM until the refunding date authorized by the bond covenants.
- Yield to put: Same calculation as the YTM but using the put date. Used only when the bond sells at a discount.
- Cash flow yield: Used for MBS and ABS and computed based on monthly cash flow and prepay- ment assumptions.

### BOND SPREADS

**Nominal Spread**

As outlined earlier, the nominal spread is simply the difference between the YTM of two bonds. It does not take the shape of the yield curve into consideration.

For example, the YTM of our TD Bank bond is 2.306% and the YTM of a Treasury note of the same three years maturity is 0.95%. Therefore the nominal spread is 2.306% – 0.95% = 1.356%

**Zero-Volatility Spread (Z-spread)**

When we actually computed the price of out TD Bank bond, we added a spread of 0.71% to all treas- ury spot rates to discount all six cash flows from this bond. This is actually the zero volatility spread which is a spread that is added to all maturities. This spread is superior to the nominal spread since it accounts for the structure of the yield curve.

For the level 1 exam, you are not expected to be able to compute the Z-spread. However, you know how to interpret it and use it to find the price of a bond just like we did in the TD Bank bond example.

**Option-Adjusted Spread (OAS)**

Since the addition of a call feature is detrimental to the bondholders, a higher yield will be required on a callable bond relative to an option-free bond. This extra yield will be captured in the Z-spread but is not related to the credit risk of the bond. Therefore we cannot compare the Z-spread of a call- able bond and regular bond. The OAS takes the option component out of the Z-spread for a callable or putable bond so that it can be compared to the Z-spread of option-free bonds.

When the bond is putable, the OAS adds the put component to the Z-spread. Remember that the put feature is favorable to bondholders and is therefore decreasing the yield required, thus decreasing the spread. To compare it to a normal bond, the spread must be increased to what it would have been without the embedded put.

## SPOT AND FORWARD RATES

We saw how spot rates are the rates earned on investments starting today at t=0. Forward rates are agreements to earn (or borrow at) a rate in the future. An investor who is expecting to receive some funds at a future date and is afraid interest rates will decline may enter the forward market to lock in a rate. Similarly, a borrower who expects to need funds in 1 year might lock in a forward rate in order to protect herself from rising rates.

Determining forward rates given spot rates does not require any formula. You should just find two equivalent investing strategies using the rates given. For example, if we are provided with the spot rates for 3 years and are asked to find the 1 year forwards ending year 2 and year 3.

The resulting time line would look like this:

An investor could implement the two equivalent investing strategies:

1. Invest for 2 year at y2 and receive (1 + y2)2 at the end of the 2 years, or

2. Invest at y1 for 1 year and reinvest the proceeds at f2 for another year. At the end of the 2 years pe- riod, the investor would get (1 + y1)(1 + f2 )

Therefore: (1 + y2)2 = (1 + y1)(1 + f2 ) (1 + 0.06)2 = (1 + 0.04)(1 + f2 )

f2 = 8.04%

Applying the same logic to find f3:

(1 + 0.07)3 = (1 + 0.06)2(1 + f3) f3 = 9.02%

## MEASUREMENT OF INTEREST RATE RISK

In study session 15, we mention the interest risk as the potential decline in bond value following an increase in interest rate. In order to assess the magnitude of that risk, we need to know how much will the bond price drop with a given increase in rates. Duration is the preferred method since it is easy to use. However, the full valuation approach provides a more accurate picture of the interest rate risk of a bond or a portfolio of bonds.

**Full Valuation Approach**

This approach aims at computing the price of the bond given an interest rate scenario. For example, if the following non-parallel shift in the Treasury rates occurs, let us compute the change in price of our TD Bank bond:

Full Valuation Given Non-Parallel Shift in the Yield Curve

The value of our TD Bank bonds would therefore decrease from 102.13 to 101.11 if the illustrated yield curve shift occurs. This represents a drop of approximately 1%. The same approach can be used for a portfolio of bonds. The new rates will then be used to discount cashflows for all the bonds in the portfolio.

**Effective Duration Approach**

Duration is the change in bond price given a change in yield. However, the decrease in price given a yield increase is smaller than the increase in bond price following a drop in yield. This property is the convexity of the bond.

Since duration is different if the yield increases and decreases, effective duration computes it as the average between the two. There are various way to compute the effective duration of a bond but given the limited time given on exam day, the following method suffices:

For our TD Bank bond the effective duration would be:

With effective duration, we can quickly compute the change in bond value given an interest rate shift. For instance, if we expect yields to rise by 1%, the new price of our TD Bank bond would be:

Where D is the effective duration and it is negative because of the inverse relationship between bond prices and yields.

If it is easier for you, you may simply compute the price change in % given by and apply it to the old price. For instance, if the interest rates increase by 200 basis points, the new price will be computed as such.

**Macaulay Duration**

Macaulay duration is computed as the weighted average time period a bondholder receives his cash (including coupon payment and principal). If the bond has large coupon payments, the bondholder gets a larger share of his cashflow sooner and the duration will be smaller. If it is a zero coupon bond, all the cash will occur at maturity and the duration will be equal to maturity. This duration method is inferior to effective duration since it does not take into account the embedded options of the bond.

**Modified Duration**

Modified duration is calculated as Macaulay duration/(1+yield) and is therefore more accurate than the Macaulay duration. It is still not appropriate to use for callable or putable bonds. Effective dura- tion is the preferred method to compute duration and is the only method you should know how to com- pute for the exam.

**Portfolio Duration**

The duration of a portfolio of bonds is simply the weighted average of the durations of individual bonds in the portfolio. It is calculated the same way as portfolio beta if you recall from the equity ses- sion.

As we saw in our discussion on the full valuation approach, a non-parallel shift in the yield curve can make the duration measure inaccurate. For instance, in the current low interest rate environment, there is no scope for short term rates to decrease. Therefore, a decrease in interest rates would only affect longer maturity. Since duration offers a single measure of interest rate risk, it does not account for the fact that short term bonds` price will increase only slightly ad longer term bonds will increase more substantially. This limitation is exacerbated when analyzing the risk of a portfolio of bonds since all bonds will react differently to a change in yield depending on the maturity and other bond fea- tures. As such, portfolio duration only gives an approximate idea of the interest risk of the bond portfo- lio.

## Convexity

While duration assumes a linear relationship between bond prices and yields, convexity measures the curvature of the curve. Therefore, a more accurate estimate of the bond price change given a shift in yields would include convexity. Candidates are not expected to compute convexity but they should know how to use it to find an estimate of the price change of a bond.

As with duration, effective convexity takes into account embedded options such as a call or a put while modified convexity does not consider options.

**Price Value of a Basis Point (PVBP**)

This measures the dollar change in value of a bond given a 1 basis point change in yield (0.01%). Since duration already gives us the percentage bond price change in given a 1% yield, it is easy to compute the PVBP given the duration.

**Yield Volatility vs. Duration**

Duration measures the sensitivity of bond prices given a change in yield. On the other hand, yield volatility measures the frequency and magnitude of the change in yield. If a bond has a high duration and but a low yield volatility, it may be less risky than a bond with a lower duration but more volatile yield.

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