Formulas & Methods– use half year y and consistent. Double to find bond-equivalent yield (market YTM/BEY)
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𝑭𝑽 = 𝑃𝑉 ∗ (1 + 𝑟)+ 𝑷𝑽 𝑨𝒏𝒏𝒖𝒊𝒕𝒚 = 𝐶 ∗ 9& − &($%&)! ; ∗ (1 + r)
$
$
$
𝑷𝑽 = ($%&)! 𝑷𝑽 𝒄𝒐𝒏𝒔𝒐𝒍 = &)*
$%* +
𝑷𝑽 𝒈𝒓𝒐𝒘. 𝑨𝒏𝒏𝒖𝒊𝒕𝒚 = 𝐶 ∗ [(&)* − (&)* ∗ B$%& C )] 𝑭𝑽 𝒂𝒏𝒏𝒖𝒊𝒕𝒚 = 𝐶 ∗ 9
($%&)! )($%*)!
(
&)*
& ,+
) 𝑭𝑽 𝑪𝒐𝒎𝒑𝒐𝒖𝒏𝒅𝒊𝒏𝒈 = 𝑃𝑉 ∗ B1 + ,C
𝑬𝒒𝒖𝒊𝒗𝒂𝒍𝒆𝒏𝒕 𝒕𝒂𝒙𝒂𝒃𝒍𝒆 𝒚 =
-./ 1/123- 45167
($) -./ &.-1)
-8-.6 957:
($%&)! )$
&
$
; 𝑭𝑽 𝒈𝒓𝒐𝒘 𝑨𝒏𝒏𝒖𝒊𝒕𝒚 = 𝐶 ∗
𝒗𝒂𝒍𝒖𝒆 𝒐𝒇 𝒕𝒂𝒙 𝒔𝒉𝒊𝒆𝒍𝒅 = 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑒𝑥𝑝𝑒𝑛𝑠𝑒 ∗ 𝑡𝑎𝑥 𝑟𝑎𝑡𝑒
munis in U.S. tax-less 𝑨𝒇𝒕𝒆𝒓 𝒕𝒂𝒙 𝒚 = 𝑝𝑟𝑒𝑡𝑎𝑥 𝑦𝑖𝑒𝑙𝑑 ∗ (1 − 𝑡𝑎𝑥)
.,,;.6
𝑩𝒊𝒅 𝒕𝒐 𝑪𝒐𝒗𝒆𝒓 𝑹𝒂𝒕𝒊𝒐 = 5::;1 :5<1 𝑩𝑬𝒀 = 31&587: 𝒓𝒆𝒍𝒂𝒕𝒊𝒗𝒆 𝒚𝒊𝒆𝒍𝒅 𝒔𝒑𝒓𝒆𝒂𝒅 =
@"
(4 8, 5::;1&)4 8, 91,=>2.&?)
4 8, 91,=>2.&?
𝑷𝑴𝑻 = (#$(#%&'(&)*+)$- Blended payment of principal and interest, where PV loan 𝒓 𝒑𝒆𝒓𝒊𝒐𝒅 = B1 +
&'(&)*+
.
&:-.-17 /
C −1
2
Where m is the annual compounding frequency, f is frequency of payout interest, for different compounding/payouts
&:-.-17 2
𝒓 𝒆𝒇𝒇𝒆𝒄𝒕. 𝒂𝒏𝒏𝒖𝒂𝒍 = B1 + 2 C − 1 Where m is the annual compounding frequency
Between coupons valuation Bonds quoted clean-price, settlement date not included in calcs, dirty includes interest
7.4: 91-A11, :1--6121,- .,7 ,1/- =8;38,
=8;38,
=8;38,
3&5,=53.6
𝒘 𝒑𝒆𝒓𝒊𝒐𝒅𝒔 = 7.4: 5, =8;38, 31&587 (=8228, $BC)
𝑷 = ($%&)0 + ($%&)0%# + ($%&)1$#%0
𝑨𝒄𝒄𝒓𝒖𝒆𝒅 𝒊𝒏𝒕𝒆𝒓𝒆𝒔𝒕 = 𝐹𝑎𝑐𝑒 ∗ 𝐶𝑜𝑢𝑝𝑜𝑛 ∗ (𝑑𝑎𝑦𝑠 𝑠𝑖𝑛𝑐𝑒 𝑙𝑎𝑠𝑡 𝑐𝑜𝑢𝑝𝑜𝑛/365) 𝑪𝒍𝒆𝒂𝒏 𝒑𝒓𝒊𝒄𝒆 = 𝐷𝑖𝑟𝑡𝑦 − 𝑎𝑐𝑐𝑟𝑢𝑒𝑑 𝑖𝑛𝑡.
Nominal Yield Curve 𝑩𝒐𝒏𝒅 𝒀𝑻𝑴 = 𝐵𝑒𝑛𝑐ℎ. 𝐺𝑜𝑣 𝐵𝑜𝑛𝑑 𝑌𝑖𝑒𝑙𝑑 + 𝐶𝑟𝑒𝑑𝑖𝑡 𝑆𝑝𝑟𝑒𝑎𝑑 𝒏𝒐𝒎𝒊𝒏𝒂𝒍 = 𝑟𝑒𝑎𝑙 +
Short (<2yrs) is monetary policy, Mid (3-10yrs), Long (10yrs+) | ST rise if: rate hike | LT rise if: future policy hike,
quantitative tightening (selling bonds) | LT lowers if: quantitative easing (buying bonds) | BPS=0.0001,100bps is 1%
->pure expectations theory (market sets yields based solely on expectations for future rates), Liquidity Preference
Theory (includes previous), Market Segmentation Theory (uses/needs)
Spot Rate Curve made-up of spot rates and maturities , YTM of a pure discount is a spot rate, PV final CF = Price
Bootstrapping: solve y1, then use y1 to solve y2, etc., if P is not = PV CFs, then arbitrage
Par-Yield Curve curve where coupon=YTM, coupon rate that a bond will pay to sell at par today, like on-the-run
Par-Yield to Spot Curve: 100 = each CF, use the discounts you have and solve for the last d, use semi-annual y
Forwards & Curve $𝑋(1 + 𝑧1)(1 + 𝐹1) = $𝑋(1 + 𝑧2), z are spots and F is a forward (rate between 2 future dates)
Upward sloping, forwards are higher than spots, rates expected to rise. Forward rate agreement hedge, future action
FRA 𝒔𝒆𝒕𝒕𝒍𝒆𝒎𝒆𝒏𝒕 = 𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 ∗
!
G
234
1
$%H&∗F
GI
234
(&)D)∗F
5,-1&1:- 3.421,- 8, ->1 !JK
= @" D.=-8& .- ->1 2.&?1- &.-1 , where f=FRA rate, T=loan in days
+⋆)+$
Linear Interpolation 𝑻 ⋆ 𝒚 = 𝑦1 + vB+M)+$C ∗ (𝑦2 − 𝑦1)w + 𝑠𝑝𝑟𝑒𝑎𝑑 | Ts are dates, T1 is the earliest, in days
Sources of Return 𝑪𝑰 = 𝐶 ∗ B
𝑯𝑷𝑹 ,
($%&)- )$
&
C Coupon on the bond + I on I, n=holding 𝒄𝒐𝒖𝒑𝒐𝒏𝒔 𝒗𝒂𝒍𝒖𝒆 = 𝐶 ∗ 𝑛
𝑃0 B1 + 2 C = 𝐶𝐼 + 𝑃𝑡 | effective return/period, P0=initial price, Pt=selling price, m=annual compound freq.
Treasury bill: zeros, maturities of 4/8/13/26/52 weeks Treasury bonds: semi-annual, maturities 20/30 years
Treasury notes: semi-annual, maturities 2/3/5/7/10 Default/counterparty: not contractual obligations, credit rating
Bullet bond: principal at maturity Sinking fund: $ in account to retire principal
Amortizing bond: partial repayment of principal with coupons between markets: intramarket yield spreads
Deferred coupon bond: interest accrues, pay later (deferred-interest, step-up bonds (increase), PIK bonds)
BEY: Bond Equivalent Yield, non-compound annual yield, does not account for reinvestment, underestimate return
Primary market & Primary dealers vs Secondary Market On-the-run: new Off-the-run: old vintage, less liquid
Ex dividend date: cutoff date to get div Settlement date: date payment due Mapple: foreign bond CAD + CAN
Credit Default Swap: insurance against credit default Vintage problem: bonds high coupons long ago still out
Re-opened bonds: bonds that offer same coupon and maturity date as an existing bond, adjusted for accrued
interest, mitigate liquidity risk, refinancing risk + Discriminatory price auction: winning bidders pay their price d
Dutch Auction: winners pay the stop-out price (minimum price that exhausts supply)
Find tax savings: find each after-tax cost of debt, subtract the smallest, that difference is the PMT
Quoting convention: 1/32 for US treasuries, a + adds 1/64 Liquidity: benchmark, lowers yield, bid/ask about 1bps
I: Price & yield = inverse, yield & PV of CFs = inverse, coupon rate/frequency & duration = inverse
D: Coupon rate & price = direct, maturity & duration = direct, modified duration & % price change = direct, yield
and reinvestment income = direct, coupon rate & reinvestment risk = direct, maturity & price volatility = direct,
zero-coupon maturity & duration = direct, credit spread & yield = direct DCM CAN: 5/10yr popular (38/46bn each)