## Discounted Cash Flow Valuation

- estimate the cash flows that the asset will generate: coupons & principle (watch out for common issues)
- Determining the appropriate rate or rates: Discount rate for the bond = Treasury yield + Yield spread
- the risk-free rate: Yield on the on-the-run Treasury security with the same maturity is viewed as the starting point for the minimum interest rate that an investor should require when investing in a bond.
- yield spread: the risk premium that investors will require reflects the additional risks the investor faces by acquiring a security that is not issued by the US government.

- Discounting the expected cash flows: V(0) = (C/2)/(1 + i/2) + (C/2)/(1 + i/2)^2 + ... + (C/2 + P)/(1 + i/2)2^N

## Price vs Yield vs Value

- price/yield relationship for an option-free bond is convex.
- value/time (assuming constant discount rate): pull to par value. More specifically, a bond's value decreases over time if the bond is selling at a premium, increases if discount, unchanged if par value. At the maturity date, the bond's value is equal to its par value ("pull to par value").
- Fixed-rate bond yield measures
- Current Yield = annual dollar coupon interest / price. Fails to recognize any capital gain or loss and reinvestment income.
- Yield to Maturity (YTM): the interest rate that will make the present value of the cash flows from a bond equal to the price plus accrued interest. the rate of return on a bond if an investor buys and holds the bond to its maturity date. Assumes that the bond will be held to maturity. Not suitable for callable bonds.
- Par bond (selling at par): coupon rate = current yield = yield to maturity.
- Premium bond: coupon rate > current yield > yield to maturity.
- Discount bond: coupon rate < current yield < yield to maturity.

- Fixed-rate Callable bond yields: callable bonds typically have multiple call dates, each with its own call price.
- yield to first call: if the callable bond is called on the first call date)
- yield to first par call: if called on the first date it is callable at par
- yield to worst: the lowest yield from amount all possible yield to calls, yield to puts, and yield to maturity. The Downside is it does not recognize that yields used in determining the yield to worst have different exposures to reinvestment risk. Reinvestment risk is affected by the timing and amount of the cash flows from a bond. The cash flows used to calculate the YTM and the yield on every call date and put date differ in their timing and amount.

- Fixed-rate MBS/ABS yield / Cash Flow Yield (CFY)
- assumes that all cash flows (principal payments and interest payments) can be reinvested at the calculated yield and that the assumed prepayment rate (ie. the rate at which prepayments will occur) will be realized over the security's life.
- a monthly measure: MBS/ABS cash flows are monthly.
- annualize the CFY: to make the CFY comparable with the YTM on a semiannual-pay bond: 2 * (1 + monthly yield)^6 - 1
- assumptions: it will be held until the final cash flow is paid. The principal of these securities are prepaid according to the assumed prepayment rate. Cash flows of these securities are reinvested at the cash flow yield. This assumption is particularly important as their cash flows are paid on a monthly basis, and include the scheduled payments and prepayments of principal.

- Floating-rate security yields: measured by spreads over the reference rate. This includes simple margin, adjusted simple margin, adjusted total margin, and discount margin (most popular)
- Zero-coupon bonds /Treasury Bills yield on a discount basis: d = (1 - p) x 360/N, where p is the price of the bill as a percentage of par value, and N is the number of days between settlement date and the maturity date. Downside include it's based on a maturity value investment rather than the actual dollar amount invested. Annualized according to a 360-day year rather than a 365-day year.

## Spreads

- Yield spread is the difference in yields between a target bond and a benchmark bond. The target bond and the benchmark bond typically have the same maturity but different credit quality. Treasury securities are usually used as the benchmark bonds.
- nominal spread is the difference between the yield for a non-Treasury bond and a comparable-maturity Treasury coupon security. two limitations
- For both bonds, it fails to consider the term structure of the spot rates. YTM is a single interest rate only.
- For bonds with embedded options, future interest rate volatility may alter the cash flows of the non-Treasury bond.

- zero-volatility spread (Z-spread / static spread): the spread that the investor would realize over the entire Treasury spot rate curve if the bond is held to maturity, thereby recognize the term structure of interest rates. Unlike the nominal spread, the Z-spread is not a spread off one point on the Treasury yield curve but a spread over the entire spot rate curve.
- The slope of the Treasury yield curve determines the z-spread / nominal spread divergence. The steeper the yield curve (either upward sloping or inverted), the greater the divergence (If the yield curve is flat, all spot rates and yields to maturity are the same, thereby eliminating the divergence.) If the yield curve is not flat,
- Principal repayment: the faster the principal is repaid, the greater the divergence will be. Therefore, for bullet bonds, unless the yield curve is very steep, the nominal spread will not differ significantly from the Z-spread; for securities where principal is repaid over time (ie. MBS) there can be a significant difference, particularly in a steep yield curve environment.
- Coupon rate: the higher the coupon rate, the greater the divergence. In particular, there is no divergence for zero-coupon bonds.

- OAS = Z-spread - embedded option cost: option adjusted because it allows for future interest rate volatility to affect the cash flows. removes the effect of the embedded option on a bond's yield.
- OAS < Z-Spread: for callable bonds and bonds with prepayment options (ie MBS/ABS), option cost > 0 since the options are a detriment to bondholders.
- OAS > Z-Spread: for putable bonds since the options are a detriment to investors.
- Unlike z-spread, a series of curves are used in the OAS calculation to reflect the changes in interest rates caused by the embedded options.

## Spot rate

The observed yields on stripped Treasury securities cannot be used to measure the spot rate since:- The liquidity is less than T-bonds and part of the yield is for liquidity.
- Many sectors that attract specific investors, thus would distort spot rate.
- The yield reflects different tax-treatment of stripped T-securities.

The par yield curve of the on-the-run Treasury securities represents the adjusted on-the-run Treasury yield curve where the coupon issues are at par value and the coupon rate is therefore equal to the yield to maturity. Default-free spot rates can be derived from the Treasury par yield curve by a method called bootstrapping.

## Forward rate

- refers to the interest rate on a loan beginning some time in the future. In contrast, a spot rate is the interest rate on a loan beginning now. For example, the 2-year forward rate one year from now is 4%. This means if you borrow a 2-year loan 1 year from now, you will pay an interest of 4%.
- Using arbitrage arguments, forward rates can be extrapolated from the Treasury yield curve or the Treasury spot rate curve: (1 + f(j, k))^(k-j) = (1+r(k))^k/(1+r(j))^j where j < maturities < k
- spot / forward rates are always annually quoted

## Interest rate risk

is the volatility of a bond's price due to changes in interest rates.- Option-free bonds exhibit positive convexity
- the price change is greater (smaller) when the level of required yield is low (high).
- The longer the term to maturity, the greater the price volatility.
- The lower the coupon rate, the greater the price volatility.

- Callable/prepayable bond exhibits positive convexity at high yield levels and negative convexity at low yield levels (levels relative to coupon rate).
- Negative convexity: the price change is greater(smaller) when the level of required yield is high(low).
- When the required yield > coupon: similar to comparable option-free bond.
- When the required yield < coupon: the option value goes up as it is more likely to get retired at the call price, which sets an upper limit.

- Putable bond value = an option-free bond + the value of the put option.Typically the put price is par value.
- When the required yield > coupon: the option value goes up as it is more likely to get sold to the issuer at the put price.
- When the required yield < coupon: similar to comparable option-free bond (option worth 0)

- Duration measures the approximate % price change for a 1% (100 basis points) change in rate, assuming that the price/yield relationship is a straight line. Also the weighted average time to receive the present value of each of the bond's coupon and principal payments.
- Duration = (V- - V+) / (2 * V0 * yield_change)
- approximate percentage price change = - duration x change in yield x 100 (the negative sign due to the inverse relationship between price change and yield change - when yields increase, bond prices fall)
- assumes a linear relationship between price and rates, whereas the relationship is non-linear. As the yield shock increases, the price change predicted by duration diverges further from the actual price change. This explains why duration is more effective in explaining the effects of small yield shocks rather than large changes in interest rates.
- Modified duration: assume the bond's expected cash flows don't change when the yield changes. ie. the change in the bond's price when the yield is changed is due solely to discounting at the new yield level. Only appropriate for option-free bonds (For bonds with embedded options, a change in yield may significantly affect the expected cash flows.)
- Macaulay Duration: Modified Duration = Macaulay Duration / (1 + yield/k)

- Effective duration: assume that the bond's expected cash flows do change when the yield changes. aka option-adjusted duration. Only appropriate for bonds with embedded options.
- Portfolio Duration = w(1)D(1) + w(2)D(2) + ... + w(k)D(k), Where w(i) is the market value of bond i / market value of the portfolio (each yield must change by 100 basis points for the duration measure to be useful.)

- Convexity = (V(+) + V(-) - 2V(0)) / (2 * V(0) * yield_change^2).
- Convexity adjustment = convexity x yield_change^2 x 100: takes into account the curvature of the price/yield relationship.
- Modified convexity: assume that yield changes have no effect on the bond's expected cash flows. Only appropriate for option-free bonds
- Effective convexity: includes the eff ects of yield changes on the cash flows. Only appropriate for bonds with embedded options. requires an adjustment in the estimated bond values (V-, V+)

- Estimated % price change = Duration estimate + Convexity adjustment
- PVBP/Price Value of a Basis Point = abs(Initial price - price if yield is changed by 1 basis point) : the absolute change (rather than % change) in the price of a bond for a 1 basis point change in yield. measured in dollars. aka. dollar value of an 01.
- Dollar Duration = Duration x Bond Price / 100: he approximate dollar change in a bond's price for a 100 basis point change in yield
- PVBP = the dollar duration of a bond for a 1 basis point change in yield.

*Category: C++ Quant > Tutorials*

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