🔎 Exam 2 Formulas

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L6 - Bonds

Name

Equation

Example

After-Tax Return

$r_{\text{after-tax}} = r_{\text{corp}} × (1-t)$

= 6.5% × (1-24%) = 4.94% after tax

Equivalent Taxable Yield

$r_{\text{taxable}} = \frac{r_{\text{muni}}}{1-t}$

= 4.94% / (1-24%) = 6.5%

General Bond Price

$P_B = \frac{Fc}{1+i} + \frac{Fc}{1+i^2} + \frac{Fc}{1+i^3} + ... + \frac{Fc+F}{1+i^T}$

= (10/(1.1)) + (10/(1.1)^2) + (110/(1.1)^3) = 100

Zero Coupon Bond Price

$P_{\text{ZCB}} = \frac{F}{(1+i)^T}$

= (100/(1.1)^3) = 75.1314

Consol Price

$P_C = \frac{Fc}{i}$

= $500 / 10% = $5,000

Consol Yield
(Spreadsheets don't have a function for calculating yield of consols.)

$i = \frac{Fc}{PB}$
Fc is annual payment.

= $500 / $5,000 = 10%

Current Yield

$= \frac{Fc}{PB}$

= 35/910 = 3.85%

Yield to Maturity $(YTM)$

Write down a bond pricing formula and solve for i.

Notation:
$P_B$ = Bond Price
$c$ = coupon Rate
$F$ = Face Value
$i$ = Discount rate

$rm$ = municipal yield (tax free)
$r$ = taxable yield
$n$ = number of years to maturity
$t$ = Tax rate
$PDV$ = Present Discounted Value

$P_B < F$	⇔	$YTM > c$	⇔	“Discount Bond”
$P_B = F$	⇔	$YTM = c$	⇔	“Par Bond”
$P_B > F$	⇔	$YTM < c$	⇔	“Premium Bond”

• Callable bonds - Issuer may buy the bond from the holder at a stipulated price
• Convertible bonds - Bondholder may convert each bond into a stipulated number of shares of stock
• Puttable Bonds - Give the holder an option to retire and/or extend the bond
• Floating-rate bonds - Adjustable coupon rate
• Eurobond - a bond denominated in a currency different from the country where the bond is issued. For example, a bond issued in Singapore but denominated in Japanese yen is a “Euroyen” bond.
• Foreign Bond - a bond issued in a foreign country, denominated in the currency of that country (Examples: Yankee bond, Samurai bond, Bulldog bond)

References: 6 Jul 17.pptx and L6-Bonds 1

Bond Spreadsheet Formulas

Spreadsheet built-in formulas

Background

Use of spreadsheets to calculate YTM came up in Section. Google Sheets is a powerful, free spreadsheet that I demoed then.
These functions work both in Excel and Free Google Sheets. PV, NPV, Rate, and IRR are typically also available on financial calculators. Questions? Email me.

Spreadsheet Bond Price Formulas

=PV(i, T, c*F, F) = P_B(7.5%,7,61,1000) when i=7.5%, c=6.1%, F=$1000
=NPV(i,Range of cells with cash flows, starting with first coupon payment)

Spreadsheet YTM functions

=RATE(#periods, coupon pmt, PB, Face)(use the I/Y button on a financial calculator)
=IRR(Range of cells with cash flows including initial price)
=YIELD (settlement date, maturity date, coupon, PB, Face, paymts per yr.)
In general, use Rate() or IRR() in this class rather than Yield().

Duration in Excel

=Duration(Settlement, Maturity, Coupon, Yield, Frequency)
Duration of a 30 year bond with a 3% coupon, a face value of $1000, and a yield of 4%. Settlement=1/1/2000, Maturity=1/1/2030, Coupon=3%, Yield=4%, Frequency=1

References: 6 Jul 17.pptx and L6-Bonds 1

L7 - Bonds 2 (Duration and Interest Rate Risk)

Duration (DUR) - 4 Steps:

1. Calculate PDV all payments
2. Divide each PDV by current Bond Price
3. Multiple PDV/PB ratio by the correlating period
4. Sum individual durations

Name

Equation

Example

Duration in Excel

See above

See above

Duration = DUR (Macaulay)

$\%ΔP ≈ - \frac{DUR}{(1+y)} × Δy$

Assume Dur = 6.76 and y ↑ 10% to 11%:
$= -\frac{6.76}{(1+.1)} × .01 =-6.145\%$

Modified Duration = $DUR\raisebox{0.5ex}{*}$

$\%ΔP ≈ -DUR\raisebox{0.5ex}{*} × Δy$
where $DUR\raisebox{0.5ex}{*} = \frac{DUR}{(1+y)}$

$DUR\raisebox{0.5ex}{*} = \frac{6.76}{(1+10\%)} = 6.145$
$= 6.145×.01 = -6.145\%$

Duration for Perpetuity

$= \frac{1+y}{y}$

$= \frac{1+ .05}{.05} = 21\text{ years}$

6 Principles of Interest Rate Sensitivity	5 Rules of Duration
Prices and yields are inversely related Convexity: increases in yields have a smaller effect than a decrease of equal magnitude Long-term bonds tend to be more price sensitive than short As maturity increases, price sensitivity increases at a decreasing rate Interest rate risk is inversely related to c Price sensitivity is inversely related to YTM	Duration for zero coupon bonds = maturity Duration & Coupons are inversely related Duration increases with maturity Duration & Yields are inversely related Duration for perpetuity = (1+y)/y

High Maturity (T)	Increases Duration and Increases Interest Rate Risk
High YTM	Decreases Duration and Decreases Interest Rate Risk
High Coupon Rate (c)	Decreases Duration and Decreases Interest Rate RiskE
A bond is Callable	Decreases “effective duration” and price volatility
Dur of Zero Cpn $= T$	Duration of Consol $= \frac{1+y}{y}$

References: 7 Jul 22.pptx and L7-Duration and Yield Curve

L8 - Monetary Policy

Old Monetary Policy (Pre-2008 - "Limited-Reserves")

Old: 3 tools of Conventional Monetary Policy:

1. Discount Rate
2. Set reserve requirement percent: R
3. Open Market Operations (OMOs were primary tools used to keep Fed Fund rate in the target range

Old: If Fed sells bonds, the money supply falls and interest rates rise.

New Monetary Policy (Post-2008 - "Ample Reserves")

New: The Fed instead uses "Administered Rates"

As of July 26, to keep the Fed Funds rate in the $5.25%-5.5% target range:

• Discount rate = 5.5%
• IORB = Interest on Reserve Balances = 5.4%
• ON RRP = Overnight Reserve RePurchase = 5.3%

New: Because there are already so many bank reserves, the only way to control the FFR is by changing the administered rates (ON RRP, IORB, and Discount Rate.

Open Market Operations (OMOs)

Open Market Operation (OMO) Formulas:

ΔTotal Deposits = Initial Δ in Reserves × (1/(R+E))

ΔMS = ΔTotalDeposits + ΔCashHeldByPublic

R = Required Reserve Ratio.

E = Excess Reserve Ratio.

OMO Example: Suppose the Fed sells a $100,000 T-Bond to a bond dealer. Bond dealer pays $100,000 from checking account.

ΔTotal Deposits = Initial Δ in Reserves × (1/(R+E))

= -$100,000×1/(10%+0%) = -$1M

ΔMS = ΔTotalDep + ΔCashHeldByPublic

= -$1M + $0 = -$1M

Therefore, the money supply decreases, and interest rates rise, as shown in the diagram above. (Contractionary)

Repo

Repo (Repurchase Agreement)

Day 1: Fed buys a US govt. security from a NBFI

Day 2: NBFI buys the security back from the Fed at a slightly higher price

Repo means that the NBFI is borrowing money, using the govt security as collateral. (Correspondingly, the Fed is lending.)

Reverse Repo(Reverse Repurchase Agreement)

Day 1: Fed sells a US govt. security to a NBFI

Day 2: Fed buys the security back from the NBFI at a slightly higher price

Reverse Repo means that the NBFI is lending money. (Correspondingly, the Fed is borrowing.)

Reverse repo is like paying IORB, except that NBFI can use it. With both IORB and ON RRP, reserves that would have been lent to the private sector are instead lent to the Fed, slowing economic activity. This is the only way to fight inflation.

References: 8 Jul 24.pptx and L8 - Monetary Policy

L10 - International Investing

1452

Name

Formula

Example

Interest Rate on Dollar Deposits

$R_\$$

$R_\$=2\%$

Interest Rate on Euro Deposits

$R_€$

$R_€=4\%$

Current Exchange Rate

$E_{\$/€}$

$E_{\$/€}=\$1$ per Euro

Expected Future Exchange Rate

$E^e_{\$/€}$

$E^e_{\$/€}=\$0.97$ per Euro

Expected Rate of appreciation of the euro

$\%ΔE_{\$/€} = \frac{E^e_{\$/€}-E_{\$/€}}{E_{\$/€}}$

$\%ΔE_{\$/€} = \frac{.97-1}{1} = -3\%$

Approximate Expected Return on Euro Deposits = Interest rate + gains/losses from currency appreciation

$$R_€ + \frac{E^e_{\$/€}-E_{\$/€}}{E_{\$/€}}$$

$$4\% + (-3\%) =1\%$$

Continued example: A US investor's return on Eurozone deposits is predicted to be: $$R_{€\text{ (from US)}} = R_€ + \frac{E^e_{\$/€} - E_{{\$}/{€}}}{E_{\$/€}} \\ = 4\%- 3\% = 1\%$$ For a US investor, the difference between dollar vs. euro deposits is therefore: (compare this to the previous equation and to Slide 23) $$R_\$ - R_{€\text{ (from US)}}$$ $$= R_\$ - \left(R_€ + \frac{E^e_{\$/€} - E_{{\$}/{€}}}{E_{\$/€}}\right)$$ $$= 2\% - \left(4\%- 3\%\right) = 1\%$$ Investors tend to put their money where they expect the highest return.

Interest rate parity: Investor's tendency to put money where they expect the highest return will cause the returns on domestic and foreign deposits to equalize.
In our example: $$R_\$ = R_€ + \frac{E^e_{\$/€} - E_{\$/€}}{E_{\$/€}}$$

Appreciation = Stronger currency = E↓ - an increase in the value of a currency relative to another currency.
Depreciation = Weaker currency = E↑ - a decrease in the value of a currency relative to another currency.

Exchange Rate Regimes:

1. Flexible (Floating) Exchange Rate
2. Fixed Exchange Rate
3. Currency Board converts between local currency and a reference currency
4. Dollarization means simply using the reference currency as your own currency

References: 10 Jul 31.pptx and L10-International Investing