note_valuation.rmd

---
title: "Fix Income Securities Formula Sheet"
author: "Alex Dou"
date: "March 10, 2019"
output:
  pdf_document: default
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```


### Discounted Cash Flow Approach

$$ P=\sum_{t=1}^{n}{\frac{C_t}{(1+r)^t} + \frac{FV}{(1+r)^n}} $$


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### Discount factor

$$ P=\sum_{t=1}^{n}{d(t)CF_t} $$


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### Compounding frequencies

$$ FV_n=PV_0(1+\frac{r}{m})^{mn} $$

$$R_c = m ln(1+\frac{R_m}{m})$$

$$ R_m=m(e^{\frac{R_c}{m}}-1)  $$

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### Annuity Factor

$$ P_n=A[\frac{(1+r)^n-1}{r}]  $$

$$ PV=A[\frac{1-\frac{1}{(1+r)^n}}{r}]  $$

$$ A_T=\sum_{t=1}^{2n}{d(\frac{t}{2})} $$


$$ Perpetuity=\frac{C}{y}$$



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### Spot Rate and Discount Factor

$$ d(t)=\frac{1}{(1+\frac{z(t)}{2})^{2t}}   $$


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### Pricing bonds with spot rate

$$ P=\sum_{t=1}^{n}{\frac{C}{(1+Z_t)^t} + \frac{FV}{(1+Z_n)^n}}  $$


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### Forward rates and spot rates

$$ (1+Z_{t1})^{t1}(1+IFR_{t1,t2})^{t2-t1} = (1+Z_{t2})^{t2}   $$

$$ r_{12} = \frac{r_{02}t_{02}-r_{01}t_{01}}{t_{02}-t_{01}}  $$


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### Par Rate

$$ \frac{c}{2}\sum_{t=1}^{2t}{d(\frac{t}{2})}+d(T)=1 $$

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### Yield to Maturity
$$ P=\sum_{t=1}^{n}{\frac{PMT_t}{(1+YTM)^t} + \frac{FV}{(1+FTM)^n}} $$



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### Full Price
$$ Full price=[\sum_{t=1}^{n}{\frac{PMT_t}{(1+r)^t} + \frac{FV}{(1+r)^n}}](1+r)^{\frac{t}{T}} $$


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### Spread of Bond

$$ P=\sum_{t=1}^{n}{\frac{CF_t}{(1+Z_t+Spread)^t}} $$


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### Macaulay Duration

$$ MacDur = \frac{\sum_{t=1}^{n}{t*PVCF_t}}{\sum{PVCF_t}}$$


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### Modified Duration

$$ D^* = ModDur=-\frac{\Delta{\%P}}{\Delta{y}} = \frac{MacDur}{1+\frac{r}{m}} $$

$$ \frac{\Delta{P}}{P} = D^*\Delta{y} $$

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### Effective Duration

$$ EffectiveDur = \frac{P_{-\Delta{y}}-P_{+\Delta{y}}}{2P_0\Delta{y}}  $$

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### DV01

$$ DV01 = D*P*{0.01\%} $$
$$ \Delta{y}=1bp $$

$$ DV01=\frac{P_{-1bp}-P_{+1bp}}{2}  $$

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### Duration hedge

$$ -\frac{V_pD_p}{V_hD_h}$$

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### DV01 Hedging

$$ HedgeRatio= -\frac{DV01(UnderlyingAsset)}{DV01(Hedging Instrument)} $$

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### Convexity 

$$ Convexity = \frac{1}{P}\frac{d^2P}{dy^2}$$

$$ Convexity \approx \frac{P_{-\Delta{y}}+P_{+\Delta{y}}-2P_0}{P_0\Delta{y}^2} $$

$$ \Delta{\%P}  \approx -D*\Delta{y} + \frac{1}{2} Convexity (\Delta{y})^2   $$

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### Key Rate Duration and Key rate 01

$$ D^k = - \frac{\Delta{\%P}}{\Delta{y}^k} $$

$$ DV01^k = - \frac{1}{10000} \frac{\Delta{P}}{\Delta{y}^k} $$



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### Portfolio Duration and Convexity

$$ D_p = \sum_{i=1}^{n}{w_iD_i} $$

$$ Convexity_p = \sum_{i=1}^{n}{w_iC_i} $$

$$ DV01_p =\sum{DV01_i}$$

### FRA

#### Cash flow

$$ C_{FRA}=L(R_k-R_m)(T_2-T_1) $$

$$ CPV_{FRA} =\frac{L(R_k-R_m)(T_2-T_1)}{1+R_m(T_2-T_1)} $$

#### Pricing

$$V_{FRA}=L(R_k-R_f)(T_2-T_1)e^{-r_2t_2}$$

### Interest Rate Futures

$$ P_{cash} = P_{quoted} + AI $$

#### Cheapest-to-Deliver Bond  
Bond with minimal 

$$ B_{quoted} - (F_{quoted} \times CF)   $$

$$ CF = \frac{B_{discounted} - AI}{FV}$$

#### Treasury bond futures price
$$ F_0=(S_0-I)e^{rt}  $$

$$ B_{dirty} = F \times CF+AI $$

#### T Bill
**n** days to maturity and a cash price of **Y**
$$ P=\frac{360}{n}(100-Y)  $$

#### Euro-dollar

**Z** is quoted price of a eurodollar futures contract

$$LIBOR_{farword} = 100-Z$$

$$ 10000[100-0.25(100-Z)] $$

$$ V_{1bp} = 25USD $$

### Swap

#### Value of Swap
$$ V_{swap} = B_{fix} - B_{float}  $$

$$ B_{float} = (L+K)e^{-rt}  $$

$$ V_{swap} = B_d-B_f S_0  $$