Analyst Dojo

CFA Level I Fixed Income formulas

Bond pricing, yield measures, duration and convexity, spreads, credit losses and securitization. After statements, this is the topic with the most to calculate.

167 formulas from the Fixed Income lessons (11–14% of the exam), 87 to memorize. Each says how deeply to learn it and opens to a quick worked example with the keystrokes for the TI BA II Plus and the HP 12C.

How deeply to learn each part
Know the idea
Understand it in general. It frames other topics, but you will not calculate with it.
Use it
Understand it well enough to solve a question with it. Recognize when it applies.
Memorize
Learn it by heart. The exam will not give it to you.

Module 1 Fixed-Income Instrument Features

The features of a bond

Fixed coupon payment Memorize
C=c×ParmC = \frac{c \times \text{Par}}{m}
Quick example

NOK 300,000 par, 4.4% coupon, paid quarterly (m = 4). 300,000 × 0.044 ÷ 4 = NOK 3,300 a quarter.

On your calculator
  1. 300000×0.044÷4= 3300
  1. 300000ENTER0.044×4÷ 3300

ENTER enter (on the HP, puts the number on the stack)

Floating-rate note coupon Memorize
C=(MRR+s)×ParmC = \frac{(\text{MRR} + s) \times \text{Par}}{m}
Quick example

EUR 80m FRN, MRR 2.10% plus 95 bps, semiannual. (0.0210 + 0.0095) × 80m ÷ 2 = EUR 1,220,000.

On your calculator
  1. 0.021+0.0095×80000000÷2= 1220000
  1. 0.021ENTER0.0095+80000000×2÷ 1220000

ENTER enter (on the HP, puts the number on the stack)

Current yield, yield to maturity and the yield curve

Current yield Memorize
CY=annual couponprice\text{CY} = \frac{\text{annual coupon}}{\text{price}}
Quick example

A 7% annual coupon bond trades at 108.50 per 100 par. 7 ÷ 108.50 = 6.45%.

On your calculator
  1. 7÷108.5= 0.0645
  1. 7ENTER108.5÷ 0.0645

ENTER enter (on the HP, puts the number on the stack)

Yield to maturity (price equation) Use it
P=∑t=1nCFt(1+r)tP = \sum_{t=1}^{n} \frac{CF_t}{(1 + r)^t}
Quick example

5 years, 4% annual coupon, price 96.00 per 100 par. Solve for r: 4.92%.

On your calculator
  1. 2NDCLR TVM with P/Y set to 1
  2. 5N96+|-PV4PMT100FV
  3. CPTI/Y 4.92

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · +|- change sign · PV present value · PMT payment · FV future value · CPT compute · I/Y interest rate per year (per period when P/Y is 1)

  1. fCLEAR FIN
  2. 5n96CHSPV4PMT100FV
  3. i 4.92

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · CHS change sign · PV present value · PMT payment · FV future value · i interest rate per period

Credit spread Use it
s=yissuer−ygov(same maturity)s = y_{issuer} - y_{gov} \quad \text{(same maturity)}
Quick example

Issuer 10-year yield 5.80%, government 10-year yield 3.95%. 5.80 − 3.95 = 1.85 points = 185 bps.

On your calculator
  1. 5.8−3.95=×100= 185
  1. 5.8ENTER3.95−100× 185

ENTER enter (on the HP, puts the number on the stack)

Bond indentures and covenants

Recovery under strict priority Use it
R=min⁡(100%, max⁡(0, Cash−S)U)R = \min\left(100\%,\ \frac{\max(0,\ \text{Cash} - S)}{U}\right)
Quick example

Cash 250m. Secured claims 100m, senior unsecured 200m. (250 − 100) ÷ 200 = 75% for senior unsecured.

On your calculator
  1. 250−100=÷200= 0.75
  1. 250ENTER100−200÷ 0.75

ENTER enter (on the HP, puts the number on the stack)

Module 2 Fixed-Income Cash Flows and Types

Bullet, amortizing and sinking fund bonds

Level payment on an amortizing loan Memorize
A=r×Principal1−(1+r)−NA = \frac{r \times \text{Principal}}{1 - (1 + r)^{-N}}
Quick example

GBP 1,200,000 at 3% a year, three equal annual payments. 0.03 × 1,200,000 ÷ (1 − 1.03⁻³) = GBP 424,236.

On your calculator
  1. 2NDCLR TVM with P/Y set to 1
  2. 3N3I/Y1200000+|-PV
  3. CPTPMT 424,236.44

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · I/Y interest rate per year (per period when P/Y is 1) · +|- change sign · PV present value · CPT compute · PMT payment

  1. fCLEAR FIN
  2. 3n3i1200000CHSPV
  3. PMT 424,236.44

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · CHS change sign · PV present value · PMT payment

Payment when a balloon is left at maturity Use it
A=r×[Principal−B(1+r)−N]1−(1+r)−NA = \frac{r \times \left[\text{Principal} - B(1 + r)^{-N}\right]}{1 - (1 + r)^{-N}}
Quick example

EUR 80 million at 5% a year, seven annual payments, EUR 25 million balloon at the end. EUR 10.755 million a year.

On your calculator
  1. 2NDCLR TVM with P/Y set to 1
  2. 7N5I/Y80+|-PV25FV
  3. CPTPMT 10.7551

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · I/Y interest rate per year (per period when P/Y is 1) · +|- change sign · PV present value · FV future value · CPT compute · PMT payment

  1. fCLEAR FIN
  2. 7n5i80CHSPV25FV
  3. PMT 10.7551 (with four decimals shown)

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · CHS change sign · PV present value · FV future value · PMT payment

Floating, step-up, inflation-linked and zero-coupon bonds

Capital-indexed coupon Memorize
C=creal×Par×I1/I0mC = \frac{c_{real} \times \text{Par} \times I_1 / I_0}{m}
Quick example

GBP 30 million, 1.2% real coupon, semiannual; the index rises from 250.0 to 254.0. Adjusted principal 30 × 1.016 = 30.48 million; 1.2% × 30.48m ÷ 2 = GBP 182,880.

On your calculator
  1. 254÷250×30000000×0.012÷2= 182,880
  1. 254ENTER250÷30000000×0.012×2÷ 182,880

ENTER enter (on the HP, puts the number on the stack)

Payment-in-kind balance Use it
Pt=P0(1+c)tP_t = P_0 (1 + c)^{t}
Quick example

USD 100 million at a 10% PIK coupon owes 100 × 1.10 = USD 110 million after one year, and 133.1 after three.

On your calculator
  1. 1.1y^x3×100= 133.1

y^x power

  1. 1.1ENTER3y^x100× 133.1

ENTER enter (on the HP, puts the number on the stack) · y^x power

Callable, putable and convertible bonds

Conversion ratio Memorize
CR=ParConversion price\text{CR} = \frac{\text{Par}}{\text{Conversion price}}
Quick example

Par EUR 1,000, conversion price EUR 80. 1,000 ÷ 80 = 12.5 shares per bond.

On your calculator
  1. 1000÷80= 12.5
  1. 1000ENTER80÷ 12.5

ENTER enter (on the HP, puts the number on the stack)

Conversion value Memorize
CV=CR×Share price\text{CV} = \text{CR} \times \text{Share price}
Quick example

Par EUR 1,000, conversion price EUR 50, shares at EUR 62. Ratio 20; 20 × 62 = EUR 1,240.

On your calculator
  1. 1000÷50×62= 1,240
  1. 1000ENTER50÷62× 1,240

ENTER enter (on the HP, puts the number on the stack)

Conversion premium Use it
Prem=Conversion price−Share priceShare price\text{Prem} = \frac{\text{Conversion price} - \text{Share price}}{\text{Share price}}
Quick example

Conversion price EUR 25, shares at EUR 20 at issue. (25 − 20) ÷ 20 = 25%.

On your calculator
  1. 25−20=÷20= 0.25
  1. 25ENTER20−20÷ 0.25

ENTER enter (on the HP, puts the number on the stack)

Domestic, foreign and Eurobonds, and how bonds are taxed

Original issue discount Memorize
OID=Par−Issue price\text{OID} = \text{Par} - \text{Issue price}
Quick example

INR 80 million par, zero-coupon, issued at 78.50 per 100. 80m × (1 − 0.785) = INR 17.2 million.

On your calculator
  1. 1−0.785=×80= 17.2, in millions
  1. 1ENTER0.785−80× 17.2, in millions

ENTER enter (on the HP, puts the number on the stack)

Module 3 Fixed-Income Issuance and Trading

Bond indexes and how they differ from equity indexes

Market value weight of a bond Use it
wi=Pari×Pi/100∑jParj×Pj/100w_i = \frac{\text{Par}_i \times P_i / 100}{\sum_j \text{Par}_j \times P_j / 100}
Quick example

Two bonds: 250m par at 108 and 750m par at 92. Market values 270 and 690, total 960. First bond: 270 ÷ 960 = 28.1% (not 25% by par).

On your calculator
  1. 250×1.08=STO1
  2. 750×0.92=STO2
  3. RCL1÷(RCL1+RCL2)= 0.28125

STO store · RCL recall a stored number

  1. 250ENTER1.08×STO1
  2. 750ENTER0.92×RCL1+1/xRCL1× 0.28125

ENTER enter (on the HP, puts the number on the stack) · STO store · RCL recall a stored number · 1/x reciprocal

Primary and secondary bond markets vs equity markets

Bid–offer spread in basis points Know the idea
S=Poffer−Pbid(Poffer+Pbid)/2×10,000S = \frac{P_{offer} - P_{bid}}{(P_{offer} + P_{bid})/2} \times 10{,}000
Quick example

Bid 96.80, offer 97.05. Spread 0.25; mid 96.925. 0.25 ÷ 96.925 × 10,000 = 25.8 bps (25 bps of par).

On your calculator
  1. 97.05+96.8=÷2=STO1 96.925
  2. 97.05−96.8=÷RCL1=×10000= 25.79

STO store · RCL recall a stored number

  1. 97.05ENTER96.8+2÷STO1 96.925
  2. 97.05ENTER96.8−RCL1÷10000× 25.79

ENTER enter (on the HP, puts the number on the stack) · STO store · RCL recall a stored number

Module 4 Fixed-Income Markets for Corporate Issuers

Short-term funding for companies: credit lines, secured loans and commercial paper

Annual cost of a committed line Use it
Cost=D×(MRR+s)+(L−D)×f\text{Cost} = D \times (\text{MRR} + s) + (L - D) \times f
Quick example

EUR 60m line, EUR 25m drawn all year at 5.10%, fee 0.20% on the unused part. 25m × 5.10% = 1,275,000; (60m − 25m) × 0.20% = 70,000. Total EUR 1,345,000.

On your calculator
  1. 25×0.051=STO1 interest 1.275 (EUR m)
  2. 60−25=×0.002+RCL1= 1.345

STO store · RCL recall a stored number

  1. 25ENTER0.051× interest 1.275 (EUR m)
  2. 60ENTER25−0.002×+ 1.345

ENTER enter (on the HP, puts the number on the stack)

All-in cost on the money used Use it
rall-in=CostDr_{\text{all-in}} = \frac{\text{Cost}}{D}
Quick example

Total cost EUR 1,345,000 on EUR 25m drawn: 1.345 ÷ 25 = 0.0538, so 5.38%, above the 5.10% interest rate because of the fee.

On your calculator
  1. 1.345÷25= 0.0538
  1. 1.345ENTER25÷ 0.0538

ENTER enter (on the HP, puts the number on the stack)

How banks fund themselves short term

Net interest margin Use it
NIM=rassets−rliabilities\text{NIM} = r_{\text{assets}} - r_{\text{liabilities}}
Quick example

Assets yield 7.15% and funding costs 3.60%. 7.15% − 3.60% = 3.55%.

On your calculator
  1. 7.15−3.6= 3.55
  1. 7.15ENTER3.6− 3.55

ENTER enter (on the HP, puts the number on the stack)

Repurchase agreements (repos)

Repurchase price Memorize
Repurchase price=P0×(1+r×days360)\text{Repurchase price} = P_0 \times \left(1 + r \times \frac{\text{days}}{360}\right)
Quick example

Borrow USD 15,000,000 for 45 days at a 2.40% repo rate. 15,000,000 × (1 + 0.024 × 45 ÷ 360) = USD 15,045,000.

On your calculator
  1. 0.024×45÷360+1×15000000= 15,045,000
  1. 0.024ENTER45×360÷1+15000000× 15,045,000

ENTER enter (on the HP, puts the number on the stack)

Initial margin Memorize
Initial margin=S0P0\text{Initial margin} = \frac{S_0}{P_0}
Quick example

Security worth 55 million, loan of 50 million: 55 ÷ 50 = 110%.

On your calculator
  1. 55÷50= 1.1
  1. 55ENTER50÷ 1.1

ENTER enter (on the HP, puts the number on the stack)

Haircut Memorize
Haircut=S0−P0S0\text{Haircut} = \frac{S_0 - P_0}{S_0}
Quick example

Same trade: (55 − 50) ÷ 55 = 9.09%. Equivalent: 1 − 1 ÷ initial margin.

On your calculator
  1. 55−50=÷55= 0.0909
  1. 55ENTER50−55÷ 0.0909

ENTER enter (on the HP, puts the number on the stack)

Variation margin Use it
VMt=(Initial margin×Pt)−St\text{VM}_t = (\text{Initial margin} \times P_t) - S_t
Quick example

Margin 103%, loan with accrued interest 50,010,000, security now 50,600,000. 1.03 × 50,010,000 − 50,600,000 = +910,300: the borrower posts more collateral.

On your calculator
  1. 1.03×50010000−50600000= 910,300
  1. 1.03ENTER50010000×50600000− 910,300

ENTER enter (on the HP, puts the number on the stack)

Profit on a short sale through a reverse repo Use it
Profit=P0×(1+r×days360)−buyback price\text{Profit} = P_0 \times \left(1 + r \times \frac{\text{days}}{360}\right) - \text{buyback price}
Quick example

Lend 10,000,000 for 90 days at 2.00%, sell the bond for 10,000,000, buy it back at 9,700,000. Repurchase price 10,050,000, so profit 350,000 before costs.

On your calculator
  1. 0.02×90÷360+1×10000000−9700000= 350,000
  1. 0.02ENTER90×360÷1+10000000×9700000− 350,000

ENTER enter (on the HP, puts the number on the stack)

Long-term funding: investment-grade vs high-yield issuers

Yield to maturity as benchmark plus spread Know the idea
YTM=ygov+scredit\text{YTM} = y_{\text{gov}} + s_{\text{credit}}
Quick example

A BBB bond for which the government yield is 3.20% and the credit spread 1.10% has YTM 4.30%. The spread is 1.10 ÷ 4.30 = 25.6% of the yield. For a B bond with a 4.80% spread, it is 4.80 ÷ 8.00 = 60%.

On your calculator
  1. 3.2+1.1=1/x×1.1= 0.2558

1/x reciprocal

  1. 3.2ENTER1.1+1/x1.1× 0.2558

ENTER enter (on the HP, puts the number on the stack) · 1/x reciprocal

Module 5 Fixed-Income Markets for Government Issuers

Government bond auctions and trading vs corporate bonds

Amount left for competitive bids Use it
C=Offering−Non-competitive bidsC = \text{Offering} - \text{Non-competitive bids}
Quick example

Offering 8.0 billion, non-competitive bids 1.4 billion. 8.0 − 1.4 = 6.6 billion must be filled by competitive bids.

On your calculator
  1. 8−1.4= 6.6
  1. 8ENTER1.4− 6.6

ENTER enter (on the HP, puts the number on the stack)

Fill at the cut-off yield Use it
F=C−BAF = \dfrac{C - B}{A}
Quick example

C = 6.6 billion; bids below the cut-off total B = 5.0; bids at the cut-off A = 3.2. (6.6 − 5.0) ÷ 3.2 = 50% of each cut-off bid is filled.

On your calculator
  1. 6.6−5=÷3.2= 0.5
  1. 6.6ENTER5−3.2÷ 0.5

ENTER enter (on the HP, puts the number on the stack)

Module 6 Bond Valuation: Prices and Yields

Pricing a bond from its yield to maturity

Price of a bond on a coupon date Memorize
PV=PMT(1+r)1+PMT(1+r)2+⋯+PMT+FV(1+r)NPV = \dfrac{PMT}{(1 + r)^{1}} + \dfrac{PMT}{(1 + r)^{2}} + \cdots + \dfrac{PMT + FV}{(1 + r)^{N}}
Quick example

4 years left, 6% coupon paid annually, par 100, yield 5%. N = 4, I/Y = 5, PMT = 6, FV = 100 gives 103.55, a premium.

On your calculator
  1. 2NDCLR TVM with P/Y set to 1
  2. 4N5I/Y6PMT100FV
  3. CPTPV −103.5460

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · I/Y interest rate per year (per period when P/Y is 1) · PMT payment · FV future value · CPT compute · PV present value

  1. fCLEAR FIN
  2. 4n5i6PMT100FV
  3. PV −103.5460

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PMT payment · FV future value · PV present value

Inputs per period Memorize
N=m×years,r=ym,PMT=coupon rate×FVmN = m \times \text{years}, \quad r = \dfrac{y}{m}, \quad PMT = \dfrac{\text{coupon rate} \times FV}{m}
Quick example

5 years, 3% coupon paid twice a year, yield 4% stated annually. N = 10, r = 2%, PMT = 1.5. Price = 95.51.

On your calculator
  1. 2NDCLR TVM with P/Y set to 1
  2. 10N2I/Y1.5PMT100FV
  3. CPTPV −95.5087

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · I/Y interest rate per year (per period when P/Y is 1) · PMT payment · FV future value · CPT compute · PV present value

  1. fCLEAR FIN
  2. 10n2i1.5PMT100FV
  3. PV −95.5087

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PMT payment · FV future value · PV present value

Yield to maturity Memorize
PVmarket=∑k=1NPMT(1+r)k+FV(1+r)N  ⇒  solve for r,y=m×rPV_{\text{market}} = \sum_{k=1}^{N} \dfrac{PMT}{(1 + r)^{k}} + \dfrac{FV}{(1 + r)^{N}} \;\Rightarrow\; \text{solve for } r, \quad y = m \times r
Quick example

5 years, 6% coupon paid annually, price 104.30. Enter N = 5, PV = −104.30, PMT = 6, FV = 100 and compute I/Y: 5.01%.

On your calculator
  1. 2NDCLR TVM with P/Y set to 1
  2. 5N104.3+|-PV6PMT100FV
  3. CPTI/Y 5.0066

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · +|- change sign · PV present value · PMT payment · FV future value · CPT compute · I/Y interest rate per year (per period when P/Y is 1)

  1. fCLEAR FIN
  2. 5n104.3CHSPV6PMT100FV
  3. i 5.0066

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · CHS change sign · PV present value · PMT payment · FV future value · i interest rate per period

Yield of a zero-coupon bond Use it
r=(FVPV)1/N−1r = \left(\dfrac{FV}{PV}\right)^{1/N} - 1
Quick example

A 3-year zero priced at 103.00 per 100 of par. (100 ÷ 103)^(1/3) − 1 = −0.98%, a negative yield.

On your calculator
  1. 2NDCLR TVM with P/Y set to 1
  2. 3N103+|-PV100FV
  3. CPTI/Y −0.9805

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · +|- change sign · PV present value · FV future value · CPT compute · I/Y interest rate per year (per period when P/Y is 1)

  1. fCLEAR FIN
  2. 3n103CHSPV100FV
  3. i −0.9805

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · CHS change sign · PV present value · FV future value · i interest rate per period

Accrued interest, flat price and full price

Full price between coupon dates Memorize
PVFull=PV×(1+r)t/TPV^{Full} = PV \times (1 + r)^{t/T}
Quick example

A bond is worth 98.40 on its last coupon date. Yield 2% per period, settlement 110 days into a 182-day period. 98.40 × 1.02^(110/182) = 99.58.

On your calculator
  1. 1.02y^x(110÷182)×98.4= 99.585

y^x power

  1. 1.02ENTER110ENTER182÷y^x98.4× 99.585

ENTER enter (on the HP, puts the number on the stack) · y^x power

Accrued interest Memorize
AI=tT×PMTAI = \dfrac{t}{T} \times PMT
Quick example

Coupon of 1.85 per half-year, settlement 112 days into a 180-day period (30/360). (112 ÷ 180) × 1.85 = 1.151.

On your calculator
  1. 112÷180×1.85= 1.1511
  1. 112ENTER180÷1.85× 1.1511

ENTER enter (on the HP, puts the number on the stack)

Flat price Memorize
PVFlat=PVFull−AIPV^{Flat} = PV^{Full} - AI
Quick example

Full price 99.585, accrued interest 1.330. Flat price = 98.255.

On your calculator
  1. 99.585−1.33= 98.255
  1. 99.585ENTER1.33− 98.255

ENTER enter (on the HP, puts the number on the stack)

30/360 day count Use it
D=360(Y2−Y1)+30(M2−M1)+(D2−D1)D = 360 (Y_2 - Y_1) + 30 (M_2 - M_1) + (D_2 - D_1)
Quick example

12 April to 3 July in the same year: 30 × 3 + (3 − 12) = 81 days, out of 180 for a half-year.

On your calculator
  1. 7−4=×30+3−12= 81
  1. 7ENTER4−30×3+12− 81

ENTER enter (on the HP, puts the number on the stack)

How price, coupon, maturity and yield are related

Percentage price change Use it
ΔP%=P1P0−1\Delta P\% = \dfrac{P_1}{P_0} - 1
Quick example

A 9-year 5% bond at par falls to 95.85 when its yield rises to 5.6%. 95.85 ÷ 100 − 1 = −4.15%.

On your calculator
  1. 95.847÷100−1= −0.0415
  1. 95.847ENTER100÷1− −0.0415

ENTER enter (on the HP, puts the number on the stack)

Price at a constant yield, with time Use it
PVN−k=∑j=1N−kPMT(1+r)j+FV(1+r)N−kPV_{N-k} = \sum_{j=1}^{N-k} \dfrac{PMT}{(1 + r)^{j}} + \dfrac{FV}{(1 + r)^{N-k}}
Quick example

A 10-year 6% bond priced at a constant 5% yield is 107.72 today and 106.46 two years later (8 years left). Same formula, fewer periods.

On your calculator
  1. 2NDCLR TVM with P/Y set to 1
  2. 8N5I/Y6PMT100FV
  3. CPTPV −106.4632

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · I/Y interest rate per year (per period when P/Y is 1) · PMT payment · FV future value · CPT compute · PV present value

  1. fCLEAR FIN
  2. 8n5i6PMT100FV
  3. PV −106.4632

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PMT payment · FV future value · PV present value

Matrix pricing

Linear interpolation of a yield Use it
y=y1+t−t1t2−t1×(y2−y1)y = y_1 + \dfrac{t - t_1}{t_2 - t_1} \times (y_2 - y_1)
Quick example

Comparables yield 5.10% at 5 years and 5.55% at 8 years. At 7 years: 5.10 + (2 ÷ 3) × 0.45 = 5.40%.

On your calculator
  1. 2÷3×0.45+5.1= 5.4
  1. 2ENTER3÷0.45×5.1+ 5.4

ENTER enter (on the HP, puts the number on the stack)

Required yield spread Use it
s=ybond−ybenchmarks = y_{\text{bond}} - y_{\text{benchmark}}
Quick example

An issuer’s 6-year bond yields 4.78%. The interpolated 6-year government yield is 4.35%. The spread is 43 bps.

On your calculator
  1. 4.78−4.35=×100= 43 bps
  1. 4.78ENTER4.35−100× 43 bps

ENTER enter (on the HP, puts the number on the stack)

Module 7 Yield and Yield Spread Measures for Fixed-Rate Bonds

Periodicity: converting yields between compounding frequencies

Stated annual yield from a periodic yield Memorize
APRm=m×rperiodAPR_m = m \times r_{\text{period}}
Quick example

A bond earns 2.10% every six months. Periodicity 2 gives a semiannual bond basis yield of 2 × 2.10% = 4.20%.

Converting between periodicities Memorize
(1+APRmm)m=(1+APRnn)n\left(1 + \dfrac{APR_m}{m}\right)^{m} = \left(1 + \dfrac{APR_n}{n}\right)^{n}
Quick example

6.00% on a quarterly basis restated semiannually: 2 × [(1 + 0.06 ÷ 4)² − 1] = 2 × 0.030225 = 6.045%.

On your calculator
  1. 1.015y^x2=−1=×2= 0.06045

y^x power

  1. 1.015ENTER2y^x1−2× 0.06045

ENTER enter (on the HP, puts the number on the stack) · y^x power

Effective annual rate Memorize
EAR=(1+APRmm)m−1EAR = \left(1 + \dfrac{APR_m}{m}\right)^{m} - 1
Quick example

4.00% on a semiannual basis: (1 + 0.04 ÷ 2)² − 1 = 4.04%.

On your calculator
  1. 1.02y^x2=−1= 0.0404

y^x power

  1. 1.02ENTER2y^x1− 0.0404

ENTER enter (on the HP, puts the number on the stack) · y^x power

Yield of a zero-coupon bond Use it
EAR=(FVPV)1/T−1,APR2=2[(FVPV)1/(2T)−1]EAR = \left(\dfrac{FV}{PV}\right)^{1/T} - 1, \qquad APR_2 = 2\left[\left(\dfrac{FV}{PV}\right)^{1/(2T)} - 1\right]
Quick example

A 10-year zero at 61.00 per 100. EAR = (100 ÷ 61)^(1/10) − 1 = 5.07%; semiannual basis = 2 × [(100 ÷ 61)^(1/20) − 1] = 5.01%.

On your calculator
  1. 2NDCLR TVM with P/Y set to 1
  2. 20N61+|-PV100FV
  3. CPTI/Y 2.5023 per half-year
  4. ×2= 5.0046

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · +|- change sign · PV present value · FV future value · CPT compute · I/Y interest rate per year (per period when P/Y is 1)

  1. fCLEAR FIN
  2. 20n61CHSPV100FV
  3. i 2.5023 per half-year
  4. 2× 5.0046

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · CHS change sign · PV present value · FV future value · i interest rate per period

Current yield, simple yield and yield conventions

Current yield Use it
ycurrent=annual couponflat pricey_{\text{current}} = \dfrac{\text{annual coupon}}{\text{flat price}}
Quick example

A 3% annual coupon bond at a flat price of 94.00. 3 ÷ 94 = 3.19%. It ignores the 6.00 gain to par.

On your calculator
  1. 3÷94= 0.031915
  1. 3ENTER94÷ 0.031915

ENTER enter (on the HP, puts the number on the stack)

Simple yield Use it
ysimple=C+100−PTPy_{\text{simple}} = \dfrac{C + \dfrac{100 - P}{T}}{P}
Quick example

A 3-year bond, 2.5% annual coupon, flat price 101.50. (2.5 + (100 − 101.5) ÷ 3) ÷ 101.5 = (2.5 − 0.5) ÷ 101.5 = 1.97%.

On your calculator
  1. 100−101.5=÷3+2.5=÷101.5= 0.019704
  1. 100ENTER101.5−3÷2.5+101.5÷ 0.019704

ENTER enter (on the HP, puts the number on the stack)

Government equivalent yield Memorize
yact/act≈y30/360×365360y_{\text{act/act}} \approx y_{30/360} \times \dfrac{365}{360}
Quick example

A corporate yield of 6.00% on 30/360 is 6.00% × 365 ÷ 360 = 6.083% on actual/actual.

On your calculator
  1. 0.06×365÷360= 0.060833
  1. 0.06ENTER365×360÷ 0.060833

ENTER enter (on the HP, puts the number on the stack)

Yield to call and yield to worst

Yield to call Memorize
P=∑t=1NPMT(1+r)t+K(1+r)N,YTC=m×rP = \sum_{t=1}^{N} \dfrac{PMT}{(1 + r)^{t}} + \dfrac{K}{(1 + r)^{N}}, \qquad YTC = m \times r
Quick example

A 5% semiannual bond at 102.00, callable in 3 years at 101. N = 6, present value (PV) = −102, PMT = 2.5, FV = 101 gives 2.2967% per half-year, so YTC = 4.59%.

On your calculator
  1. 2NDCLR TVM with P/Y set to 1
  2. 6N102+|-PV2.5PMT101FV
  3. CPTI/Y 2.2967 per half-year
  4. ×2= 4.5934

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · +|- change sign · PV present value · PMT payment · FV future value · CPT compute · I/Y interest rate per year (per period when P/Y is 1)

  1. fCLEAR FIN
  2. 6n102CHSPV2.5PMT101FV
  3. i 2.2967 per half-year
  4. 2× 4.5934

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · CHS change sign · PV present value · PMT payment · FV future value · i interest rate per period

Yield to worst Memorize
YTW=min⁡(YTC1, YTC2, …, YTM)YTW = \min\left(YTC_1,\, YTC_2,\, \ldots,\, YTM\right)
Quick example

Yields to first call, second call and maturity of 4.20%, 4.55% and 4.80%. YTW = 4.20%.

Option-adjusted price Use it
POA=Pcallable+Vcall,Vcall=Poption-free−PcallableP_{OA} = P_{\text{callable}} + V_{\text{call}}, \qquad V_{\text{call}} = P_{\text{option-free}} - P_{\text{callable}}
Quick example

A callable bond trades at 101.20 and its call is worth 1.35. Option-adjusted price = 101.20 + 1.35 = 102.55.

On your calculator
  1. 101.2+1.35= 102.55
  1. 101.2ENTER1.35+ 102.55

ENTER enter (on the HP, puts the number on the stack)

G-spread, I-spread, Z-spread and option-adjusted spread

Yield spread Memorize
yield spread=YTM−benchmark yield\text{yield spread} = YTM - \text{benchmark yield}
Quick example

A bond yields 4.85% and the benchmark yields 3.95%. Spread = 90 bps.

On your calculator
  1. 0.0485−0.0395=×10000= 90
  1. 0.0485ENTER0.0395−10000× 90

ENTER enter (on the HP, puts the number on the stack)

G-spread with interpolation Memorize
ygov=y1+T−T1T2−T1 (y2−y1),G=YTM−ygovy_{gov} = y_1 + \dfrac{T - T_1}{T_2 - T_1}\,(y_2 - y_1), \qquad G = YTM - y_{gov}
Quick example

A 6-year bond yields 5.18%; governments yield 3.80% at 5 years and 4.45% at 10. y_gov = 3.80% + (1 ÷ 5) × 0.65% = 3.93%. G-spread = 125 bps.

On your calculator
  1. (6−5)÷(10−5)×(4.45−3.8)+3.8= 3.93, the government yield
  2. +|-+5.18=×100= 125

+|- change sign

  1. 6ENTER5−10ENTER5−÷4.45ENTER3.8−×3.8+ 3.93, the government yield
  2. CHS5.18+100× 125

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

I-spread Memorize
I=YTM−swap rate (same currency, same tenor)I = YTM - \text{swap rate (same currency, same tenor)}
Quick example

A 5-year euro corporate bond yields 4.08% and the 5-year euro swap rate is 3.51%. I-spread = 57 bps.

On your calculator
  1. 0.0408−0.0351=×10000= 57
  1. 0.0408ENTER0.0351−10000× 57

ENTER enter (on the HP, puts the number on the stack)

Z-spread Use it
PV=PMT(1+z1+Z)1+PMT(1+z2+Z)2+⋯+PMT+FV(1+zN+Z)NPV = \dfrac{PMT}{(1 + z_1 + Z)^{1}} + \dfrac{PMT}{(1 + z_2 + Z)^{2}} + \cdots + \dfrac{PMT + FV}{(1 + z_N + Z)^{N}}
Quick example

A 3-year 3.5% annual bond, spot rates 2.8%, 3.1%, 3.5%, Z = 0.80%. 3.5 ÷ 1.036 + 3.5 ÷ 1.039² + 103.5 ÷ 1.043³ = 97.84.

On your calculator
  1. 3.5÷1.036=STO1
  2. 3.5÷(1.039y^x2)=+RCL1=STO1
  3. 103.5÷(1.043y^x3)=+RCL1= 97.8400

STO store · y^x power · RCL recall a stored number

  1. 3.5ENTER1.036÷
  2. 3.5ENTER1.039ENTER2y^x÷+
  3. 103.5ENTER1.043ENTER3y^x÷+ 97.8400

ENTER enter (on the HP, puts the number on the stack) · y^x power

Option-adjusted spread Memorize
OAS=Z-spread−option value (bps a year)OAS = Z\text{-spread} - \text{option value (bps a year)}
Quick example

Z-spread 210 bps, embedded call worth 55 bps a year. OAS = 210 − 55 = 155 bps.

On your calculator
  1. 210−55= 155
  1. 210ENTER55− 155

ENTER enter (on the HP, puts the number on the stack)

Module 8 Yield and Yield Spread Measures for Floating-Rate Instruments

Floating-rate notes: quoted margin and discount margin

FRN coupon per period Memorize
PMT=(MRR+QM)×FVmPMT = \dfrac{(MRR + QM) \times FV}{m}
Quick example

Semiannual FRN at MRR + 70 bps with MRR at 2.50%: (2.50% + 0.70%) × 100 ÷ 2 = 1.60 per 100 of par.

On your calculator
  1. 2.5+0.7=÷2= 1.6
  1. 2.5ENTER0.7+2÷ 1.6

ENTER enter (on the HP, puts the number on the stack)

Simplified FRN price on a reset date Use it
PV=∑t=1N(MRR+QM) FV/m(1+MRR+DMm)t+FV(1+MRR+DMm)NPV = \sum_{t=1}^{N} \dfrac{(MRR + QM)\,FV/m}{\left(1 + \frac{MRR + DM}{m}\right)^{t}} + \dfrac{FV}{\left(1 + \frac{MRR + DM}{m}\right)^{N}}
Quick example

Two-year quarterly FRN, MRR 2.00%, QM 100 bps, DM 130 bps. PMT = 0.75; r = 0.825% per period; N = 8. Price = 99.42.

On your calculator
  1. 2NDCLR TVM with P/Y set to 1
  2. 8N
  3. 0.825I/Y
  4. 0.75PMT
  5. 100FV
  6. CPTPV −99.4217

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · I/Y interest rate per year (per period when P/Y is 1) · PMT payment · FV future value · CPT compute · PV present value

  1. fCLEAR FIN
  2. 8n
  3. 0.825i
  4. 0.75PMT
  5. 100FV
  6. PV −99.4217

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PMT payment · FV future value · PV present value

Discount margin from a price Use it
DM=m×r−MRR,r solved from PV,PMT,FV,NDM = m \times r - MRR, \quad r \text{ solved from } PV, PMT, FV, N
Quick example

Three-year semiannual FRN, MRR 3.00%, QM 70 bps, price 99.00. PMT = 1.85; N = 6; r = 2.0287% per period. DM = 2 × 2.0287% − 3.00% = 1.057% (106 bps).

On your calculator
  1. 2NDCLR TVM with P/Y set to 1
  2. 6N
  3. 99+|-PV
  4. 1.85PMT
  5. 100FV
  6. CPTI/Y 2.0287
  7. ×2−3= 1.057

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · +|- change sign · PV present value · PMT payment · FV future value · CPT compute · I/Y interest rate per year (per period when P/Y is 1)

  1. fCLEAR FIN
  2. 6n
  3. 99CHSPV
  4. 1.85PMT
  5. 100FV
  6. i 2.0287
  7. 2×3− 1.057

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · CHS change sign · PV present value · PMT payment · FV future value · i interest rate per period

Money market rates: discount basis and add-on basis

Price on a discount basis Memorize
PV=FV×(1−DaysYear×DR)PV = FV \times \left(1 - \dfrac{Days}{Year} \times DR\right)
Quick example

A 182-day bill with face value 2,000,000 at a 3.90% discount rate (360-day year): 2,000,000 × (1 − 182/360 × 0.039) = 1,960,567.

On your calculator
  1. 182÷360×0.039=+|-+1=×2000000= 1960566.67

+|- change sign

  1. 182ENTER360÷0.039×CHS1+2000000× 1960566.67

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Discount rate from a price Memorize
DR=YearDays×FV−PVFVDR = \dfrac{Year}{Days} \times \dfrac{FV - PV}{FV}
Quick example

A 126-day bill with face value 8,000,000 costs 7,900,000 (360-day year): 360/126 × 100,000 ÷ 8,000,000 = 3.571%.

On your calculator
  1. 8000000−7900000=÷8000000×360÷126= 0.035714
  1. 8000000ENTER7900000−8000000÷360×126÷ 0.035714

ENTER enter (on the HP, puts the number on the stack)

Amount at maturity on an add-on basis Memorize
FV=PV×(1+DaysYear×AOR)FV = PV \times \left(1 + \dfrac{Days}{Year} \times AOR\right)
Quick example

30,000,000 placed for 270 days at a 4.20% add-on rate (365-day year): 30,000,000 × (1 + 270/365 × 0.042) = 30,932,055.

On your calculator
  1. 270÷365×0.042+1=×30000000= 30932054.8
  1. 270ENTER365÷0.042×1+30000000× 30932054.8

ENTER enter (on the HP, puts the number on the stack)

Price on an add-on basis Memorize
PV=FV1+DaysYear×AORPV = \dfrac{FV}{1 + \frac{Days}{Year} \times AOR}
Quick example

A deposit will pay 5,000,000 in 60 days. At a 3.00% add-on rate (365-day year) it is worth 5,000,000 ÷ (1 + 60/365 × 0.03) = 4,975,463.

On your calculator
  1. 60÷365×0.03+1=1/x×5000000= 4975463.47

1/x reciprocal

  1. 60ENTER365÷0.03×1+1/x5000000× 4975463.47

ENTER enter (on the HP, puts the number on the stack) · 1/x reciprocal

Add-on rate from a price Memorize
AOR=YearDays×FV−PVPVAOR = \dfrac{Year}{Days} \times \dfrac{FV - PV}{PV}
Quick example

A 270-day deposit of 30,000,000 pays back 30,900,000 (365-day year): 365/270 × 900,000 ÷ 30,000,000 = 4.056%.

On your calculator
  1. 30900000−30000000=÷30000000×365÷270= 0.040556
  1. 30900000ENTER30000000−30000000÷365×270÷ 0.040556

ENTER enter (on the HP, puts the number on the stack)

Comparing money market instruments: bond equivalent yield

Price per 100 from a discount quote Memorize
PV=100×(1−DaysYear×DR)PV = 100 \times \left(1 - \dfrac{Days}{Year} \times DR\right)
Quick example

A 91-day bill at a 3.70% discount rate (360-day year): 100 × (1 − 91/360 × 0.037) = 99.06.

On your calculator
  1. 91÷360×0.037=+|-+1=×100= 99.0647

+|- change sign

  1. 91ENTER360÷0.037×CHS1+100× 99.0647

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Bond equivalent yield Memorize
BEY=365Days×FV−PVPVBEY = \dfrac{365}{Days} \times \dfrac{FV - PV}{PV}
Quick example

A 150-day bill priced at 98.00 per 100: 365/150 × 2 ÷ 98 = 4.966%.

On your calculator
  1. 100−98=÷98×365÷150= 0.049660
  1. 100ENTER98−98÷365×150÷ 0.049660

ENTER enter (on the HP, puts the number on the stack)

Rescaling an add-on rate to a 365-day year Use it
BEY=AOR360×365360BEY = AOR_{360} \times \dfrac{365}{360}
Quick example

A CD at a 3.90% add-on rate on a 360-day year: 3.90% × 365/360 = 3.954%.

On your calculator
  1. 0.039×365÷360= 0.039542
  1. 0.039ENTER365×360÷ 0.039542

ENTER enter (on the HP, puts the number on the stack)

Periodicity of a money market rate Memorize
m=365Daysm = \dfrac{365}{Days}
Quick example

A 75-day instrument has periodicity 365 ÷ 75 = 4.87.

On your calculator
  1. 365÷75= 4.8667
  1. 365ENTER75÷ 4.8667

ENTER enter (on the HP, puts the number on the stack)

Restating on a semiannual bond basis Use it
APR2=2[(1+BEYm)m/2−1],m=365DaysAPR_2 = 2\left[\left(1 + \dfrac{BEY}{m}\right)^{m/2} - 1\right], \quad m = \dfrac{365}{Days}
Quick example

A 60-day BEY of 5.27%: m = 365/60 = 6.0833, so 2 × [(1 + 0.0527 ÷ 6.0833)^(6.0833/2) − 1] = 5.317%.

On your calculator
  1. 365÷60=STO1 m
  2. 0.0527÷RCL1+1=y^x(RCL1÷2)=−1=×2= 0.053167

STO store · RCL recall a stored number · y^x power

  1. 365ENTER60÷STO1 m
  2. 0.0527RCL1÷1+RCL1ENTER2÷y^x1−2× 0.053167

ENTER enter (on the HP, puts the number on the stack) · STO store · RCL recall a stored number · y^x power

Module 9 Term Structure: Spot, Par and Forward Curves

Spot rates and pricing a bond with the spot curve

Price of a bond from spot rates Memorize
PV=PMT(1+z1)1+PMT(1+z2)2+⋯+PMT+FV(1+zN)NPV = \dfrac{PMT}{(1 + z_1)^{1}} + \dfrac{PMT}{(1 + z_2)^{2}} + \cdots + \dfrac{PMT + FV}{(1 + z_N)^{N}}
Quick example

Spot rates 2.0%, 2.5%, 3.0% for 1, 2, 3 years. A 3-year bond with a 4% annual coupon: 4 ÷ 1.02 + 4 ÷ 1.025² + 104 ÷ 1.03³ = 3.922 + 3.807 + 95.175 = 102.904.

On your calculator
  1. 4÷1.02=STO1
  2. 4÷(1.025y^x2)=STO2
  3. 104÷(1.03y^x3)+RCL1+RCL2= 102.9036

STO store · y^x power · RCL recall a stored number

  1. 4ENTER1.02÷
  2. 4ENTER1.025ENTER2y^x÷+
  3. 104ENTER1.03ENTER3y^x÷+ 102.9036

ENTER enter (on the HP, puts the number on the stack) · y^x power

Spot rate from a zero-coupon price Use it
zN=(FVPV)1/N−1z_N = \left(\dfrac{FV}{PV}\right)^{1/N} - 1
Quick example

A 2-year default-free zero trades at 93.00 per 100 of par. (100 ÷ 93)^(1/2) − 1 = 3.695%.

On your calculator
  1. 2NDCLR TVM with P/Y set to 1
  2. 2N93+|-PV100FV
  3. CPTI/Y 3.6952

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · +|- change sign · PV present value · FV future value · CPT compute · I/Y interest rate per year (per period when P/Y is 1)

  1. fCLEAR FIN
  2. 2n93CHSPV100FV
  3. i 3.6952

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · CHS change sign · PV present value · FV future value · i interest rate per period

Par rates from spot rates

Discount factor Use it
dt=1(1+zt)td_t = \dfrac{1}{(1 + z_t)^{t}}
Quick example

The 3-year spot rate is 3.6%. d₃ = 1 ÷ 1.036³ = 0.8993: one unit paid in 3 years is worth 0.8993 today.

On your calculator
  1. 1.036y^x3=1/x 0.8993

y^x power · 1/x reciprocal

  1. 1.036ENTER3y^x1/x 0.8993

ENTER enter (on the HP, puts the number on the stack) · y^x power · 1/x reciprocal

Par rate from spot rates Memorize
cN=1−dNd1+d2+⋯+dNc_N = \dfrac{1 - d_N}{d_1 + d_2 + \cdots + d_N}
Quick example

Spot rates 2.0% (1 year) and 2.6% (2 years). d₁ = 0.98039, d₂ = 0.94996. Par rate = (1 − 0.94996) ÷ (0.98039 + 0.94996) = 2.592%.

On your calculator
  1. 1.021/xSTO1
  2. 1.026y^x2=1/xSTO2
  3. 1−RCL2=÷(RCL1+RCL2)=×100= 2.5923

1/x reciprocal · STO store · y^x power · RCL recall a stored number

  1. 1.021/xSTO1
  2. 1.026ENTER2y^x1/xSTO2
  3. 1RCL2−RCL1RCL2+÷100× 2.5923

1/x reciprocal · STO store · ENTER enter (on the HP, puts the number on the stack) · y^x power · RCL recall a stored number

Implied forward rates: from spot to forward and back

Implied forward rate Memorize
(1+zA)A (1+fA,B)B=(1+zA+B)A+B  ⇒  fA,B=[(1+zA+B)A+B(1+zA)A]1/B−1(1 + z_A)^{A}\,(1 + f_{A,B})^{B} = (1 + z_{A+B})^{A+B} \;\Rightarrow\; f_{A,B} = \left[\dfrac{(1 + z_{A+B})^{A+B}}{(1 + z_A)^{A}}\right]^{1/B} - 1
Quick example

The 1-year spot rate is 2.0% and the 3-year spot rate is 3.0%. The 1y2y forward is (1.03³ ÷ 1.02)^(1/2) − 1 = 3.504%.

On your calculator
  1. 1.03y^x3=÷1.02=y^x0.5=−1= 0.0350

y^x power

  1. 1.03ENTER3y^x1.02÷0.5y^x1− 0.0350

ENTER enter (on the HP, puts the number on the stack) · y^x power

Spot rate from one-year forward rates Memorize
(1+zn)n=(1+f0,1)(1+f1,1)⋯(1+fn−1,1)(1 + z_n)^{n} = (1 + f_{0,1})(1 + f_{1,1})\cdots(1 + f_{n-1,1})
Quick example

One-year forwards today, in 1 year and in 2 years: 1.8%, 2.4%, 3.0%. The 3-year spot rate is (1.018 × 1.024 × 1.030)^(1/3) − 1 = 2.399%.

On your calculator
  1. 1.018×1.024×1.03=y^x(1÷3)=−1= 0.0240

y^x power

  1. 1.018ENTER1.024×1.03×31/xy^x1− 0.0240

ENTER enter (on the HP, puts the number on the stack) · 1/x reciprocal · y^x power

Bond price from forward rates Use it
PV=CF11+f0,1+CF2(1+f0,1)(1+f1,1)+CF3(1+f0,1)(1+f1,1)(1+f2,1)+⋯PV = \dfrac{CF_1}{1 + f_{0,1}} + \dfrac{CF_2}{(1 + f_{0,1})(1 + f_{1,1})} + \dfrac{CF_3}{(1 + f_{0,1})(1 + f_{1,1})(1 + f_{2,1})} + \cdots
Quick example

A 3-year, 5% annual-coupon bond with one-year forwards of 1.8%, 2.4% and 3.0%: 5 ÷ 1.018 + 5 ÷ (1.018 × 1.024) + 105 ÷ (1.018 × 1.024 × 1.030) = 107.500.

On your calculator
  1. 105÷1.03+5=÷1.024+5=÷1.018= 107.5003
  1. 105ENTER1.03÷5+1.024÷5+1.018÷ 107.5003

ENTER enter (on the HP, puts the number on the stack)

How the spot, par and forward curves relate

One-year forward from adjacent spot rates Use it
fn−1,1=(1+zn)n(1+zn−1)n−1−1f_{n-1,1} = \dfrac{(1 + z_n)^{n}}{(1 + z_{n-1})^{n-1}} - 1
Quick example

The 1-year spot is 6.0% and the 2-year spot is 5.2%. The 1y1y forward is 1.052² ÷ 1.06 − 1 = 4.41%, below both spot rates: an inverted curve.

On your calculator
  1. 1.052y^x2=÷1.06−1= 0.0441

y^x power

  1. 1.052ENTER2y^x1.06÷1− 0.0441

ENTER enter (on the HP, puts the number on the stack) · y^x power

Module 10 Interest Rate Risk and Return

The sources of return from a fixed-rate bond

Future value of reinvested coupons Use it
FVcoupons=C×(1+r)H−1rFV_{coupons} = C \times \dfrac{(1 + r)^{H} - 1}{r}
Quick example

4 annual coupons of 5, each reinvested at 6%. 5 × (1.06⁴ − 1) ÷ 0.06 = 21.873: 20.000 of coupons plus 1.873 of interest on interest.

On your calculator
  1. 2NDCLR TVM with P/Y set to 1
  2. 4N6I/Y5PMT
  3. CPTFV −21.873, negative by the TVM sign convention

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · I/Y interest rate per year (per period when P/Y is 1) · PMT payment · CPT compute · FV future value

  1. fCLEAR FIN
  2. 4n6i5PMT
  3. FV −21.873

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PMT payment · FV future value

Horizon yield Memorize
r=(FVcoupons+PsaleP0)1/H−1r = \left(\dfrac{FV_{coupons} + P_{sale}}{P_0}\right)^{1/H} - 1
Quick example

Buy a 6-year, 4% annual bond at par. Yields drop to 3% at once and you sell after 2 years. Coupons 8.12, sale price 103.717. (111.837 ÷ 100)^(1/2) − 1 = 5.75%.

On your calculator
  1. 2NDCLR TVM with P/Y set to 1
  2. 2N100+|-PV8.12+103.717=FV
  3. CPTI/Y 5.753

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · +|- change sign · PV present value · FV future value · CPT compute · I/Y interest rate per year (per period when P/Y is 1)

  1. fCLEAR FIN
  2. 2n100CHSPV8.12ENTER103.717+FV
  3. i 5.753

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · CHS change sign · PV present value · ENTER enter (on the HP, puts the number on the stack) · FV future value · i interest rate per period

Carrying value Memorize
CV=PV of the remaining cash flows at the original YTMCV = \text{PV of the remaining cash flows at the original YTM}
Quick example

A 6-year, 3% annual bond bought at a 4% yield. After 2 years, 4 years remain. At 4%: N = 4, I/Y = 4, PMT = 3, FV = 100 gives 96.370.

On your calculator
  1. 2NDCLR TVM with P/Y set to 1
  2. 4N4I/Y3PMT100FV
  3. CPTPV −96.370

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · I/Y interest rate per year (per period when P/Y is 1) · PMT payment · FV future value · CPT compute · PV present value

  1. fCLEAR FIN
  2. 4n4i3PMT100FV
  3. PV −96.370

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PMT payment · FV future value · PV present value

Capital gain or loss Use it
G=Psale−CVG = P_{sale} - CV
Quick example

The bond above is sold for 94.80 while its carrying value is 96.370. 94.80 − 96.37 = −1.57, a capital loss.

On your calculator
  1. 94.8−96.37= −1.57
  1. 94.8ENTER96.37− −1.57

ENTER enter (on the HP, puts the number on the stack)

Investment horizon, price risk vs reinvestment risk, and duration gap

Duration gap Memorize
Gap=MacDur−H\text{Gap} = \text{MacDur} - H
Quick example

A bond has a Macaulay duration of 4.6 years and the investor’s horizon is 6 years. 4.6 − 6 = −1.4: a negative gap, so falling rates are the risk.

On your calculator
  1. 4.6−6= −1.4
  1. 4.6ENTER6− −1.4

ENTER enter (on the HP, puts the number on the stack)

Macaulay duration: definition and calculation

Macaulay duration Memorize
MacDur=∑t=1Nt×PV(CFt)PVFull\text{MacDur} = \sum_{t=1}^{N} t \times \dfrac{PV(CF_t)}{PV^{Full}}
Quick example

A 4-year, 3% annual bond at a 4% yield has price 96.370. Weighting each year by its share of that price gives 3.825 years.

On your calculator
  1. (1.04y^x4−1)×0.03+0.04=STO1 the denominator, 0.045096
  2. 4×(0.03−0.04)+1.04=÷RCL1= 22.1750
  3. +|-+(1.04÷0.04)= 3.8250

y^x power · STO store · RCL recall a stored number · +|- change sign

  1. 1.04ENTER4y^x1−0.03×0.04+STO1 the denominator, 0.045096
  2. 0.03ENTER0.04−4×1.04+RCL1÷ 22.1750
  3. 1.04ENTER0.04÷−CHS 3.8250

ENTER enter (on the HP, puts the number on the stack) · y^x power · STO store · RCL recall a stored number · CHS change sign

Annualized Macaulay duration Memorize
MacDur (years)=MacDur (periods)m\text{MacDur (years)} = \dfrac{\text{MacDur (periods)}}{m}
Quick example

A semiannual bond has a weighted average time of 7.64 half-year periods. 7.64 ÷ 2 = 3.82 years.

On your calculator
  1. 7.64÷2= 3.82
  1. 7.64ENTER2÷ 3.82

ENTER enter (on the HP, puts the number on the stack)

Closed-form Macaulay duration Use it
MacDur=1+rr−1+r+N(c−r)c[(1+r)N−1]+r−tT\text{MacDur} = \dfrac{1 + r}{r} - \dfrac{1 + r + N(c - r)}{c\left[(1 + r)^{N} - 1\right] + r} - \dfrac{t}{T}
Quick example

6-year, 4% annual bond at a 5% yield, on a coupon date (t/T = 0). 21.0000 − 0.99 ÷ 0.063604 = 5.435 years.

On your calculator
  1. (1.05y^x6−1)×0.04+0.05=STO1 the denominator, 0.063604
  2. 6×(0.04−0.05)+1.05=÷RCL1= 15.5651
  3. +|-+(1.05÷0.05)= 5.4349

y^x power · STO store · RCL recall a stored number · +|- change sign

  1. 1.05ENTER6y^x1−0.04×0.05+STO1 the denominator, 0.063604
  2. 0.04ENTER0.05−6×1.05+RCL1÷ 15.5651
  3. 1.05ENTER0.05÷−CHS 5.4349

ENTER enter (on the HP, puts the number on the stack) · y^x power · STO store · RCL recall a stored number · CHS change sign

Macaulay duration between coupon dates Use it
MacDurt=MacDur0−tT\text{MacDur}_{t} = \text{MacDur}_{0} - \dfrac{t}{T}
Quick example

A bond’s Macaulay duration is 6.10 years on its last coupon date (annual coupons). 0.4 years later, with the yield unchanged: 6.10 − 0.4 = 5.70 years.

On your calculator
  1. 6.1−0.4= 5.7
  1. 6.1ENTER0.4− 5.7

ENTER enter (on the HP, puts the number on the stack)

Module 11 Modified Duration, Money Duration and PVBP

Modified duration

Modified duration Memorize
ModDur=MacDur1+r\text{ModDur} = \dfrac{\text{MacDur}}{1 + r}
Quick example

A semiannual bond yielding 5% a year has an annualized Macaulay duration of 7.45. Divide by the periodic factor, 1 + 0.05/2 = 1.025: 7.45 ÷ 1.025 = 7.268.

On your calculator
  1. 7.45÷1.025= 7.2683
  1. 7.45ENTER1.025÷ 7.2683

ENTER enter (on the HP, puts the number on the stack)

Estimated price change Memorize
%ΔPVFull≈−AnnModDur×Δy\%\Delta PV^{Full} \approx -\text{AnnModDur} \times \Delta y
Quick example

A bond with an annualized modified duration of 4.80 sees its yield rise 30 bp. Estimated change: −4.80 × 0.0030 = −1.44%.

On your calculator
  1. 4.8×0.003=+|- −0.0144

+|- change sign

  1. 4.8ENTER0.003×CHS −0.0144

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Approximate modified duration Memorize
ApproxModDur=PV−−PV+2×Δy×PV0\text{ApproxModDur} = \dfrac{PV_- - PV_+}{2 \times \Delta y \times PV_0}
Quick example

A bond is priced at 103.00. At a yield 5 bp lower it is 103.28 and at a yield 5 bp higher it is 102.77. (103.28 − 102.77) ÷ (2 × 0.0005 × 103.00) = 4.95.

On your calculator
  1. (103.28−102.77)÷(2×0.0005×103)= 4.9515
  1. 103.28ENTER102.77−2ENTER0.0005×103×÷ 4.9515

ENTER enter (on the HP, puts the number on the stack)

Approximate Macaulay duration Use it
ApproxMacDur=ApproxModDur×(1+r)\text{ApproxMacDur} = \text{ApproxModDur} \times (1 + r)
Quick example

An annual-pay bond yielding 4% has an approximate modified duration of 4.9515. Approximate Macaulay duration: 4.9515 × 1.04 = 5.15 years.

On your calculator
  1. 4.9515×1.04= 5.1496
  1. 4.9515ENTER1.04× 5.1496

ENTER enter (on the HP, puts the number on the stack)

Money duration and PVBP

Money duration Memorize
MoneyDur=AnnModDur×PVFull\text{MoneyDur} = \text{AnnModDur} \times PV^{Full}
Quick example

A holding of USD 10,000,000 par is priced at 98.00 (market value 9,800,000) with an annualized modified duration of 6.10. Money duration = 6.10 × 9,800,000 = USD 59,780,000.

On your calculator
  1. 10000000×0.98×6.1= 59,780,000
  1. 10000000ENTER0.98×6.1× 59,780,000

ENTER enter (on the HP, puts the number on the stack)

Estimated change in value Memorize
ΔPVFull≈−MoneyDur×Δy\Delta PV^{Full} \approx -\text{MoneyDur} \times \Delta y
Quick example

The same position, with a money duration of 59,780,000, and a yield rise of 15 bp: −59,780,000 × 0.0015 = −USD 89,670.

On your calculator
  1. 59780000×0.0015=+|- −89,670

+|- change sign

  1. 59780000ENTER0.0015×CHS −89,670

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Price value of a basis point Memorize
PVBP=PV−−PV+2PVBP = \dfrac{PV_- - PV_+}{2}
Quick example

A bond is priced at 100.0562 at a yield 1 bp lower and 99.9438 at a yield 1 bp higher. (100.0562 − 99.9438) ÷ 2 = 0.0562 per 100 of par.

On your calculator
  1. 100.0562−99.9438=÷2= 0.0562
  1. 100.0562ENTER99.9438−2÷ 0.0562

ENTER enter (on the HP, puts the number on the stack)

Basis point value Use it
BPV=MoneyDur×0.0001BPV = \text{MoneyDur} \times 0.0001
Quick example

A position has a money duration of 59,780,000. Its basis point value is 59,780,000 × 0.0001 = USD 5,978 per basis point.

On your calculator
  1. 59780000×0.0001= 5,978
  1. 59780000ENTER0.0001× 5,978

ENTER enter (on the HP, puts the number on the stack)

What makes duration higher or lower

Zero-coupon bond duration Memorize
MacDur=T,ModDur=T1+y\text{MacDur} = T, \qquad \text{ModDur} = \dfrac{T}{1 + y}
Quick example

A 10-year zero-coupon bond yields 6% (annual). Macaulay duration is 10.00 years. Modified duration = 10 ÷ 1.06 = 9.434.

On your calculator
  1. 10÷1.06= 9.4340
  1. 10ENTER1.06÷ 9.4340

ENTER enter (on the HP, puts the number on the stack)

Perpetual bond duration Memorize
MacDur=1+rr,ModDur=1r\text{MacDur} = \dfrac{1 + r}{r}, \qquad \text{ModDur} = \dfrac{1}{r}
Quick example

A non-callable perpetual bond is priced to yield 3%. MacDur = 1.03 ÷ 0.03 = 34.33 years; ModDur = 1 ÷ 0.03 = 33.33.

On your calculator
  1. 1.03÷0.03= 34.3333
  1. 1.03ENTER0.03÷ 34.3333

ENTER enter (on the HP, puts the number on the stack)

Floating-rate note duration Memorize
MacDur=T−tT periods\text{MacDur} = \dfrac{T - t}{T} \text{ periods}
Quick example

A quarterly-reset floater is 25 days into a 91-day period. (91 − 25) ÷ 91 = 0.7253 quarters, so 0.7253 ÷ 4 = 0.181 years.

On your calculator
  1. (91−25)÷91÷4= 0.1813
  1. 91ENTER25−91÷4÷ 0.1813

ENTER enter (on the HP, puts the number on the stack)

Module 12 Convexity and Portfolio Duration

Convexity and the convexity adjustment

Approximate modified duration (recap) Use it
ApproxModDur=PV−−PV+2×Δy×PV0\text{ApproxModDur} = \dfrac{PV_- - PV_+}{2 \times \Delta y \times PV_0}
Quick example

A 5-year, 4% annual coupon bond priced at 100.00 would be 101.1208 at a yield 25 bp lower and 98.8948 at a yield 25 bp higher. (101.1208 − 98.8948) ÷ (2 × 0.0025 × 100) = 4.452.

On your calculator
  1. 101.1208−98.8948=÷0.005÷100= 4.452
  1. 101.1208ENTER98.8948−0.005÷100÷ 4.452

ENTER enter (on the HP, puts the number on the stack)

Approximate convexity Memorize
ApproxCon=PV−+PV+−2×PV0(Δy)2×PV0\text{ApproxCon} = \dfrac{PV_- + PV_+ - 2 \times PV_0}{(\Delta y)^2 \times PV_0}
Quick example

Same bond: PV₀ = 100.00, PV₋ = 101.1208, PV₊ = 98.8948, Δy = 0.0025. (101.1208 + 98.8948 − 200) ÷ (0.0025² × 100) = 0.0156 ÷ 0.000625 = 24.96.

On your calculator
  1. 101.1208+98.8948−200=÷0.0025÷0.0025÷100= 24.96
  1. 101.1208ENTER98.8948+200−0.0025÷0.0025÷100÷ 24.96

ENTER enter (on the HP, puts the number on the stack)

Convexity adjustment Memorize
Convexity adjustment=12×AnnConvexity×(Δy)2\text{Convexity adjustment} = \tfrac{1}{2} \times \text{AnnConvexity} \times (\Delta y)^2
Quick example

Convexity 24.0 and a 150 bp yield move (either direction): ½ × 24.0 × 0.015² = +0.27% added to the duration estimate.

On your calculator
  1. 0.5×24×0.015×0.015= 0.0027
  1. 0.5ENTER24×0.015×0.015× 0.0027

ENTER enter (on the HP, puts the number on the stack)

Estimating price change with duration and convexity

Price change with duration and convexity Memorize
%ΔPVFull≈(−AnnModDur×Δy)+[12×AnnConvexity×(Δy)2]\%\Delta PV^{Full} \approx (-\text{AnnModDur} \times \Delta y) + \left[\tfrac{1}{2} \times \text{AnnConvexity} \times (\Delta y)^2\right]
Quick example

Modified duration 3.80, convexity 18.5, yield falls 75 bp. Duration: +3.80 × 0.0075 = 2.850%. Convexity: ½ × 18.5 × 0.0075² = 0.052%. Total +2.902%.

On your calculator
  1. 3.8×0.0075=STO1
  2. 0.5×18.5×0.0075×0.0075+RCL1= 0.02902

STO store · RCL recall a stored number

  1. 3.8ENTER0.0075×
  2. 0.5ENTER18.5×0.0075×0.0075×+ 0.02902

ENTER enter (on the HP, puts the number on the stack)

Money convexity Memorize
MoneyCon=AnnConvexity×PVFull\text{MoneyCon} = \text{AnnConvexity} \times PV^{Full}
Quick example

AUD 10,000,000 par at a full price of 98.50 (market value 9,850,000) with convexity 40.0. MoneyCon = 40.0 × 9,850,000 = AUD 394,000,000.

On your calculator
  1. 10000000×0.985×40= 394,000,000
  1. 10000000ENTER0.985×40× 394,000,000

ENTER enter (on the HP, puts the number on the stack)

Change in value with money duration and money convexity Memorize
ΔPVFull≈(−MoneyDur×Δy)+[12×MoneyCon×(Δy)2]\Delta PV^{Full} \approx (-\text{MoneyDur} \times \Delta y) + \left[\tfrac{1}{2} \times \text{MoneyCon} \times (\Delta y)^2\right]
Quick example

MoneyDur 41,000,000, MoneyCon 394,000,000, yield +50 bp. −41,000,000 × 0.005 = −205,000; ½ × 394,000,000 × 0.005² = +4,925. Total −200,075.

On your calculator
  1. 41000000×0.005=+|-STO1
  2. 0.5×394000000×0.005×0.005+RCL1= −200,075

+|- change sign · STO store · RCL recall a stored number

  1. 41000000ENTER0.005×CHS
  2. 0.5ENTER394000000×0.005×0.005×+ −200,075

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Portfolio duration and convexity

Market-value weight Use it
wi=Full pricei×Pari∑Full pricej×Parjw_i = \dfrac{\text{Full price}_i \times \text{Par}_i}{\sum \text{Full price}_j \times \text{Par}_j}
Quick example

USD 7,000,000 par of a bond at 90.00 and USD 3,000,000 par of a bond at 70.00. Market values 6,300,000 and 2,100,000, so the first bond has weight 6,300,000 ÷ 8,400,000 = 0.75.

On your calculator
  1. 3000000×0.7=+6300000=STO1 8,400,000
  2. 6300000÷RCL1= 0.75

STO store · RCL recall a stored number

  1. 3000000ENTER0.7×6300000+STO1 8,400,000
  2. 6300000RCL1÷ 0.75

ENTER enter (on the HP, puts the number on the stack) · STO store · RCL recall a stored number

Portfolio modified duration Memorize
ModDurP≈∑iwi×ModDuri\text{ModDur}_P \approx \sum_i w_i \times \text{ModDur}_i
Quick example

Weights 50%, 30% and 20% with modified durations 3.0, 7.0 and 10.0: 0.50 × 3.0 + 0.30 × 7.0 + 0.20 × 10.0 = 1.5 + 2.1 + 2.0 = 5.6.

On your calculator
  1. 0.5×3=STO1
  2. 0.3×7+RCL1=STO1
  3. 0.2×10+RCL1= 5.6

STO store · RCL recall a stored number

  1. 0.5ENTER3×0.3ENTER7×+0.2ENTER10×+ 5.6

ENTER enter (on the HP, puts the number on the stack)

Portfolio convexity Memorize
ConvexityP≈∑iwi×Convexityi\text{Convexity}_P \approx \sum_i w_i \times \text{Convexity}_i
Quick example

Same weights (50%, 30%, 20%) with convexities 12, 60 and 110: 0.50 × 12 + 0.30 × 60 + 0.20 × 110 = 6 + 18 + 22 = 46.

On your calculator
  1. 0.5×12=STO1
  2. 0.3×60+RCL1=STO1
  3. 0.2×110+RCL1= 46

STO store · RCL recall a stored number

  1. 0.5ENTER12×0.3ENTER60×+0.2ENTER110×+ 46

ENTER enter (on the HP, puts the number on the stack)

Module 13 Effective Duration, Key Rate Duration and Empirical Duration

Effective duration and effective convexity

Effective duration Memorize
EffDur=PV−−PV+2×ΔCurve×PV0\text{EffDur} = \dfrac{PV_- - PV_+}{2 \times \Delta Curve \times PV_0}
Quick example

Full price 100.80. A model gives 102.00 with the benchmark curve 15 bp lower and 99.55 with it 15 bp higher. (102.00 − 99.55) ÷ (2 × 0.0015 × 100.80) = 2.45 ÷ 0.3024 = 8.10.

On your calculator
  1. 102−99.55=÷2÷0.0015÷100.8= 8.10
  1. 102ENTER99.55−2÷0.0015÷100.8÷ 8.10

ENTER enter (on the HP, puts the number on the stack)

Effective convexity Memorize
EffCon=PV−+PV+−2×PV0(ΔCurve)2×PV0\text{EffCon} = \dfrac{PV_- + PV_+ - 2 \times PV_0}{(\Delta Curve)^2 \times PV_0}
Quick example

Same prices: (102.00 + 99.55 − 2 × 100.80) ÷ (0.0015² × 100.80) = −0.05 ÷ 0.0002268 = −220.5. Negative, so the price gains less on the way down in rates than it loses on the way up.

On your calculator
  1. 102+99.55−201.6=÷0.0015÷0.0015÷100.8= −220.5
  1. 102ENTER99.55+201.6−0.0015÷0.0015÷100.8÷ −220.5

ENTER enter (on the HP, puts the number on the stack)

Price change with effective duration and convexity

Price change with effective duration and convexity Memorize
%ΔPVFull≈(−EffDur×ΔCurve)+[12×EffCon×(ΔCurve)2]\%\Delta PV^{Full} \approx (-\text{EffDur} \times \Delta Curve) + \left[\tfrac{1}{2} \times \text{EffCon} \times (\Delta Curve)^2\right]
Quick example

Effective duration 7.20, effective convexity −210, benchmark curve up 75 bp. Duration: −7.20 × 0.0075 = −5.400%. Convexity: ½ × (−210) × 0.0075² = −0.591%. Total −5.99%.

On your calculator
  1. 7.2×0.0075=+|-STO1
  2. 0.5×210+|-×0.0075×0.0075+RCL1= −0.059906

+|- change sign · STO store · RCL recall a stored number

  1. 7.2ENTER0.0075×CHS
  2. 0.5ENTER210CHS×0.0075×0.0075×+ −0.059906

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Key rate duration

Key rate duration Use it
KRDk=−1PV×ΔPVΔrkKRD_k = -\frac{1}{PV} \times \frac{\Delta PV}{\Delta r_k}
Quick example

Shift only the 5-year benchmark rate by ±1 bp. Prices: PV₋ 100.0190, PV₀ 100.0000, PV₊ 99.9810. ΔPV ÷ Δr is estimated by (100.019 − 99.981) ÷ (2 × 0.0001), so KRD₅ = 0.038 ÷ (2 × 0.0001 × 100) = 1.90.

On your calculator
  1. 100.019−99.981=÷2÷0.0001÷100= 1.9
  1. 100.019ENTER99.981−2÷0.0001÷100÷ 1.9

ENTER enter (on the HP, puts the number on the stack)

Sum of key rate durations Memorize
∑k=1nKRDk=EffDur\sum_{k=1}^{n} KRD_k = \text{EffDur}
Quick example

Key rate durations of 0.50 (2-year), 1.20 (5-year), 2.30 (10-year) and 1.65 (30-year) give an effective duration of 0.50 + 1.20 + 2.30 + 1.65 = 5.65.

On your calculator
  1. 0.5+1.2+2.3+1.65= 5.65
  1. 0.5ENTER1.2+2.3+1.65+ 5.65

ENTER enter (on the HP, puts the number on the stack)

Price change from key rate durations Memorize
%ΔPVFull≈−∑kKRDk×Δrk\%\Delta PV^{Full} \approx -\sum_{k} KRD_k \times \Delta r_k
Quick example

KRDs 0.60 (2-year), 2.20 (5-year), 3.40 (10-year). Rates move +15, 0 and −25 bp. −[0.60 × 0.0015 + 2.20 × 0 + 3.40 × (−0.0025)] = −[0.0009 − 0.0085] = +0.76%.

On your calculator
  1. 0.6×0.0015=STO1
  2. 3.4×0.0025+|-+RCL1=+|- 0.0076

STO store · +|- change sign · RCL recall a stored number

  1. 0.6ENTER0.0015×3.4ENTER0.0025CHS×+CHS 0.0076

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Empirical vs analytical duration

Empirical duration over a stress period (illustration) Know the idea
Dempirical=−%ΔPΔybenchmarkD_{empirical} = \frac{-\%\Delta P}{\Delta y_{benchmark}}
Quick example

Analytical duration 5.0. In a stress month the benchmark yield falls 60 bp and the credit spread widens 45 bp, so the bond’s own yield falls 15 bp and its price rises about 5.0 × 0.0015 = 0.75%. Against the benchmark move: 0.0075 ÷ 0.0060 = 1.25.

On your calculator
  1. 5×0.0015=÷0.006= 1.25, both yields fall so the signs cancel
  1. 5ENTER0.0015×0.006÷ 1.25, both yields fall so the signs cancel

ENTER enter (on the HP, puts the number on the stack)

Module 14 Credit Risk

Credit risk: probability of default and loss given default

Expected exposure Memorize
EE=face value+accrued interest−collateral valueEE = \text{face value} + \text{accrued interest} - \text{collateral value}
Quick example

A GBP 3,000,000 secured bond with GBP 45,000 of accrued interest, backed by assets now worth GBP 1,100,000. 3,000,000 + 45,000 − 1,100,000 = GBP 1,945,000.

On your calculator
  1. 3000000+45000−1100000= 1,945,000
  1. 3000000ENTER45000+1100000− 1,945,000

ENTER enter (on the HP, puts the number on the stack)

Loss given default Memorize
LGD=EE×(1−RR)LGD = EE \times (1 - RR)
Quick example

Expected exposure GBP 1,945,000 and a recovery rate of 45%. 1,945,000 × (1 − 0.45) = GBP 1,069,750.

On your calculator
  1. 1−0.45=×1945000= 1,069,750
  1. 1ENTER0.45−1945000× 1,069,750

ENTER enter (on the HP, puts the number on the stack)

Expected loss Memorize
EL=POD×LGD=POD×EE×(1−RR)EL = POD \times LGD = POD \times EE \times (1 - RR)
Quick example

A one-year probability of default of 2% on a loss given default of GBP 1,069,750. 0.02 × 1,069,750 = GBP 21,395.

On your calculator
  1. 1069750×0.02= 21,395
  1. 1069750ENTER0.02× 21,395

ENTER enter (on the HP, puts the number on the stack)

Credit spread and expected loss Memorize
credit spread≈POD×LGD\text{credit spread} \approx POD \times LGD
Quick example

POD 1.2% a year and LGD 45% give an expected loss of 0.54%, or 54 bp. A bond paying a 70 bp spread is more than fairly compensated (70 > 54).

On your calculator
  1. 0.012×0.45= 0.0054, so 54 bp
  1. 0.012ENTER0.45× 0.0054, so 54 bp

ENTER enter (on the HP, puts the number on the stack)

What drives credit spreads

Yield of a zero-coupon bond from its price Use it
y=(100P)1/N−1y = \left(\frac{100}{P}\right)^{1/N} - 1
Quick example

A 5-year zero-coupon bond priced at 82.10 per 100 of par. (100 ÷ 82.10)^(1/5) − 1 = 4.0235%.

On your calculator
  1. 100÷82.1=y^x51/x=−1= 0.040235

y^x power · 1/x reciprocal

  1. 100ENTER82.1÷51/xy^x1− 0.040235

ENTER enter (on the HP, puts the number on the stack) · 1/x reciprocal · y^x power

Liquidity spread Memorize
liquidity spread=ybid−yoffer\text{liquidity spread} = y_{bid} - y_{offer}
Quick example

A 5-year zero is quoted 82.10 bid and 82.60 offer. Bid yield 4.0235%, offer yield 3.8972%. Difference 0.1262%, or 12.6 bp.

On your calculator
  1. 100÷82.1=y^x51/x=STO1
  2. 100÷82.6=y^x51/x=+|-+RCL1= 0.001262

y^x power · 1/x reciprocal · STO store · +|- change sign · RCL recall a stored number

  1. 100ENTER82.1÷51/xy^xSTO1
  2. 100ENTER82.6÷51/xy^xCHSRCL1+ 0.001262

ENTER enter (on the HP, puts the number on the stack) · 1/x reciprocal · y^x power · STO store · CHS change sign · RCL recall a stored number

Credit part of a yield spread Use it
credit spread≈yield spread−liquidity spread\text{credit spread} \approx \text{yield spread} - \text{liquidity spread}
Quick example

A bond yields 4.62% against a 2.35% benchmark: a 227 bp yield spread. With a liquidity spread of 8 bp, the credit part is about 219 bp.

On your calculator
  1. 4.62−2.35=×100=−8= 219
  1. 4.62ENTER2.35−100×8− 219

ENTER enter (on the HP, puts the number on the stack)

Price impact of spread changes

Price change from a small spread change Use it
%ΔPVFull≈−AnnModDur×ΔSpread\%\Delta PV^{Full} \approx -\text{AnnModDur} \times \Delta\text{Spread}
Quick example

Modified duration 9.1 and a spread that tightens by 35 bp. −9.1 × (−0.0035) = +3.185%.

On your calculator
  1. 9.1×0.0035= 0.03185, so +3.185%
  1. 9.1ENTER0.0035× 0.03185, so +3.185%

ENTER enter (on the HP, puts the number on the stack)

Price change from a spread change, with convexity Memorize
%ΔPVFull≈−(AnnModDur×ΔSpread)+12×AnnConvexity×(ΔSpread)2\%\Delta PV^{Full} \approx -(\text{AnnModDur} \times \Delta\text{Spread}) + \tfrac{1}{2} \times \text{AnnConvexity} \times (\Delta\text{Spread})^2
Quick example

Modified duration 4.8, convexity 30 (already rescaled), spread widens 60 bp. −4.8 × 0.006 = −2.880%; ½ × 30 × 0.006² = +0.054%. Total −2.826%.

On your calculator
  1. 4.8×0.006=+|-STO1
  2. 0.5×30×0.006×0.006+RCL1= −0.02826

+|- change sign · STO store · RCL recall a stored number

  1. 4.8ENTER0.006×CHS
  2. 0.5ENTER30×0.006×0.006×+ −0.02826

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Rescaling quoted convexity Memorize
AnnConvexityused=AnnConvexityquoted×100\text{AnnConvexity}_{used} = \text{AnnConvexity}_{quoted} \times 100
Quick example

A data service shows convexity 0.62 for a bond with modified duration 7.4. Used in the formula: 0.62 × 100 = 62, the same order as 7.4² ≈ 55.

On your calculator
  1. 0.62×100= 62
  1. 0.62ENTER100× 62

ENTER enter (on the HP, puts the number on the stack)

Module 15 Credit Analysis for Government Issuers

Sovereign credit analysis: the key ratios

Debt to GDP Use it
Debt to GDP=general government debtGDP\text{Debt to GDP} = \dfrac{\text{general government debt}}{\text{GDP}}
Quick example

Debt of 195 and GDP of 300 (billions). 195 ÷ 300 = 0.65, so 65%. Higher is weaker.

On your calculator
  1. 195÷300= 0.65
  1. 195ENTER300÷ 0.65

ENTER enter (on the HP, puts the number on the stack)

Debt to revenue Use it
Debt to revenue=general government debtgovernment revenue\text{Debt to revenue} = \dfrac{\text{general government debt}}{\text{government revenue}}
Quick example

Debt of 195 and revenue of 78. 195 ÷ 78 = 2.5, so 250%. Higher is weaker.

On your calculator
  1. 195÷78= 2.5
  1. 195ENTER78÷ 2.5

ENTER enter (on the HP, puts the number on the stack)

Interest to GDP Use it
Interest to GDP=government interest paymentsGDP\text{Interest to GDP} = \dfrac{\text{government interest payments}}{\text{GDP}}
Quick example

Interest of 7.8 on GDP of 300. 7.8 ÷ 300 = 0.026, so 2.6%. Higher is weaker.

On your calculator
  1. 7.8÷300= 0.026
  1. 7.8ENTER300÷ 0.026

ENTER enter (on the HP, puts the number on the stack)

Interest to revenue Use it
Interest to revenue=government interest paymentsgovernment revenue\text{Interest to revenue} = \dfrac{\text{government interest payments}}{\text{government revenue}}
Quick example

Interest of 7.8 on revenue of 78. 7.8 ÷ 78 = 0.10, so 10% of revenue goes on interest. Higher is weaker.

On your calculator
  1. 7.8÷78= 0.1
  1. 7.8ENTER78÷ 0.1

ENTER enter (on the HP, puts the number on the stack)

GDP per capita Use it
GDP per capita=GDPpopulation\text{GDP per capita} = \dfrac{\text{GDP}}{\text{population}}
Quick example

GDP of USD 300 billion and 12 million people. 300,000 million ÷ 12 million = USD 25,000. Higher is stronger.

On your calculator
  1. 300000÷12= 25,000
  1. 300000ENTER12÷ 25,000

ENTER enter (on the HP, puts the number on the stack)

FX reserves to GDP Use it
FX reserves to GDP=FX reservesGDP\text{FX reserves to GDP} = \dfrac{\text{FX reserves}}{\text{GDP}}
Quick example

Reserves of 45 and GDP of 300. 45 ÷ 300 = 0.15, so 15%. Higher is stronger.

On your calculator
  1. 45÷300= 0.15
  1. 45ENTER300÷ 0.15

ENTER enter (on the HP, puts the number on the stack)

Reserve ratio Use it
Reserve ratio=FX reservesexternal debt\text{Reserve ratio} = \dfrac{\text{FX reserves}}{\text{external debt}}
Quick example

Reserves of 45 and external debt of 90. 45 ÷ 90 = 0.50: reserves cover half of what is owed abroad. Higher is stronger.

On your calculator
  1. 45÷90= 0.5
  1. 45ENTER90÷ 0.5

ENTER enter (on the HP, puts the number on the stack)

External debt burden Use it
External debt burden=long-term external debtGDP\text{External debt burden} = \dfrac{\text{long-term external debt}}{\text{GDP}}
Quick example

Long-term external debt of 60 and GDP of 300. 60 ÷ 300 = 0.20, so 20%. Higher is weaker.

On your calculator
  1. 60÷300= 0.2
  1. 60ENTER300÷ 0.2

ENTER enter (on the HP, puts the number on the stack)

External debt due Use it
External debt due=external debt due within 12 monthsGDP\text{External debt due} = \dfrac{\text{external debt due within 12 months}}{\text{GDP}}
Quick example

External debt of 15 falls due within a year and GDP is 300. 15 ÷ 300 = 0.05, so 5%. Higher is weaker.

On your calculator
  1. 15÷300= 0.05
  1. 15ENTER300÷ 0.05

ENTER enter (on the HP, puts the number on the stack)

Agencies, supranationals and local government bonds

Debt service coverage ratio Memorize
DSCR=revenue−operating expensesdebt service\text{DSCR} = \dfrac{\text{revenue} - \text{operating expenses}}{\text{debt service}}
Quick example

Revenue 70, operating expenses 30, debt service (principal plus interest) 25. (70 − 30) ÷ 25 = 1.60×. Higher is stronger.

On your calculator
  1. 70−30=÷25= 1.6
  1. 70ENTER30−25÷ 1.6

ENTER enter (on the HP, puts the number on the stack)

Revenue at the covenant limit Use it
Revmin=Opex+covenant DSCR×DS\text{Rev}_{min} = \text{Opex} + \text{covenant DSCR} \times \text{DS}
Quick example

Operating expenses 30, debt service 25 and a 1.40× covenant. 30 + 1.40 × 25 = 65; revenue of 70 can fall 7.1% before a breach.

On your calculator
  1. 1.4×25+30= 65
  1. 1.4ENTER25×30+ 65

ENTER enter (on the HP, puts the number on the stack)

Module 16 Credit Analysis for Corporate Issuers

Financial ratios in credit analysis

EBIT margin Memorize
EBIT margin=EBITRevenue\text{EBIT margin} = \dfrac{EBIT}{\text{Revenue}}
Quick example

EBIT 84 on revenue 700. 84 ÷ 700 = 12.0%. Higher is better.

On your calculator
  1. 84÷700= 0.12
  1. 84ENTER700÷ 0.12

ENTER enter (on the HP, puts the number on the stack)

EBIT to interest expense Memorize
Coverage=EBITInterest expense\text{Coverage} = \dfrac{EBIT}{\text{Interest expense}}
Quick example

EBIT 180 and interest expense 45. 180 ÷ 45 = 4.0×. Higher is better.

On your calculator
  1. 180÷45= 4
  1. 180ENTER45÷ 4

ENTER enter (on the HP, puts the number on the stack)

EBITDA Memorize
EBITDA=EBIT+D&AEBITDA = EBIT + D\&A
Quick example

EBIT 210 plus D&A 65. 210 + 65 = 275.

On your calculator
  1. 210+65= 275
  1. 210ENTER65+ 275

ENTER enter (on the HP, puts the number on the stack)

Debt to EBITDA Memorize
DebtEBITDA=DebtEBIT+D&A\dfrac{\text{Debt}}{EBITDA} = \dfrac{\text{Debt}}{EBIT + D\&A}
Quick example

Debt 1,320, EBIT 210, D&A 65. 1,320 ÷ (210 + 65) = 4.8×. Lower is better.

On your calculator
  1. 1320÷(210+65)= 4.8
  1. 1320ENTER210ENTER65+÷ 4.8

ENTER enter (on the HP, puts the number on the stack)

Net debt Memorize
Net debt=Debt−cash and securities\text{Net debt} = \text{Debt} - \text{cash and securities}
Quick example

Debt 700 and cash and marketable securities 100. 700 − 100 = 600.

On your calculator
  1. 700−100= 600
  1. 700ENTER100− 600

ENTER enter (on the HP, puts the number on the stack)

Retained cash flow Memorize
RCF=CFO−dividends paidRCF = CFO - \text{dividends paid}
Quick example

Cash flow from operations 170 and dividends 40. 170 − 40 = 130.

On your calculator
  1. 170−40= 130
  1. 170ENTER40− 130

ENTER enter (on the HP, puts the number on the stack)

RCF to net debt Memorize
RCFNet debt=CFO−dividendsDebt−cash\dfrac{RCF}{\text{Net debt}} = \dfrac{CFO - \text{dividends}}{\text{Debt} - \text{cash}}
Quick example

CFO 170, dividends 40, debt 700, cash 100. 130 ÷ 600 = 21.7%. Higher is better.

On your calculator
  1. 170−40=÷(700−100)= 0.2167
  1. 170ENTER40−700ENTER100−÷ 0.2167

ENTER enter (on the HP, puts the number on the stack)

Funds from operations Use it
FFO=NIcont+D&A+DT+NCFFO = NI_{cont} + D\&A + DT + NC
Quick example

Net income from continuing operations 70, D&A 30, deferred tax 6, other non-cash 4. 70 + 30 + 6 + 4 = 110.

On your calculator
  1. 70+30+6+4= 110
  1. 70ENTER30+6+4+ 110

ENTER enter (on the HP, puts the number on the stack)

FFO to debt Use it
FFODebt\dfrac{FFO}{\text{Debt}}
Quick example

FFO 110 and debt 500. 110 ÷ 500 = 22.0%. Higher is better.

On your calculator
  1. 110÷500= 0.22
  1. 110ENTER500÷ 0.22

ENTER enter (on the HP, puts the number on the stack)

Free cash flow Know the idea
FCF=NI−ΔWC−capex−InetFCF = NI - \Delta WC - \text{capex} - I_{net}
Quick example

Net income 120, working capital up 15, fixed-asset investment 60, net interest paid 10. 120 − 15 − 60 − 10 = 35.

On your calculator
  1. 120−15−60−10= 35
  1. 120ENTER15−60−10− 35

ENTER enter (on the HP, puts the number on the stack)

Seniority, recovery rates and notching

Senior unsecured recovery after a secured shortfall Use it
RR=V−min⁡(C,S)U+max⁡(S−C, 0)RR = \dfrac{V - \min(C, S)}{U + \max(S - C,\, 0)}
Quick example

Liquidation value V = 350. Secured claim S = 180 backed by collateral C = 130. Senior unsecured claims U = 220. Shortfall 50 joins U. (350 − 130) ÷ (220 + 50) = 220 ÷ 270 = 81.5%.

On your calculator
  1. (350−130)÷(220+180−130)= 0.8148
  1. 350ENTER130−220ENTER180+130−÷ 0.8148

ENTER enter (on the HP, puts the number on the stack)

Expected loss for a class of debt Memorize
EL=POD×(1−RR)EL = POD \times (1 - RR)
Quick example

One-year POD 3%; the class recovers 25% in default. 0.03 × (1 − 0.25) = 2.25% a year.

On your calculator
  1. 1−0.25=×0.03= 0.0225
  1. 1ENTER0.25−0.03× 0.0225

ENTER enter (on the HP, puts the number on the stack)

Module 17 Securitization: How It Works and Why

Why securitize? Benefits for issuers, investors and markets

Blended securitization spread Use it
Blend=∑iwi si\text{Blend} = \sum_i w_i \, s_i
Quick example

Notes of 70% at +50 bp, 20% at +180 bp and 10% at +520 bp. 0.70 × 50 + 0.20 × 180 + 0.10 × 520 = 35 + 36 + 52 = 123 bp.

On your calculator
  1. (0.7×50)+(0.2×180)+(0.1×520)= 123
  1. 0.7ENTER50×0.2ENTER180×+0.1ENTER520×+ 123

ENTER enter (on the HP, puts the number on the stack)

Funding saving from securitizing Use it
Saving=c−Blend\text{Saving} = c - \text{Blend}
Quick example

Corporate bond at +340 bp against a blend of 123 bp. 340 − 123 = 217 bp, or 2.17% a year on the amount raised.

On your calculator
  1. 340−123= 217
  1. 340ENTER123− 217

ENTER enter (on the HP, puts the number on the stack)

Module 18 Asset-Backed Securities, Covered Bonds and CDOs

Covered bonds

Covered bond overcollateralization Use it
OC=PoolBonds−1OC = \frac{\text{Pool}}{\text{Bonds}} - 1
Quick example

A cover pool worth 2,142 backs covered bonds with a face value of 2,100. 2,142 ÷ 2,100 − 1 = 2.0% overcollateralization.

On your calculator
  1. 2142÷2100−1= 0.02
  1. 2142ENTER2100÷1− 0.02

ENTER enter (on the HP, puts the number on the stack)

Credit enhancement and credit tranching

Overcollateralization amount Use it
OC=Pool−NotesOC = \text{Pool} - \text{Notes}
Quick example

Loans worth 462 back notes of 420. 462 − 420 = 42 of overcollateralization.

On your calculator
  1. 462−420= 42
  1. 462ENTER420− 42

ENTER enter (on the HP, puts the number on the stack)

Overcollateralization as a share of the notes Use it
OCNotes=Pool−NotesNotes\frac{OC}{\text{Notes}} = \frac{\text{Pool} - \text{Notes}}{\text{Notes}}
Quick example

Loans worth 618 back notes of 600. (618 − 600) ÷ 600 = 3%.

On your calculator
  1. 618−600=÷600= 0.03
  1. 618ENTER600−600÷ 0.03

ENTER enter (on the HP, puts the number on the stack)

Excess spread Use it
ES=y−c−fES = y - c - f
Quick example

Pool yield 8.4%, note coupon 5.1%, fees 0.6%. 8.4% − 5.1% − 0.6% = 2.7% a year.

On your calculator
  1. 0.084−0.051−0.006= 0.027
  1. 0.084ENTER0.051−0.006− 0.027

ENTER enter (on the HP, puts the number on the stack)

Loss borne by a tranche Use it
Loss=min⁡(Size, max⁡(0, L−tranches below))\text{Loss} = \min\bigl(\text{Size},\ \max(0,\ L - \text{tranches below})\bigr)
Quick example

Senior 200, mezzanine 50, junior 30; pool losses 65. The junior absorbs 30; the mezzanine bears 65 − 30 = 35 and keeps 15.

On your calculator
  1. 65−30= 35
  1. 65ENTER30− 35

ENTER enter (on the HP, puts the number on the stack)

Losses the senior tranche can absorb untouched Use it
K=Mezzanine+JuniorK = \text{Mezzanine} + \text{Junior}
Quick example

Mezzanine 50 and junior 30 sit below the senior tranche. The senior tranche loses nothing until pool losses exceed 50 + 30 = 80.

On your calculator
  1. 50+30= 80
  1. 50ENTER30+ 80

ENTER enter (on the HP, puts the number on the stack)

Credit card, auto loan and solar ABS

Monthly interest to noteholders Use it
I=Notes×c12I = \frac{\text{Notes} \times c}{12}
Quick example

Card ABS notes of 600 pay 5.4% a year. 600 × 0.054 ÷ 12 = 2.7 of interest a month.

On your calculator
  1. 600×0.054=÷12= 2.7
  1. 600ENTER0.054×12÷ 2.7

ENTER enter (on the HP, puts the number on the stack)

Principal in an amortizing payment Use it
Principal=Payment−Balance×r\text{Principal} = \text{Payment} - \text{Balance} \times r
Quick example

A pool of 200 at 0.5% a month with a payment of 3.87. Interest is 200 × 0.005 = 1.00, so principal is 3.87 − 1.00 = 2.87.

On your calculator
  1. 200×0.005= 1
  2. +|-+3.87= 2.87

+|- change sign

  1. 3.87ENTER200ENTER0.005×− 2.87

ENTER enter (on the HP, puts the number on the stack)

Collateralized debt obligations and CLOs

Cash left for the CLO equity tranche Use it
X=P y−D c−P f−P lX = P\,y - D\,c - P\,f - P\,l
Quick example

Loans of 400 yield 6.5% (26), debt of 360 costs 4.0% (14.4), the manager fee is 0.5% of the pool (2), no defaults. 26 − 14.4 − 2 = 9.6 left for equity.

On your calculator
  1. (400×0.065)−(360×0.04)−(400×0.005)= 9.6
  1. 400ENTER0.065×360ENTER0.04×−400ENTER0.005×− 9.6

ENTER enter (on the HP, puts the number on the stack)

CLO equity return Use it
R=XER = \frac{X}{E}
Quick example

Cash left for equity of 9.6 on an equity tranche of 40. 9.6 ÷ 40 = 24%.

On your calculator
  1. 9.6÷40= 0.24
  1. 9.6ENTER40÷ 0.24

ENTER enter (on the HP, puts the number on the stack)

Overcollateralization test ratio Use it
OC ratio=Collateral parPar of the notes tested\text{OC ratio} = \frac{\text{Collateral par}}{\text{Par of the notes tested}}
Quick example

Collateral of 690 against 600 of notes. 690 ÷ 600 = 1.15. If this falls below the trigger, cash goes to the senior notes.

On your calculator
  1. 690÷600= 1.15
  1. 690ENTER600÷ 1.15

ENTER enter (on the HP, puts the number on the stack)

Module 19 Mortgage-Backed Securities: RMBS, CMOs and CMBS

Prepayment risk and time tranching

Weighted average life Use it
WAL=∑tt×PrincipaltTotal principal\text{WAL} = \frac{\sum_t t \times \text{Principal}_t}{\text{Total principal}}
Quick example

A tranche of 150 receives 50 in year 1, 40 in year 2, 35 in year 3 and 25 in year 4. (1 × 50 + 2 × 40 + 3 × 35 + 4 × 25) ÷ 150 = 2.23 years.

On your calculator
  1. (1×50)+(2×40)+(3×35)+(4×25)= 335
  2. ÷150= 2.2333
  1. 1ENTER50×2ENTER40×+3ENTER35×+4ENTER25×+ 335
  2. 150÷ 2.2333

ENTER enter (on the HP, puts the number on the stack)

Residential mortgage loans: the features that matter

Loan-to-value ratio Memorize
LTV=Loan amountProperty value\text{LTV} = \frac{\text{Loan amount}}{\text{Property value}}
Quick example

A loan of JPY 26,000,000 on an apartment worth JPY 40,000,000. 26,000,000 ÷ 40,000,000 = 65.0%.

On your calculator
  1. 26000000÷40000000= 0.65
  1. 26000000ENTER40000000÷ 0.65

ENTER enter (on the HP, puts the number on the stack)

Debt-to-income ratio Memorize
DTI=Monthly debt paymentsMonthly gross income\text{DTI} = \frac{\text{Monthly debt payments}}{\text{Monthly gross income}}
Quick example

Mortgage payment 1,800 and other debt payments 350 a month; gross income 78,000 a year. 2,150 ÷ (78,000 ÷ 12) = 33.1%.

On your calculator
  1. 1800+350= 2,150
  2. ÷(78000÷12)= 0.3308
  1. 1800ENTER350+ 2,150
  2. 78000ENTER12÷ 6,500
  3. ÷ 0.3308

ENTER enter (on the HP, puts the number on the stack)

Mortgage pass-throughs and CMOs

Weighted average coupon Use it
WAC=∑iBi ri∑iBi\text{WAC} = \frac{\sum_i B_i \, r_i}{\sum_i B_i}
Quick example

Loans of 250 at 3.2%, 350 at 3.8% and 400 at 4.7%. (250 × 3.2 + 350 × 3.8 + 400 × 4.7) ÷ 1,000 = 4.01%.

On your calculator
  1. (250×3.2)+(350×3.8)+(400×4.7)= 4,010
  2. ÷1000= 4.01
  1. 250ENTER3.2×350ENTER3.8×+400ENTER4.7×+ 4,010
  2. 1000÷ 4.01

ENTER enter (on the HP, puts the number on the stack)

Weighted average maturity Use it
WAM=∑iBi mi∑iBi\text{WAM} = \frac{\sum_i B_i \, m_i}{\sum_i B_i}
Quick example

Loans of 400 with 280 months left and 600 with 180 months left. (400 × 280 + 600 × 180) ÷ 1,000 = 220 months.

On your calculator
  1. (400×280)+(600×180)= 220,000
  2. ÷1000= 220
  1. 400ENTER280×600ENTER180×+ 220,000
  2. 1000÷ 220

ENTER enter (on the HP, puts the number on the stack)

Pass-through rate Memorize
Pass-through rate=WAC−Fees\text{Pass-through rate} = \text{WAC} - \text{Fees}
Quick example

WAC of 4.55% and fees of 0.48%. 4.55 − 0.48 = 4.07%.

On your calculator
  1. 4.55−0.48= 4.07
  1. 4.55ENTER0.48− 4.07

ENTER enter (on the HP, puts the number on the stack)

Sequential-pay tranche balance Use it
B1=B0−(Scheduled+Prepaid)B_1 = B_0 - (\text{Scheduled} + \text{Prepaid})
Quick example

Tranche A of 160 is first in line. Pool principal in year 1: 14 scheduled and 36 prepaid. 160 − 14 − 36 = 110.

On your calculator
  1. 160−14−36= 110
  1. 160ENTER14−36− 110

ENTER enter (on the HP, puts the number on the stack)

Commercial mortgage-backed securities (CMBS)

Net operating income Memorize
NOI=Rent−Opex−Reserves\text{NOI} = \text{Rent} - \text{Opex} - \text{Reserves}
Quick example

Rent EUR 4.1 million, cash operating expenses 1.6 million, replacement reserves 0.3 million. (4.1 − 1.6) − 0.3 = EUR 2.2 million.

On your calculator
  1. 4.1−1.6−0.3= 2.2
  1. 4.1ENTER1.6−0.3− 2.2

ENTER enter (on the HP, puts the number on the stack)

Debt service coverage ratio Memorize
DSCR=NOIAnnual debt service\text{DSCR} = \frac{\text{NOI}}{\text{Annual debt service}}
Quick example

NOI of 2.2 and annual debt service (interest plus principal) of 1.8. 2.2 ÷ 1.8 = 1.22×.

On your calculator
  1. 2.2÷1.8= 1.2222
  1. 2.2ENTER1.8÷ 1.2222

ENTER enter (on the HP, puts the number on the stack)

Prepayment penalty Use it
Penalty=Points×1%×Balance\text{Penalty} = \text{Points} \times 1\% \times \text{Balance}
Quick example

Three points to refinance a loan with GBP 6.5 million outstanding. 3 × 1% × 6,500,000 = GBP 195,000.

On your calculator
  1. 3×0.01×6500000= 195,000
  1. 3ENTER0.01×6500000× 195,000

ENTER enter (on the HP, puts the number on the stack)

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