## Spread / Margin Measures for Floating-Rate Securities

We can't really compute yield to maturity for floaters because we don't really know the cash flows ahead of time. Instead, we go after approximating the margin for these floaters.

### Spread for Life

Adjusts the quoted spread for any discount or premium in the price of the security.

A = 100 ( 100 - Price of the Floater)

B = A / Years left to the maturity of the floater

C = B + Quoted Margin of the floater

Spread for Life = C * (100 / Price of the Floater)

Part A computes the discount or premium on the floater (as percent) and converts it to Basis Points.

Part B computes the average discount/premium per years left in the term of the floater.

Part C adds the quoted margin (which is an annual thing) to the average annual basis point discount/premium. The effect here is to adjust the quoted margin by the discount or premium.

Finally, the above is multiplied by (100 / Price of Floater) - this states the adjusted quoted margin on the discount/premium basis. In other words, if the price is less than par, 100 / Price is > 1, which magnifies the adjusted margin, and vice versa for the premium.

Ignores coupon rate or time value of money.

### Discount Margin

Assumes all cash flows as if the reference rate will never change and attempts to select a margin such that reference rate + margin discounts those flows to the present value. The purpose is to adjust the margin for the effect of premium or discount - if the floater is at par then the discount margin is the same as the quoted margin. Ignores the possibility of rate changes or rate cap/floors.

## Yield to Maturity (YTM)

Assumes a reinvestment rate equal to the YTM. Why? To match the total future dollars that would be returned by a CD sold at the same price and yielding the same. Future dollars have 3 components: coupon, capital gain/loss, and reinvestment.

How to compare future dollars due to reinvestment (Interest on Interest) - compute future cash flows with and without compounding. Example: $100 bond (at par) 8% coupon, 15 years maturity, paying twice a year.

Total future dollars: 100 * (1.004)^30 = 324.34

Interest + IOI = 324.34 - 100 = 224.34

Now compare to the dollars from just the coupon: $4 * 30 = $120. Therefore IOI = $224.34 - $120 = $104.34

### Reinvestment Risks

Longer maturity means more reinvestment risk since more of return depends on IOI

Higher coupon means more reinvestment risk since more return depends on the interest on those coupons (as opposed to capital gains)

Bonds sold at premium have more reinvestment risk since they depend on interest on interest to overcome the premium. Zero-coupon bonds have no reinvestment risk - if held to maturity.

Converting between Annual-Pay basis (European) and bond-equivalent.

A = ( 1 + Annual Yield ) ^(0.5)

B = A - 1

Bond Equivalent = 2 * B

Part A "de-compounds" the annual yield into what it would earn in just 6 months.

Part B isolates the Yield from the factor.

Then convert to Bond Equiv by doubling the semi-annual yield. Bond Equivalent is always less than annual-pay since it removes compounding.

Now from Bond Equivalent to Annual-Pay

A = Bond Equivalent / 2

B = (1 + A)^2

Yield on Annual Pay Basis = B - 1

Part A splits undoes the bond equivalence into the 6-month yield.

Part B compounds that to the annual yield.

Then we remove the factor and are left with the yield. It's always bigger than the bond-equivalent yield since it accounts for compounding.

### Cash Flow Yield

Similar to YTM for mortgage-backed and asset-backed, where prepayment is relevant and which pay on monthly basis. Conversion to bond-equivalence:

A = (1 + Monthly Cash Flow)^6 -1

Bond Equivalence = 2*A

Part A compounds the monthly payment for 6 months, removes the factor. Then double it for Bond-Equivalence.

--- **Jan 2008** ---

**Sharpe Ratio** - Indicates whether returns of a portfolio are due to good investments or excessive risks. Bigger values of the Sharpe ratio indicate better risk-adjusted returns. Sharpe Ratio = (Expected Portfolio Return - Risk Free Rate) / ( Portfolio Standard Deviation). In other words, the numerator is the return the portfolio earns on top of the risk free rate. It's divided by the risk of the portfolio - its standard deviation. The higher the risk given the same level of excess returns, the lower the Sharpe ratio. Similarly, given the same risk, the portfolio earning the higher return over the RFR will have a higher Sharpe ratio.

**Skew** - Measures the asymmetry of a distribution of a random variable. Positive or negative skew mean that one of the "tails" is longer than it would have been in a normal distribution. If the right tail is longer, the distribution is right-skewed or positively skewed.

**Conditional Probability** - P(A|B) is the probability of A given B. P(A|B) = P(A and B)/P(B).

**Power, Type I and Type II errors** - We have a Null Hypothesis which represents the "default" state of nature, and a Alternative Hypothesis that we're trying to prove. Type I error is a "false positive" - that is we reject the Null when we should not have and thus accept a false alternative hypothesis. Type II error is a "false negative" - we fail to reject the null and thus do not accept the alternative, even though the alternative is true. The power of the test is its ability to reject false nulls. Greater power reduces Type II errors. Increasing power increases Type I errors however - reducing the chance of false negatives increases the chance of false positives.

**Keynesian Expansionary** - Economy is stimulated by a reduction in interest rates and an increase in government spendings. Countercyclical fiscal policies encourage deficit spending during recession periods and tax increases/spending cuts during other times as a check on inflation.

**Interest Rate Parity** - Denies the possibility of interest rate/exchange rate arbitrage such as exchanging into a second currency, lending in that currency, and then converting back. More specifically, the return from the above series of transactions should not be different than the interest which would have been earned on the original currency.

**Crowding Out Effect **- A theoretical side effect of an expansionary fiscal policy. The government borrows in order to stimulate the economy by raising expenditures and/or lowering taxes. The borrowing leads to greater demand for money and thus a higher interest rate. This higher interest rate may discourage ("crowd out") investment by private sector, which can have long-term impact on the supply side by decreasing potential output. Additionally, the higher interest rates lure foreign investors. This in turn appreciates the currency and reduces the attractiveness of the nation's exports.

**Free Cash Flow (FCF) **- The cash that a company can generate after it pays for maintaining or expanding its assets. This money is available to the security holders of the company. Another view is that this answers "how much cash can be taken out of the company while allowing it to be productive". FCF = (NET INCOME) + (Amortization/Depreciation) - Changes in Working Capital - Capital Expenditures . Amortization/Depreciation is added on top of the Net Income because it was taken out in the Net Income calculation but is not cash flows. Changes in working capital are subtracted because they represent investment in current assets. Another way to get FCF is to subtract CAPEX or CFI from CFO. (CFI can be used to estimate CAPEX). This makes sense: we need to subtract our capital expenditures from CFO before we know how much money can be taken out of the company without impacting operations.

**Operating Cash Flow (OCF)** - EBIT + Depreciation - Taxes. EBIT is Revenue - Operating Expenses, which include depreciation - which is why we have to add it back.

**Interest Coverage Ratio **- A measure of how well a company's earnings "cover" its interest expense. The formula is EBIT/INTEREST. In other words, the more company earns (or the lower its interest expense) the easier it can pay its interest due, and thus the higher its Interest Coverage Ratio.

**LIFO Reserve** - Using LIFO reduces the company's reported and/or taxable income by raising its COGS. LIFO reserve is the amount by which one of these income reports has been decreased (cumulatively) since LIFO was adopted.

**Current Ratio / Liquidity Ratio** - Current Assets / Current Liabilities. An indicator of the firm's ability to meet its current obligations. Higher ratio indicates better ability to pay.

**Accumulated Deprecation vs. Depreciation Expense** - Depreciation Expense is the allocation of the cost of a purchase over a period. In other words, the income of a given period is decreased by that periods share of depreciation, rather than the income of the first period taking on all of the hit. Accumulated Depreciation is a contra-asset account that offsets an asset by its depreciation. In other words if an asset was bought for $50 but could only be sold for $10 today, Accumulated Depreciation would reflect $40 for the asset.

**Deferred Tax Asset (liability)** - Something carried on the books which can decrease (or will increase) a future tax.

**Amortizing Bond** - Repays a part of the principal in the coupon payments.

**Serial Bond** - Some of the outstanding issue matures at regular intervals. These are used to finance projects where regular income is expected. The partial maturities of the issue can coincide with the expected incomes.

**Intrinsic Features of a Bond **- Maturity, Serial Maturity Dates, Principal, bearer bond vs. registered.

**Eurobond** - a bond denominated in a currency not native to the country (market) of issue. For example, a Euroyen bond is denominated in yen. Important to note that this doesn't have to do with the currency being different than the currency of the issuer, but only different from the currency of the market in which it was issued.

**Binomial Distribution **- To find the probability of X successful Bernoulli trials if the probability of success in any trial is P and the total number of trials is N, we need to compute two factors. First, we need to know how many different ways we can get a sequence of N trials with X successes. That is, (N chose X) = N! / X! * (N-X!)!. Next, we need to know the probability of any such combination, which is simply (P^X)*(1-P)^(X-N) = the probability of success to the power of number of successes times probability of failure to the power of failures. Multiplying the number of sequences by the probability of such a sequence gives us the probability of X successes in N trials.

**Duration** - Sensitivity to interest rate movements, and is proportional to the percentage change in price given a change in yield. In other words, a bond whose duration is higher would raise more on lower yield and drop less on higher yield than a bond with lower duration. We can figure out the duration by observing the bond's price change in reaction to a down and up change in yield by some percentage. Duration = (Price at Lower Yield - Price at Higher Yield) / (2 * Initial Price * Decimal Change in Yield). The numerator is the difference in the two prices. We divide by 2 to get the average move (ie half of the range). We divide by original price since the higher the price, the less the price range means in percentage terms (i.e. a $2 move means more when price is $25 than when price is $100). Finally we divide by the percentage change in yield - the smaller the change in yield, the more significant the jump in price (i.e. a change in $2 is more significant when the rates change by .0001 than when they change by .01)

**Asset Beta** - A company's (Equity) Beta reflects its business risk (uncertainty of cash-flows) and financial risk (the risk of default on debt). Asset Beta reflects business risk only. When company has no debt (no leverage), Asset Beta = Equity Beta. When leverage is present, Equity Beta increases and is larger than Asset Beta. To find Asset Beta, we need to ignore the risk due to financing. (Asset Beta) = 1 / [ 1 + (Equity Beta) * (1 - Tax Rate) * (D/E) ]